\documentclass[%
 reprint,
 superscriptaddress, 
%groupedaddress,
%unsortedaddress,
%runinaddress,
%frontmatterverbose, 
%preprint,
%preprintnumbers,
%nofootinbib,
%nobibnotes,
%bibnotes,
 amsmath,amssymb,
 aps,
 pra,
%prb,
%rmp,
%prstab,
%prstper,
 floatfix,
]{revtex4-2}
% Rules of thumb that make it easier when you have to deal with journal submissions
%
% These are not all "good LaTeX" rules of thumb,
% but they help when working with antiquated journal submission systems.
%
% Have a single tex file
% Always have 0 errors and work hard to have 0 warnings - "a float is stuck" is probably the only warning it is ok to accept after an attempt or two to fix it
% Use only the automatic placement location for floats, figures, tables until the very last edit
% Have only alpha-numerical characters or underscores in your file names (no spaces).
% Do not use sub-directories, have everything in a single folder (do not put images in a separate folder).
% Use ~\cite and ~\ref (a fixed width non-breaking ~, not a generic whitespace in front of the command)
% Use \cite, not \citep or others
% Be consistent in abreviation, e.g. use only Fig. not Fig/Fig./fig/figure; only Eq. not (eq)/equation/eq./Eq
%
% Math:
% Avoid defining custom math commands
% In math formulas, use \mathrm for subscripts, unless the subscript is a dummy variable like an index
% In math formulas, use align, not eqarray, for systems of equations
% Always use numbered equations
%
% Images:
% Have a single image per figure (not two images included one after the other)
% Have your images be high DPI (>400) raster files or vector EPS files
% Avoid vector images if they are heatmaps or have more than hundred points or lines
% !IMPORTANT! Set the width of your images in your plotting software to the correct width in inches or cm not pixels, otherwise it would scale weirdly when imported in latex
% Set width=\linewidth for your images in your figures
% Set the font size of text in your images to one fixed value for all images, in pt, around 10-14pt is probably good

\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{dsfont}
\usepackage{xcolor}
\usepackage{enumitem}
\usepackage{multirow}
\usepackage{hyperref}
\usepackage{amsthm}
\usepackage{amsmath}
\usepackage[linesnumbered,ruled,vlined]{algorithm2e}
\usepackage{stackengine}
\DeclareMathOperator{\kernel}{ker}
\DeclareMathOperator{\Hom}{Hom}
\newtheorem{definition}{Definition}
\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\usepackage{array}
\usepackage{amsmath, amssymb}
\usepackage{hyperref}
\usepackage{amsmath,amsthm}
\usepackage{float}
\usepackage{xcolor}
\usepackage{epstopdf}

\usepackage{mathtools}
\usepackage{colortbl}
\usepackage{array}
\usepackage{booktabs}
\usepackage{mathtools,amssymb}
\usepackage{amsthm}
\usepackage[caption=false]{subfig}
\usepackage[dvipsnames]{xcolor}
\usepackage{nicematrix}
\usepackage{tikz}
\usepackage{amsmath,amssymb}
\usetikzlibrary{positioning,arrows.meta,decorations.pathreplacing,calligraphy,calc}
\usetikzlibrary{cd}
\DeclareMathOperator{\im}{im}
\usepackage{tikz}
\usetikzlibrary{cd}
\usepackage{graphicx}% Include figure files
\usepackage{dcolumn}% align table columns on decimal point
\usepackage{bm}% bold math
\usepackage{dsfont}
\usepackage{xcolor}
\newtheorem{statement}[lemma]{Statement}
\usepackage{enumitem}
\usepackage{multirow}
\usepackage{hyperref}
\usepackage{array}
\usepackage{float}
\usepackage{csquotes}% to use normal quotes
\usepackage[T1]{fontenc}
\MakeOuterQuote{"}

\usepackage[linesnumbered, ruled, vlined]{algorithm2e}
\SetKw{KwDownTo}{downto}
\SetKwInput{KwInput}{Input}
\SetKwInput{KwOutput}{Output}
\usepackage{graphicx} 
\usepackage{mathtools}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{makecell}
\usepackage{physics}
\newcommand\editcolor[1]{\textcolor{blue}{\textbf{#1}}}
\newcommand{\dhor}{\mathsf{h}}
\newcommand{\dver}{\mathsf{v}}
\newcommand{\hor}{\mathsf{v}}
\newcommand{\ver}{\mathsf{h}}
\newcommand{\lmp}[1]{\operatorname{L}_{#1}}
\newcommand{\ann}[1]{\operatorname{ann}(#1)}
\newcommand{\Code}{\mathcal{C}}
\newcommand{\ftwo}{\mathbb{F}_2}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\Tor}{Tor}
\newtheorem{corollary}{Corollary}[theorem]
\newtheorem{proposition}{Proposition}[section]
\newtheorem{observation}[theorem]{Observation}
\newtheorem{remark}{Remark}
\theoremstyle{definition}
\newtheorem{example}{Example}
\theoremstyle{definition}
\def\id{{\mathchoice {\rm 1\mskip-4mu l} {\rm 1\mskip-4mu l} {\rm 1\mskip-4.5mu l} {\rm 1\mskip-5mu l}}}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, positioning}

\usepackage{graphicx}
\definecolor{umassLight}{HTML}{D4737C}
\definecolor{umassMed}{HTML}{A33848}
\definecolor{umassDark}{HTML}{7A1C30}
\definecolor{bufferC}{HTML}{E8A0A0}
\definecolor{umassPale}{HTML}{F5ECEE}
\definecolor{frameC}{HTML}{111827}
\definecolor{axisC}{HTML}{4B5563}

\tikzset{
  cell/.style={draw=frameC!75, line width=0.4pt, rounded corners=1.5pt,
               minimum width=7.5mm, minimum height=7.5mm, inner sep=0pt},
  past/.style ={cell, fill=umassDark!90,    draw=umassDark!60!black},
  win/.style  ={cell, fill=umassLight!95,   draw=umassLight!60!black},
  commit/.style={cell, fill=umassMed!95,    draw=umassMed!60!black},
  buffer/.style={cell, fill=bufferC!80,     draw=bufferC!50!black},
  fut/.style  ={cell, fill=umassPale,       draw=frameC!25},
  rdlabel/.style={font=\normalsize\bfseries, text=axisC, fill=white, inner sep=0.5pt},
  leglabel/.style={font=\small\bfseries, text=frameC},
}

\newcommand{\owdcollabels}[1]{%
  \node[rdlabel] at (0, -0.4) {$1$};
  \node[rdlabel] at (7, -0.4) {$T$};
}

\newcommand{\owdpanel}[2]{%
  \begin{scope}[shift={#2}, x=8.5mm, y=7.8mm]
    \foreach \r in {0,...,7} {
      \pgfmathtruncatemacro{\ispast}{\r < #1 ? 1 : 0}
      \pgfmathtruncatemacro{\iscommit}{\r == #1 ? 1 : 0}
      \pgfmathtruncatemacro{\isbuffer}{\r == #1+1 ? 1 : 0}
      \pgfmathtruncatemacro{\isfuture}{\r > #1+1 ? 1 : 0}
      \foreach \d in {0,...,5} {
        \ifnum\iscommit=1 
          \node[commit] (p#1-\r-\d) at (\r,\d) {};
        \else\ifnum\isbuffer=1 
          \node[buffer] (p#1-\r-\d) at (\r,\d) {};
        \else\ifnum\ispast=1 
          \node[past] (p#1-\r-\d) at (\r,\d) {};
        \else 
          \node[fut] (p#1-\r-\d) at (\r,\d) {};
        \fi\fi\fi
      }
    }
    \owdcollabels{#1}
  \end{scope}
}


\usepackage{braket}

\usepackage{listings}
\usepackage{color}
\definecolor{mygreen}{rgb}{0,0.6,0}
\definecolor{myred}{rgb}{0.9,0.2,0.3}
\definecolor{mymauve}{rgb}{0.28,0,0.92}
\definecolor{mygray}{rgb}{0.5,0.5,0.5}

\lstdefinelanguage{JuliaREPL}{
  keywords={julia},
  keywordstyle=\color{mygreen}\bfseries,
  keywords=[2]{Stabilizer, GeneralizedStabilizer, naive_encoding_circuit, PauliOperator, Destabilizer, project, CliffordOperator, apply},
  keywordstyle=[2]\color{mymauve}\bfseries,
  identifierstyle=\color{black},
  sensitive=true,
  mathescape=true,
  comment=[l]{\#},
  morecomment=[s]{/*}{*/},
  commentstyle=\color{mygray}\ttfamily,
  stringstyle=\color{myred}\ttfamily,
  morestring=[b]',
  morestring=[b]"
}

\lstset{
   language=JuliaREPL,
   extendedchars=true,
   basicstyle=\footnotesize\ttfamily,
   showstringspaces=false,
   showspaces=false,
   tabsize=2,
   breaklines=true,
   showtabs=false
}

\usepackage{csquotes}% to use normal quotes

% a package for neat todo notes
% comment this out when finishing the manuscript
\usepackage[textsize=tiny]{todonotes}

% a setup to mark edited lines when resubmitting to a journal after review
% disable this when you want to hide edit colors
% colors from tableau 10 scheme
% if there are yet more rounds of reviews add yet another command
\definecolor{customcolorblue}{HTML}{4573ae}
%\newcommand\editcolor[1]{\textcolor{customcolorblue}{\textbf{#1}}}
\begin{document}

\title{Multivariate Multicycle Codes for Complete Single-Shot Decoding}

% Multivariate Multicycle Codes for Complete Single Shot Decoding

\author{Feroz Ahmed Mian}
\email{fmian@umass.edu}
\affiliation{Manning College of Information and Computer Sciences, University of Massachusetts Amherst, 140 Governors Drive
Amherst, Massachusetts 01003, USA}

\author{Owen Gwilliam}
\email{ogwilliam@umass.edu}
\affiliation{Department of Mathematics and Statistics, University of Massachusetts Amherst, Lederle Graduate Research Tower, 1654, 710 N Pleasant St, Amherst, Massachusetts 01003, USA}

\author{Stefan Krastanov}
\email{skrastanov@umass.edu}
\affiliation{Manning College of Information and Computer Sciences, University of Massachusetts Amherst, 140 Governors Drive
Amherst, Massachusetts 01003, USA}
\affiliation{Department of Physics, University of Massachusetts Amherst, 1126 Lederle Graduate Research Tower,
710 North Pleasant Street
Amherst, Massachusetts 01003, USA}

\date{\today}

% QEC 26 conference reviews:
%Additionally, it would be helpful to provide a more rigorous definition of the “confinement profile,” as this term is sparse in literature and would benefit from some clarity.

% Finally, any claims about transversal non-cliffords is tenuous as it is not discussed beyond the conclusion/outlook as future work and should probably be omitted from the (extended) abstract.

\begin{abstract}
We introduce multivariate multicycle (MM) codes, a new family of quantum error correcting codes that unifies and generalizes bivariate bicycle codes, multivariate bicycle codes, abelian two-block group algebra codes, generalized bicycle codes, trivariate tricycle codes, and n-dimensional toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-\editcolor{\textit{t+1}} chain complexes with \textit{t} $\ge 4$. \editcolor{The chief advantage of these codes is that they possess metachecks and high confinement that permit complete single-shot decoding}. We offer a framework that facilitates the construction of long-length chain complexes through the use of Koszul complex. In particular, obtaining explicit boundary maps (parity check and metacheck matrices) is particularly straightforward in our approach. This simple but very general parameterization of codes permitted us to efficiently perform a numerical search, where we identify several MM code candidates that demonstrate these capabilities at high rates and high code distances. Examples of new codes with parameters $[[n,k,d]]$ include $[[96, 12, 8]]$, $[[96, 44, 4]]$\editcolor{,} \editcolor{$[[144, 12, 12]]$}, $[[216, 12, 12]]$, $[[360, 30, 6]]$, $[[384, 80, 4]]$, $[[486, 24, 12]]$, $[[486, 66, 9]]$ and \editcolor{$[[630, 70, 9]]$}. Notably, our codes achieve confinement profiles that surpass all known single-shot decodable quantum CSS codes of practical blocksize. \editcolor{Our codes are also the first explicit instances of collapsed 5D, 6D, 7D, 8D and 9D quantum codes, with check weights significantly lower than those of recent small instances of quantum Tanner codes.}
% this work calls the codes as 'collasped' codes https://arxiv.org/pdf/2602.23307 if we have higher dimensions, and we use one variable

%\editcolor{We perform overlapping-window decoding experiments demonstrating few-shot decodability, and we show that our codes are competitive with weight-6 AMC codes under both phenomenological and circuit-level noise models. Although MM codes are non-local, we expect they can be efficiently implemented in reconfigurable atom arrays.}
\end{abstract}

\maketitle
\section{Introduction}

The development of quantum low-density parity-check (qLDPC) codes~\cite{Tillich_2014,Breuckmann_2021, gottesman2013fault, Panteleev_2021, breuckmann2021balanced,dinur2021locallytestablecodesconstant, panteleev2022asymptoticallygoodquantumlocally, Leverrier_2022, leverrier2022efficientdecodingconstantfraction, dinur2022goodquantumldpccodes, gu2022efficientdecoderlineardistance, leverrier2022quantumtannercodes, leverrier2025smallquantumtannercodes, radebold2025explicit, evra2020decodablequantumldpccodes, liang2025generalized, guemard2025lifts, guémard2025moderatelengthliftedquantumtanner, Hastings_2021, gulshen2025quantum, 10756140, liang2025planar, wang2026check} represents a pivotal advancement in quantum error correction, offering a route to high-threshold, high-rate logical memories and scalable low-overhead fault-tolerant quantum processors. One of their many advantages is the possibility of \editcolor{single-shot or few-shot} decoding~\cite{Bombin_2013, bombin2015single, Campbell_2019, Hong_2025}, which can suppress errors using only \editcolor{one or a few rounds} of noisy syndrome extraction. The frontier of the field has now shifted from memory to computation, where the central challenge is the construction of codes that retain their single-shot property during fault-tolerant logical operations. Current approaches, including limited transversal gates~\cite{eastin2009restrictions,Breuckmann_2024}, code automorphisms~\cite{sayginel2025faulttolerantlogicalcliffordgates, eberhardt2024logicaloperatorsfoldtransversalgates}, and code surgery~\cite{Ide_2025,cowtan2024css,cowtan2025parallel, Cohen_2022, zheng2025highratesurgeryconstantoverheadlogical, cross2025improvedqldpcsurgerylogical, williamson2024lowoverheadfaulttolerantquantumcomputation}, must interface with the decoding problem, making the code's performance under a single-shot decoder~\cite{breuckmann2016local, brown2016fault, Herold_2017, 9585444,Vasmer_2021, Quintavalle_2021, Higgott_2023, gu2022efficientdecoderlineardistance, lin2024single, Guo_2026} the paramount metric for a scalable architecture~\cite{tan2025single, yoder2025tourgrossmodularquantum}.

A significant line of research has focused on constructing such codes using algebraic techniques, leading to celebrated families of codes such as bivariate bicycle (BB)~\cite{Bravyi_2024, eberhardt2024logicaloperatorsfoldtransversalgates,postema2025existencecharacterisationbivariatebicycle, symons2025sequences, liang2025selfdualbivariatebicyclecodes}, abelian and non-abelian two-block group algebra (2BGA)~\cite{lin2023quantumtwoblockgroupalgebra}, and the recently introduced trivariate tricycle (TT) \cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic}. A key motivation for exploring higher-dimensional algebraic constructions (such as three-dimensional (3D) gauge color ~\cite{Brown_2016}, 4D surface, and 4D toric~\cite{Dennis_2002, breuckmann2016local, Berthusen_2024}, double homological product~\cite{Campbell_2019}, homological product~\cite{xu2024fastparallelizablelogicalcomputation}, 3D surface and 3D toric~\cite{Quintavalle_2021}, 3D subsystem toric~\cite{Kubica_2022}, abelian multicycle~\cite{lin2025abelianmulticyclecodessingleshot}, intertwined toric~\cite{Stahl_2024}, 4D lifted homological product (4D-LHP)~\cite{11249819}, 4D hyperbolic~\cite{9585444}) is the inherent presence of local redundancies among parity checks. These linear redundancies, often formalized as "checks on checks" or metachecks, are important for single-shot error correction~\cite{PhysRevX.5.031043, Campbell_2019}.
% Stefayn: I commented this out as it seems to be a bit more of a "side remark", and it is for just a very specific code.
% It feels a bit out of place to mention it in the intro.
%\editcolor{In single-shot decoding paradigm, a single round of noisy stabiliser measurements suffices because the redundancy in the checks can be used to repair the syndrome record. From a topological perspective, Ref.~\cite{breuckmann2016local} notes that noisy syndrome measurements in topological codes can produce "open strings", which single-shot decoding locally corrects using the redundancy of the checks, effectively converting them into valid "closed loops".}
This capability for complete single-shot decoding has been recently demonstrated in practice through the implementation of the [[33,1,4]] 4D surface code on a trapped-ion quantum computer~\cite{Berthusen_2024}.

In this work, we establish that most of the algebraic constructions mentioned above~\cite{Bravyi_2024, voss2025multivariate, PhysRevA.88.012311, lin2023quantumtwoblockgroupalgebra, jacob2025singleshotdecodingfaulttolerantgates, rabeti2025listdecodingnewbicycle, aydin2025cyclic} can be unified. All of them are equivalent to Koszul complexes over quotient rings with cyclic relations. Our new formalism employs tools from commutative algebra and homological algebra~\cite{eisenbud2013commutative, weibel1994introduction}. We leverage the standard isomorphism between tensor products of length-1 chain complexes and Koszul complexes to provide a systematic framework for the construction of these single-shot-capable codes (i.e.\ codes with X and Z metachecks). This new perspective not only clarifies the mathematical formalism of existing codes, but also naturally generalizes them. For example, while TT codes have been shown to support only partial single-shot decoding~\cite{jacob2025singleshotdecodingfaulttolerantgates}, our broader framework supports complete single-shot decoding for $t \ge 4$\editcolor{, where $t+1$ is the length of the Koszul complex we use, also related to other defining features of the code as discussed in the definitions in the main text}. We call the code family we introduce the Multivariate Multicycle (MM) codes.

Of note is how simple our Koszul complex construction is. Compared to the sophisticated processes of finding the boundary maps (i.e.\ parity checks and metachecks) in previous work (e.g.~\cite{lin2025abelianmulticyclecodessingleshot, jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic} or Fig.~2 of~\cite{11249819}), the Koszul complex provides simple closed-form expressions for the boundary maps. Moreover, the Koszul complex trivially extends to \editcolor{length 5}---a sufficient length for the existence of a complete set of metachecks, unlike previous more-limited constructions. Lastly, this simplicity enabled us to efficiently perform a numerical search for good single-shot-decodable quantum codes, as seen in our numerical experiments section.
%\todo{we should say that t should be greather than 4}

Our work is the first to use the Koszul complex over quotient rings for constructing novel quantum error correcting codes. Koszul complex theory~\cite{green1984koszul, aprodu2010koszul, august2015differential} has recently been applied to several other problems in fault-tolerant quantum computation. We mention other applications here for general context, noting they represent uses distinct from our approach: In context of BB codes, Ref.~\cite{Chen_2025} employs a Koszul duality property yielding an $R$-module isomorphism between the homology groups $H(a,b;g,f)$ and $H(g,f;a,b)$, where $(a,b)=(x^\ell-1,y^m-1)$ encode periodic boundary conditions and \editcolor{$(g,f)$} the stabilizer checks, relating equivalence classes of logical operators to equivalence classes of anyon configurations. In proof theory~\cite{negri2008structural}, they model cut-elimination in multiplicative proof nets as error-correction processes~\cite{murfet2024linearlogicquantumerror}. Furthermore, Ref.~\cite{breuckmann2025logical} establishes that tile codes correspond to higher global sections of the shifted Koszul complex \(K^\bullet(\mathbb{P}^1\times\mathbb{P}^1,f,g)\otimes\mathcal{S}\) of vector bundles on \(\mathbb{P}^1\times\mathbb{P}^1\), where \(f,g\in\mathbb{F}_2[x^{\pm1},y^{\pm1}]\) encode stabilizer generators and \(\mathcal{S}\) is a line bundle controlling code dimensions and boundaries. Their framework yields logical operators isomorphic to \(\mathbb{F}_2[x^{\pm1},y^{\pm1}]/(f,g)\) and enables derived automorphisms---operations implementable via fault-tolerant lattice surgery~\cite{Horsman_2012, Litinski_2019} that induce logical CNOT circuits.

This paper is structured as follows: In Section~\ref{sec:ha_preliminaries}, we provide the mathematical preliminaries, first from the perspective of homological algebra and single-shot error correction, and then from the perspective of commutative algebra in Section~\ref{sec:ca_preliminaries}. Section~\ref{sec:previous_constructions} surveys previous constructions of abelian 2BGA codes, including BB codes, multivariate bicycle (MB) codes, generalized bicycle (GB) codes, and TT codes, establishing the context for our work. Section~\ref{sec:koszul_complex_formalism_for_quantum_codes} introduces our Koszul complex formalism for quantum codes. After defining these complexes using the language of exterior algebras and tensor products, it reinterprets the code constructions from previous section as specific manifestations of this general framework. In Section~\ref{sec:multivariate_multicycle_codes}, using the method established in the previous sections, we then present our main contribution: the Multivariate Multicycle (MM) codes. We establish several examples that unify various families of previous quantum error-correcting codes within this framework. Section~\ref{sec:numerical_simulations} provides numerical simulations 
using the \editcolor{belief propagation with ordered statistics decoding (BP+OSD)}~\cite{Roffe_LDPC_Python_tools_2022} and Tesseract~\cite{beni2025tesseractsearchbaseddecoderquantum} decoders 
over MM codes found from an extensive search over MM polynomials for $t=4$. We compare these results to other metacheck-capable codes. In particular, our codes have high rates and distances and record-breaking confinement, enabling high single-shot performance. In Section~\ref{sec:discussion}, we discuss our results in relation to other codes of contemporary interest and analyze the specific instances of MM codes discovered through our search. We also provide concluding remarks and outlines directions for future research.  In Appendix~\ref{sec:ha_preliminaries}, we provide the mathematical preliminaries, first from the perspective of homological algebra and single-shot error correction, and then from the perspective of commutative algebra in Section~\ref{sec:ca_preliminaries}. The Appendix further includes additional novel instances of MM codes and also explicitly presents the generation algorithms for MM codes.
\section{Preliminaries from homological algebra\label{sec:ha_preliminaries}}
This section presents the preliminaries for our study of classical and quantum codes through the lens of homological algebra. For a standard introduction to homological algebra, the reader may refer to~\cite{eisenbud2013commutative, weibel1994introduction}. Readers familiar with other works that employ chain complexes for the creation of codes can skip this section.

\subsection{Chain Complexes and Homology\label{sec:chain_complexes_and_homology}}
We give a formal definition of chain complexes and homology. The definition below is quite dry and concise, but the next subsection restates these definitions in more familiar terms related to error correcting codes. The main intuition is that chain complexes, when used for creating stabilizer codes, are a natural way to build sequences of vector spaces with linear constraints at each step of the sequence. These constraints will be the parity checks or metachecks of our codes.

Let $R$ be a commutative ring. In the category $R\text{-}Mod$ of $R$-modules, a \emph{chain complex} $(\mathcal{C}_\bullet, \partial_\bullet)$ is a sequence of $R$-modules and $R$-linear maps
\begin{equation}
\cdots \leftarrow C_{i-1} \xleftarrow{\partial_i} C_i \xleftarrow{\partial_{i+1}} C_{i+1} \leftarrow \cdots
\end{equation}
satisfying $\partial_i \circ \partial_{i+1} = 0$ for all $i$ (see Definition 1.1.1 of~\cite{weibel1994introduction}). This perspective allows us to define the fundamental homological invariants as modules:
\begin{equation}
Z_i(\mathcal{C}) = \ker \partial_i, \quad 
B_i(\mathcal{C}) = \im \partial_{i+1}, \quad H_i(\mathcal{C}) = Z_i(\mathcal{C}) / B_i(\mathcal{C})
\end{equation}
called \emph{\(i\)-cycles}, \emph{\(i\)-boundaries}, and \emph{\(i\)-th homology}, respectively.

The contravariant functor $\Hom_R(-, R): R\text{-}Mod \to R\text{-}Mod$ yields the \emph{cochain complex} $(\mathcal{C}^\bullet, \delta^\bullet)$ with $C^i = \Hom_R(C_i, R)$ and coboundary maps $\delta^i = \partial_{i+1}^*$. The cohomology modules are:
\begin{equation}
Z^i(\mathcal{C}) = \ker \delta^i, \quad B^i(\mathcal{C}) = \im \delta^{i-1}, \quad H^i(\mathcal{C}) = Z^i(\mathcal{C}) / B^i(\mathcal{C})
\end{equation}

When $R = \mathbb{F}_2$, we work in the category of $\mathbb{F}_2$-vector spaces. If each $C_i$ is equipped with a basis, we may identify $C^i \cong C_i$ via the canonical pairing, yielding isomorphisms $H_i(\mathcal{C}) \cong H^i(\mathcal{C})$~\cite{breuckmann2021balanced}.

\subsection{Codes from Chain Complexes\label{sec:codes_from_chain_complexes}}

\subsubsection{Classical Codes\label{sec:classical_codes}}

A classical $[n,k,d]$ linear code $\mathcal{C} \subset \mathbb{F}_2^n$ corresponds to a 2-term chain complex with $R=\mathbb{F}_2$:
\begin{equation}
\mathcal{C}_{\text{classical}}: \quad \mathbb{F}_2^{n-k} \xleftarrow{P} \mathbb{F}_2^n
\end{equation}
where $P$ is a parity-check matrix. The codespace is $\mathcal{C} = Z_1(\mathcal{C}_{\text{classical}}) = \ker P$ with parameters:
\begin{equation}
n = \dim \mathbb{F}_2^n, \quad 
k = \dim H_1(\mathcal{C}_{\text{classical}})
\end{equation}
and code distance given by
\begin{equation}
d = \min\{ |c| : c \in Z_1(\mathcal{C}_{\text{classical}}) \setminus B_1(\mathcal{C}_{\text{classical}}) \}
\end{equation}

\subsubsection{Quantum CSS Codes}
A CSS \editcolor{$[[n,k,d]]$} code~\cite{calderbank1996good, steane1996simple} is specified by two binary matrices $P_X$ and $P_Z$ of sizes $m_X \times n$ and $m_Z \times n$ respectively, satisfying the orthogonality condition  
\begin{equation}
P_X P_Z^{\top} = 0.
\end{equation}
These matrices corresponds to a 3‑term chain complex of \(\mathbb{F}_2\)-vector spaces  
\begin{equation}
\label{eqn: CSS as chain complex}
\mathcal{C}_{\mathrm{CSS}}  : \mathbb{F}_2^{m_X}
\xleftarrow{\;\partial_1 = P_X\;}
\mathbb{F}_2^{n}
\xleftarrow{\;\partial_2 = P_Z^{\top}\;} 
\mathbb{F}_2^{m_Z} 
\end{equation}
where the condition $\partial_1 \partial_2 = 0$ corresponds to the orthogonality of \(P_X\) and \(P_Z\).  
The space \(\mathbb{F}_2^n\) in degree 1 corresponds to the physical qubits.

The homology group in degree 1 is the quotient vector space
\begin{equation}
H_1(\mathcal{C}_{\mathrm{CSS}}) = \frac{\ker \partial_1}{\operatorname{im} \partial_2}
   = \frac{\ker P_X}{\operatorname{im} P_Z^{\top}}.
\end{equation}
Its nonzero elements represent homology classes of nontrivial \(Z\)-type logical operators—cycles that cannot be expressed as boundaries.

Dually, the cohomology group in degree 1 is the quotient vector space
\begin{equation}
H^1(\mathcal{C}_{\mathrm{CSS}}) = \frac{\ker \partial_2^{\!\top}}{\operatorname{im} \partial_1^{\!\top}}
   = \frac{\ker P_Z}{\operatorname{im} P_X^{\top}}.
\end{equation}
Its nonzero elements correspond to cohomology classes of nontrivial \(X\)-type logical operators. 
The parameters of the code are  
\begin{equation}
n = \dim \mathbb{F}_2^n, \qquad
k = \dim H_1(\mathcal{C}_{\mathrm{CSS}}) = \dim H^1(\mathcal{C}_{\mathrm{CSS}})
\end{equation}
and the \(Z\) and \(X\) distances are defined as the minimum Hamming weights among representatives of nontrivial (co)homology classes:  
\begin{align}
d_Z = \min\bigl\{ |z| : z \in \ker P_X \setminus \operatorname{im} P_Z^{\top} \bigr\}, \\
\qquad
d_X = \min\bigl\{ |x| : x \in \ker P_Z \setminus \operatorname{im} P_X^{\top} \bigr\}.
\end{align}
The overall code distance is \(d = \min(d_X,d_Z)\).

A family of CSS quantum codes is \emph{low-density parity-check (LDPC)} if there exists a constant $w$ such that each row and column of $P_X$ and $P_Z$ has Hamming weight at most $w$.

\subsection{Tensor Product of Complexes\label{sec:balanced_product_of_complexes}}

The tensor product of complexes constructs higher-length complexes from a few lower-length complexes. As such, it is a convenient tool to create codes with extra structure (e.g.\ quantum codes or even codes with metachecks) out of simpler building blocks (e.g.\ classical codes).
 
The tensor product of complexes is built as follows~\cite{eberhardt2024logicaloperatorsfoldtransversalgates}. Consider two-term complexes in $R\text{-}Mod$:
\begin{align}
\mathcal{A}_\bullet: &\quad R \xleftarrow{a} R \\
\mathcal{B}_\bullet: &\quad R \xleftarrow{b} R
\end{align}

Their tensor product $\mathcal{A}_\bullet \otimes_R \mathcal{B}_\bullet$ is the following double complex:
\begin{equation}
\begin{array}{ccc}
R \otimes_R R & \xleftarrow{a \otimes \id} & R \otimes_R R \\
\downarrow{\scriptstyle \id \otimes b} & & \downarrow{\scriptstyle \id \otimes b} \\
R \otimes_R R & \xleftarrow{a \otimes \id} & R \otimes_R R
\end{array}
\end{equation}

Here, the differentials should follow the Koszul sign rule, but when \(R\) has characteristic \(2\) (which is the case for complexes used to build stabilizer codes), the sign distinctions vanish, simplifying the expressions.

Taking the total complex $\operatorname{Tot}(\mathcal{A}_\bullet \otimes_R \mathcal{B}_\bullet)$ yields a complex a higher length:
\begin{equation}
R \otimes_R R \xleftarrow{(a \otimes \id, \id \otimes b)} 
(R \otimes_R R) \oplus (R \otimes_R R) \xleftarrow{\begin{pmatrix} \id \otimes b \\ a \otimes \id \end{pmatrix}} R \otimes_R R.
\end{equation}
This "total complex" is frequently also called "the tensor product complex". 
A quantum code is associated to $\operatorname{Tot}(\mathcal{A}_\bullet \otimes_R \mathcal{B}_\bullet)$, where $\mathcal{A}_\bullet$ and $\mathcal{B}_\bullet$ are chain complexes representing classical linear codes.

A generalization of this construction, called $n$-fold balanced product quantum codes is discussed in Appendix B of~\cite{eberhardt2024logicaloperatorsfoldtransversalgates}.

\subsection{Single-Shot Error Correction\label{sec:single_shot_error_correction}}

Quantum codes constructed via sufficiently long chain complexes naturally have metachecks for both the X and Z parity checks. Let the code be defined on the middle space $C_q$ of the chain complex in consideration with $P_X = \partial_q$ and $P_Z = \partial_{q+1}^\top$
where $P_X$ and $P_Z$ are the X and Z-type parity check matrices, respectively. The natural candidates for metacheck matrices are the adjacent boundary maps
\begin{align}
M_X = \partial_{q-1} \quad\text{and}\quad M_Z = \partial_{q+2}^{\top}
\end{align}
so that $M_X P_X=0$ and \editcolor{$M_Z P_Z=0$} by the exactness of the complex.
When the chain complex is sufficiently long \editcolor{($t+1 \ge 5$)}, 
both $M_X$ and $M_Z$ are non-trivial; for \editcolor{$t+1=4$} only $M_Z$ 
is non-trivial, and for \editcolor{$t+1=3$} both are zero. These metachecks encode the redundancies among stabilizers that facilitate single-shot error correction~\cite{Campbell_2019, Quintavalle_2021}.

The single-shot distances associated with X- and Z-type stabilizer measurements are defined as:
\begin{align}
d_{ss}^X = \min \{ |s| : M_X s = 0,\ s \notin \im P_Z^\top \} \\
d_{ss}^Z = \min \{ |s| : M_Z s = 0,\ s \notin \im P_X^\top \}
\end{align}

These quantify the code's robustness against measurement errors during a single measurement round~\cite{Campbell_2019}. The overall single-shot performance is determined by both $d_{ss}^X$ and $d_{ss}^Z$.

\begin{remark}
It is important to note that single-shot decoding does not necessarily require higher-dimensional complexes. Quantum expander codes exhibit single-shot decoding under stochastic noise due to their expansion property of the underlying expander graphs~\cite{Leverrier_2015, leverrier2022quantumtannercodes, Gu_2024, tan2025singleshotuniversalityquantumldpc}. In addition, planar generalized bicycle codes, for example, achieve single-shot and highly effective two-shot decoding when stabilizer redundancy gives a maximum syndrome distance \(d_{S}= 3\)~\cite{lin2025singleshottwoshotdecodinggeneralized}.

For a unifying perspective on these cases, we refer the reader to the notion of confinement introduced in Ref.~\cite{Quintavalle_2021}, which captures the principle that low-weight physical errors give rise to correspondingly small measurement syndromes. It is closely related to the notion of soundness introduced earlier by Campbell~\cite{Campbell_2019}, but is strictly weaker criterion: while it constrains the syndromes produced by small errors, soundness additionally requires that any small syndrome can be generated by an error of small weight. While~\cite{Quintavalle_2021} establishes that quantum expander codes have confinement but not soundness,~\cite{sriram2025diffusioncodesselfcorrectionsmallerset} notes that strong expansion is a sufficient condition for confinement.
\end{remark}

\section{Preliminaries from commutative algebra\label{sec:ca_preliminaries}}

While the previous section describes the utility of chain complexes for the construction of codes in the abstract, this section enumerates a number of algebraic structures that are particularly useful for the explicit construction of specific instances of chain complexes.

This section presents preliminaries from commutative algebra, including polynomial rings, quotient rings, and group rings, which underlie our MM code framework and are used extensively in later sections. The tensor algebra serves as a convenient framework for constructing exterior algebras and wedge products, which are then used to define the Koszul complex over ring $R$. Readers already familiar with these algebraic tools may skip this section. For a standard introduction to commutative algebra, the reader may refer to~\cite{dummit2004abstract, eisenbud2013commutative}.

\subsection{Group Rings}

Many contemporary quantum error-correcting codes can be described using chain complexes of modules over an algebra $\mathbb{F}_2[G]$ associated to a group~$G$. 
Here, we formally define such group algebras (or group rings).

Let $R$ be a commutative ring with identity, and let $G = \{g_1, g_2, \dots, g_n\}$ be a finite group. The group ring $RG$ (also denoted $R[G]$) is the set of all formal sums
\begin{align}
\sum_{i=1}^n a_i g_i, \qquad a_i \in R
\end{align}
where the $a_i$ are coefficients from $R$ and the $g_i$ are elements of $G$. 
Addition and multiplication are defined by
\begin{align}
\sum_{i=1}^n a_i g_i + \sum_{i=1}^n b_i g_i
&= \sum_{i=1}^n (a_i+b_i) g_i \\
\left(\sum_{i=1}^n a_i g_i\right)
\left(\sum_{j=1}^n b_j g_j\right)
&= \sum_{k=1}^n
\left(\sum_{g_i g_j = g_k} a_i b_j \right) g_k 
\end{align}
With these operations, $RG$ forms a group ring~\cite{dummit2004abstract}. This ring is commutative if $G$ is an abelian group.

When the coefficient ring $R$ is a field $F$, the group ring $RG$ is often called
the $F$-algebra (a.k.a.~group algebra over $F$), denoted $F[G]$. In this case, $F[G]$ is a vector space
over $F$ with basis $G$, so that its dimension as an $F$-vector space is $|G|$, and
the elements of $F$ commute with all elements of $F[G]$. These two properties make
$F[G]$ an $F$-algebra~\cite{dummit2004abstract}.

\subsection{Polynomial Rings}
Before introducing quotient rings, we recall the definition of the polynomial ring
$R[x]$ in one indeterminate $x$ over a commutative ring $R$.
An element, known as a polynomial, is a formal sum
\begin{align}
f(x) = \sum_{i=0}^n a_i x^i
\end{align}
where $a_i \in R$ and where $n$ can be any natural number~\cite{dummit2004abstract}.

Addition is defined componentwise:
\begin{align}
\sum_{i=0}^n a_i x^i + \sum_{i=0}^n b_i x^i
= \sum_{i=0}^n (a_i + b_i)x^i
\end{align}
where coefficients may be zero to accommodate polynomials of different degrees.
Multiplication is determined by the rule
\begin{align}
(a x^i)(b x^j) = ab\, x^{i+j}
\end{align}
and extended to all polynomials by distributivity, so that
\begin{align}
\left( \sum_{i=0}^n a_i x^i \right)
\left( \sum_{j=0}^m b_j x^j \right)
= \sum_{k=0}^{n+m} \left( \sum_{i+j=k} a_i b_j \right) x^k .
\end{align}
If $a_n \neq 0$, the integer $n$ is called the degree of $f(x)$ and $a_n$ is called its
leading coefficient. 
The zero polynomial is defined to have leading coefficient~$0$. 
A polynomial is monic if its leading coefficient is $1$. Identifying each $r \in R$ with the constant polynomial $r$, the ring $R[x]$ contains $R$ as a subring and hence has an identity~\cite{dummit2004abstract}.

\begin{remark}
Let $\mathbb{K}$ be a field and let $V$ be an $n$-dimensional $\mathbb{K}$-vector space with basis ${x_1,\dots,x_n}$. Then the symmetric algebra on $V$ is isomorphic to the polynomial ring in $n$ variables over $\mathbb{K}$:
\begin{align}
 \operatorname{Sym}(V) \cong \mathbb{K}[x_1,\dots,x_n].   
\end{align}
\end{remark}

\subsection{Quotient Rings}
In quantum code construction, quotient rings are frequently used to encode periodic boundary conditions and "cyclic" algebraic relations. Here we define the quotient ring formally.

Let $R$ be a commutative ring with identity and let $I\subset R$. The subset $I$ is an ideal when $I$ is an additive subgroup of $R$ and is closed under multiplication by arbitrary elements of $R$, i.e., $ra \in I$ for all $r\in R$ and $a\in I$. Consider the set of additive cosets
\begin{align} 
r+I=\{r+a \,:\, a\in I\}
\end{align} 
with $r \in R$. 
With addition and multiplication of cosets defined by
\begin{align}
\label{eq:quotient_ring_addmul}
(r+I)+(s+I)&=(r+s)+I \\ (r+I)(s+I)&=rs+I
\end{align}
the collection of cosets becomes the quotient ring~$R/I$. 

\subsection{Tensor Algebra}
Before defining the exterior algebra, 
which plays a central role in this paper,
we first introduce the tensor algebra, which provides a convenient method for constructing the exterior algebra as a quotient algebra. 
We first review graded rings, graded ideals, and graded homomorphisms, which provide the language for formulating tensor and exterior algebras.

A ring $S$ is called graded if
\begin{align}
S &= \bigoplus_{k=0}^\infty S_k, \quad 
S_i S_j \subseteq S_{i+j}, \quad \forall i,j \ge 0
\end{align}
where $S_k$ is called the homogeneous component of degree $k$ and its elements are homogeneous of degree $k$.

An ideal $I \subseteq S$ is a graded ideal if
\begin{align}
I &= \bigoplus_{k=0}^\infty (I \cap S_k).
\end{align}
A ring homomorphism $\varphi: S \to T$ is a graded homomorphism if
\begin{align}
\varphi(S_k) &\subseteq T_k, \quad \forall k \ge 0.
\end{align}
Now we define a graded algebra  formally built from a given module.

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. For each integer $k \ge 1$, define
\begin{align}
\label{eq:tensors}
T^k(M) &= M \otimes_R \cdots \otimes_R M \quad {(k \text{ factors})}
\end{align}
and set $T^0(M) = R$~\cite{dummit2004abstract}. The tensor algebra of $M$ is a graded, non-commutative algebra with
\begin{align}
\label{eq:tensor_algebra}
T(M) &= \bigoplus_{k=0}^\infty T^k(M)
\end{align}
where multiplication is simply the tensor product itself:
\begin{align}
(m_1 \otimes \cdots &\otimes m_k) \cdot (m'_1 \otimes \cdots \otimes m'_\ell) \\
&= m_1 \otimes \cdots \otimes m_k \otimes m'_1 \otimes \cdots \otimes m'_\ell.
\end{align}
Elements of $T^k(M)$ are called $k$-tensors. 
Identifying $M$ with $T^1(M)$, the module $M$ is an $R$-submodule of $T(M)$. 
For further discussion on the multiplicative structure and the universal property of $T(M)$, see~\cite[Theorem 31]{dummit2004abstract}.

\subsection{Exterior Algebra}
Having defined the tensor algebra $T(M)$, we next construct the exterior algebra as a quotient of $T(M)$ that enforces skew-commutativity (a.k.a. anti-commutativity).

Let $M$ be an $R$-module over a commutative ring $R$.
The exterior algebra of $M$ is the quotient of the tensor algebra $T(M)$ by the two-sided ideal $J(M)$ generated by all elements of the form
\begin{align}
m \otimes m, \quad \text{with }m \in M.
\end{align}
We denote the exterior algebra by $\bigwedge M$, and write the image of $m_1 \otimes \cdots \otimes m_k$ in $\bigwedge M$ as $m_1 \wedge \cdots \wedge m_k$.

The ideal $J(M)$ enforces skew-commutativity. 
For any $x, y \in M$, the identity
\begin{align}
(x + y) \otimes (x + y) = x \otimes x + x \otimes y + y \otimes x + y \otimes y
\end{align}
holds in $T(M)$ and hence induces the identity 
\begin{align}
0 = (x + y) \wedge (x + y) = x \wedge x + x \wedge y + y \wedge x + y \wedge y
\end{align}
in~$\bigwedge M$.
Since $x \wedge x = 0 = y \wedge y$, we obtain that
\begin{align}
0 &\equiv x \wedge y + y \wedge x \\
&\implies x \wedge y = -\, y \wedge x \quad \forall\, x, y \in M
\end{align}
in the quotient ring~$\bigwedge M$.
Thus, $\bigwedge M$ is a skew-commutative algebra obtained from $T(M)$ by imposing the relations $m \otimes m = 0$ for all $m \in M$.

Of note is the fact that the exterior algebra is graded \cite[Proposition 33]{dummit2004abstract}, with $k$th homogeneous component
\editcolor{\begin{align}
\label{eq:exterior_algebra_homogeneous_component}
\bigwedge^k(M) = T^k(M) / J^k(M)
\end{align}}
where \editcolor{$J^k$ is the $k$th graded component of the ideal $J(M)$} and we define $\wedge^0(M) = R$ and $\wedge^1(M) = M$, treating $M$ as an $R$-submodule of $\bigwedge(M)$.
The wedge product (a.k.a.\ exterior product, as in "exterior algebra") that we just defined above is an alternative to the tensor product when building higher-length chain complexes. In particular, in a later section we will introduce the Kozsul complex, building it either in terms of tensor products or in terms of wedge products. We will see that the wedge product notation is useful to simplifying the process of building longer-length chain complexes.

More formally, the multiplication in $\bigwedge(M)$, called the "wedge product", is alternating: $m_1 \wedge \cdots \wedge m_k = 0$ if $m_i = m_j$ for some $i \neq j$, and simple wedge products satisfy anticommutativity $m \wedge m' = - m' \wedge m$ for $m,m' \in M$ \cite[Theorem 36]{dummit2004abstract}.

\begin{remark}
It is worth observing that in characteristic~2, 
the exterior algebra is the same as a symmetric algebra, i.e., polynomial ring.
\end{remark}
\section{Previous Explicit Constructions\label{sec:previous_constructions}}

In this section we survey previously discovered families of qLDPC codes, anticipating rederiving them as special cases of our Multivariate Multicycle (MM) codes in an upcoming section. The reader may skip this section if they are familiar with the recent discoveries of various qLDPC codes.

\subsection{Abelian Two-Block Group Algebra Codes\label{sec:abelian_2bga}}

The construction of 2BGA codes~\cite{lin2023quantumtwoblockgroupalgebra} considers two length-1 complexes $\mathcal{A}_1 \xrightarrow{a} \mathcal{A}_0$ and $\mathcal{B}_1 \xrightarrow{b} \mathcal{B}_0$. Their tensor product complex has the form:
\begin{equation}
\mathcal{A}_0 \otimes_G \mathcal{B}_0 \xleftarrow{\;\,\partial_1\;\,} (\mathcal{A}_0 \otimes_G \mathcal{B}_1) \oplus (\mathcal{A}_1 \otimes_G \mathcal{B}_0) \xleftarrow{\;\,\partial_2\;\,} \mathcal{A}_1 \otimes_G \mathcal{B}_1
\end{equation}

Using $\mathbb{F}_2[G] \otimes_G \mathbb{F}_2[G] \simeq \mathbb{F}_2[G]$, this yields the total complex which has the following boundary maps:
\begin{align}
\partial_1 = \begin{bmatrix} a & b \end{bmatrix}, \quad 
\partial_2 = \begin{bmatrix} b \\ a \end{bmatrix}
\end{align}

For $G = \mathbb{Z}_\ell \times \mathbb{Z}_m$, this construction gives BB codes with parity-check matrices $P_X = \partial_1$ and $P_Z = \partial_2^\top$.

\begin{example}
Let \(G = \mathbb{Z}_{14} \times \mathbb{Z}_2 = \langle x, s \mid x^{14} = s^2 = xsx^{-1}s^{-1} = 1\rangle\) and consider the group algebra \(\mathbb{F}_2[G]\). Denote by \(x\) and \(s\) the generators corresponding to the cyclic groups \(\mathbb{Z}_4\) and \(\mathbb{Z}_2\), respectively. Define $A(x,y) = 1 + x^7$ and $B(x,y) = 1 + x^7 + s + x^8 + s x^7 + x$. The 2BGA code built from \editcolor{$A(x,y)$} and \editcolor{$B(x,y)$} has parameters $[[56, 28, 2]]$  \cite{lin2023quantumtwoblockgroupalgebra}.
\end{example}

Notably, some papers in the literature prefer to use the notation $C_x$ instead of \(\mathbb{Z}_x\), presumably to allude to \editcolor{the cycle-graph structure of the corresponding Cayley graph.} % Slightly tightened the wording here after looking into https://math.stackexchange.com/questions/2396937/do-cyclic-groups-always-have-the-shape-of-a-cycle

\subsection{Multivariate Bicycle Codes\label{sec:mb_code}}

MB~\cite{voss2025multivariate} codes generalize the BB construction from two cyclic dimensions to an arbitrary number $r$ of cyclic groups. While BB codes are defined over the group algebra $\mathbb{F}_2[\mathbb{Z}_\ell \times \mathbb{Z}_m]$, MB codes employ the multivariate group algebra $\mathbb{F}_2[\mathbb{Z}_{l_1} \times \mathbb{Z}_{l_2} \times \cdots \times \mathbb{Z}_{l_r}]$.

For the trivariate bicycle (TB) case ($r=3$) which is the one explicitly explored in Table 2 of~\cite{voss2025multivariate}, the matrices are:
\begin{equation}
x = S_l \otimes I_m,\quad y = I_l \otimes S_m,\quad z = S_l \otimes S_m
\end{equation}
where $S_l$ and $S_m$ are cyclic shift matrices. These matrices mutually commute and generate a circulant matrux representation of the underlying group algebra.
The stabilizers generator matrices maintain the standard bicycle form:
\begin{align}
P_X = [A|B], \quad P_Z = [B^\top|A^\top] 
\end{align}
where \editcolor{$A(x,y,z)$} and \editcolor{$B(x,y,z)$} are polynomials in $x$, $y$, and $z$. Additional variables are explicitly generated through products of group algebra generators (a.k.a.\ generating polynomials), e.g.\ $z = xy$. 

\begin{example}
    \label{ex:48-4-6-code} 
    Let $x$ and $y$ be the generators of the group algebra $\mathbb{F}_2[\mathbb{Z}_4\times \mathbb{Z}_6]$. Define $z = xy$, $A(x,y, z) = x^3 + y^5$, and $B(x,y,z) = x + z^5 + y^5 + y^2$. 
    The 2BGA code constructed from \editcolor{$A(x,y,z)$} and \editcolor{$B(x,y,z)$} has parameters $[[48,4,6]]$ whose stabilizer checks have weight~6~\cite{voss2025multivariate}.
\end{example}

\subsection{Generalized Bicycle Codes\label{sec:gb_code}}

GB codes~\cite{PhysRevA.88.012311}, constitute a special case of MM codes where one cyclic dimension is trivial. The standard GB construction employs a single cyclic group $C_\ell$ to generate commuting circulant matrices $A$ and $B$, forming stabilizer matrices:
\begin{equation}
P_X = [A|B], \quad P_Z = [B^\top|A^\top]
\end{equation}
yielding quantum codes with parameters $[[2\ell, k, d]]$.

The GB code construction uses the quotient ring $\mathbb{F}_2[x,y]/\langle x^\ell-1, y-1\rangle \cong\mathbb{F}_2[x]/\langle x^\ell-1\rangle$ due to $y=1$. The generating polynomials employ only the non-trivial variable $x$:
\editcolor{\begin{align}
A(x) = \sum_{i \in a_{\text{shifts}}} x^{i \bmod \ell}\\ 
B(x) = \sum_{j \in b_{\text{shifts}}} x^{j \bmod \ell}
\end{align}}
\begin{example}
Let x be the generator of $\mathbb{F}_2[\mathbb{Z}_{35}]$ where $\mathbb{Z}_{35}$ is the cyclic group of order 35. Using \editcolor{$A(x) = \sum_{i \in a_{\text{shifts}}} x^{i \bmod 35}$} and \editcolor{$B(x) = \sum_{j \in b_{\text{shifts}}} x^{j \bmod 35}$}, where $a_{\text{shifts}} = \{0,15,16,18\}$ and $b_{\text{shifts}} = \{0,1,24,27\}$, we construct the instance of GB code with parameters $[[70, 8, 10]]$ ~\cite{lin2023quantumtwoblockgroupalgebra}.
\end{example}

\subsection{Trivariate Tricycle Codes\label{sec:tt_code}}

The work of~\cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic} extends the BB codes to three length-1 complexes $\mathcal{A}_\bullet$, $\mathcal{B}_\bullet$, $\mathcal{C}_\bullet$. Their tensor product (after some algebraic manipulations) has the form~\cite{jacob2025singleshotdecodingfaulttolerantgates}:

\editcolor{\begin{equation}
\begin{aligned}
\label{eq:equation_35}
&\bigoplus_{i+j+k=2} \mathcal{A}_i \otimes_G \mathcal{B}_j \otimes_G \mathcal{C}_k \xleftarrow{\partial_3} \mathcal{A}_1 \otimes_G \mathcal{B}_1 \otimes_G \mathcal{C}_1 \\
&\qquad \mathcal{A}_0 \otimes_G \mathcal{B}_0 \otimes_G \mathcal{C}_0 \xleftarrow{\partial_1} \bigoplus_{i+j+k=1} \mathcal{A}_i \otimes_G \mathcal{B}_j \otimes_G \mathcal{C}_k \xleftarrow{\partial_2}
\end{aligned}
\end{equation}}

The boundary maps of this tensor product complex are given by~\cite{jacob2025singleshotdecodingfaulttolerantgates}:

\begin{align}
\label{eq:tt_code}
\quad \partial_3 = \begin{bmatrix} a \\ b \\ c \end{bmatrix}, 
\quad \partial_2 = \begin{bmatrix} 0 & c & b \\ c & 0 & a \\ b & a & 0 \end{bmatrix}, 
\quad \partial_1 = \begin{bmatrix} a & b & c \end{bmatrix}
\end{align}
For TT codes, the parity-check matrices are $P_X = \partial_1$, $P_Z = \partial_2^\top$, and $M_Z = \partial_3^\top$ provides Z-metachecks. Consequently, the code construction employs \editcolor{a} circulant matrix realization of the group algebra. For $G = \mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_p$, the group is generated by the matrices:
\editcolor{\begin{align}
    x &= S_\ell\otimes I_m\otimes I_p \\
    y &= I_\ell\otimes S_m \otimes I_p \\
    z &= I_\ell \otimes I_m\otimes S_p
\end{align}}
where $S_n$ denotes the $n \times n$ cyclic shift matrix with elements $(S_n)_{i,j} = \delta_{i,j\oplus 1}$, and $\oplus$ denotes addition modulo $n$. These matrices satisfy
\begin{align}
x^\ell = y^m = z^p = I_N \quad\\ 
xy = yx, \quad xz = zx, \quad yz = zy
\end{align}
 where $N = \ell m p$. The commutation relations follow from the mixed-product property of the Kronecker product. These matrices define a faithful representation
\begin{align}
\rho: G \longrightarrow \mathrm{GL}(N, \mathbb{F}_2), \quad (g_1,g_2,g_3) \mapsto x^{g_1} y^{g_2} z^{g_3}
\end{align}
whose image is isomorphic to $G = \mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_p$~\cite{jacob2025singleshotdecodingfaulttolerantgates}.
To obtain a polynomial-ring description of the TT code and establish its identification with the algebra of circulant matrices, consider the quotient ring \editcolor{$R = \mathbb{F}_2[x,y,z]/\langle x^\ell-1, y^m-1, z^p-1\rangle \cong \mathbb{F}_2[G]$} together with the \editcolor{injective ring homomorphism}
\begin{align}
\psi: R\longrightarrow M_{\ell m p}(\mathbb{F}_2), \quad
\begin{aligned}
x &\mapsto S_\ell \otimes I_m \otimes I_p\\
y &\mapsto I_\ell \otimes S_m \otimes I_p\\
z &\mapsto I_\ell \otimes I_m \otimes S_p 
\end{aligned}
\end{align}
where $S_\ell, S_m, S_p$ are the circulant shift matrices of sizes \editcolor{$\ell \times \ell$, $m \times m$, and $p \times p$}, respectively. \editcolor{Under this map}, a polynomial \editcolor{$A(x,y,z) = \sum_{i,j,k} a_{ijk} \, x^i y^j z^k \in \mathbb{F}_2[G]$} \editcolor{(with coefficients $a_{ijk} \in \mathbb{F}_2$)} corresponds to the matrix $\psi(A(x,y,z) = \sum_{i,j,k} a_{ijk} \, S_\ell^i \otimes S_m^j \otimes S_p^k$. In particular, there is a bijection between the set of monomials $\{ x^i y^j z^k \mid 0 \le i < \ell, \ 0 \le j < m, \ 0 \le k < p \}$ and the set of $(\ell m p) \times (\ell m p)$ matrices generated by $x$, $y$, and $z$, allowing us to work with either polynomial formulation (which allows for easier characterization of new codes) or the matrix formulation~\cite{wang2025coprimebivariatebicyclecodes}.

Having defined the boundary maps \(\partial_1, \partial_2, \partial_3\) of the abstract chain complex in \editcolor{Eq.}~\ref{eq:tt_code}, we now construct the concrete parity-check matrices for the quantum CSS code. This ensures the fundamental commutativity relations \(P_X P_Z^\top = 0\) and \(M_Z P_Z = 0\) are satisfied, as they are the matrix equivalents of the chain complex condition \(\partial_1 \circ \partial_2 = 0\) and \(\partial_2 \circ \partial_3 = 0\), respectively.

\begin{example}
Let $\mathbb{F}_2[\mathbb{Z}_4 \times \mathbb{Z}_3 \times \mathbb{Z}_2] \cong S = \mathbb{F}_2[x,y,z]/\langle x^4-1, y^3-1, z^2-1\rangle$. Define the polynomials \(A(x,y,z) = 1 + y + xy^2\), \(B(x,y,z) = 1 + yz + x^2y^2\), and \(C(x,y,z) = 1 + xy^2z + x^2y\) in $S$. The TT code constructed from trivariate quotient polynomials $A(x,y,z)$, $B(x,y,z)$, $C(x,y,z)$ yields a quantum code with parameters $[[72,6,6]]$~\cite{jacob2025singleshotdecodingfaulttolerantgates}.
\end{example}
\begin{figure*}
\centering
\begin{tikzpicture}[
  font=\normalsize,
  box/.style={
    rectangle, rounded corners=6pt,
    draw,
    very thick,
    align=center,
    text width=6.4cm,
    inner sep=10pt,
    minimum height=1.1cm
  },
  amc/.style={box, fill=red!3, draw=red!60!black},
  mm/.style={box, fill=teal!3, draw=teal!60!black},
  css/.style={box, fill=violet!4, draw=violet!60!black, text width=13.5cm},
  arr/.style={-{Stealth[length=6pt]}, thick, draw=gray!60},
  header/.style={font=\bfseries\large}
]
\node[header] at (-4.2,0.7) {AMC Codes};
\node[header] at ( 4.2,0.7) {Our Codes};
\node[amc] (a1) at (-4.2,-0.8)
  {Abelian \textcolor{red}{group algebra}};
\node[amc] (a2) at (-4.2,-2.6)
  {\textcolor{red}{Regular representation} of group algebra elements};
\node[amc] (a3) at (-4.2,-4.4)
  {Multi-block complex (MBC) via \textcolor{red}{iterative tensor products of chain complexes}};
\node[mm] (m1) at (4.2,-0.8)
  {\textcolor{blue}{Polynomial ring} R (free algebra)};
\node[mm] (m2) at (4.2,-2.6)
  {\textcolor{blue}{Quotient} R by an \textcolor{blue}{ideal} I \\ (Circulant matrix representation of polynomials in R/I)};
\node[mm] (m3) at (4.2,-4.4)
  {\textcolor{blue}{Koszul complex} over R/I};
\node[css] (css) at (0,-6.4)
  {CSS quantum code};
\draw[arr] (a1) -- (a2);
\draw[arr] (a2) -- (a3);
\draw[arr] (a3) -- (a3.south |- css.north);

\draw[arr] (m1) -- (m2);
\draw[arr] (m2) -- (m3);
\draw[arr] (m3) -- (m3.south |- css.north);
\draw[gray!30, line width=0.6pt] (0,-0.4) -- (0,-5.9);
\end{tikzpicture}
\caption{\editcolor{\textbf{Two general constructions of CSS quantum codes.} Left (AMC codes~\cite{lin2025abelianmulticyclecodessingleshot}): To make a candidate for a good qLDPC CSS code, start by picking an abelian group algebra, then pick a regular representation that will be used to convert elements of the algebra to binary matrices, then pick a few elements of that algebra, and finally, use the MBC rule~\cite{lin2025abelianmulticyclecodessingleshot}) to map the binary matrix representation of these elements to an arbitrary length chain complex. If the length of the complex is greater than 4, we are guaranteed to have metachecks. One has to perform a (potentially random) search over elements of the the chosen algebra to find good code candidates. Right (our Multivariate Multicycle codes): To make a candidate code, start by picking a polynomial ring $R$ (i.e.\ picking a fixed number of symbolic variables $x,\ y,\ z,\ \dots$), then pick an ideal for it $I$ (equivalent to specifying lattice size for the physical qubits), then pick $t$ polynomials from the quotient of the polynomial ring and the ideal (potentially at random), and finally use our "Kozsul complex" construction to make a chain complex of length $t+1$ and check numerically whether that is a good code. We claim that the "Kozsul complex" rule we introduce is also simpler to implement than the iterative MBC rule. As any finite abelian group can be represented as a product of cyclic groups and as group algebras of products of cyclic groups are isomorphic to quotient rings $R/I$, the MM codes can represent any AMC code, as seen in Remark~\ref{mm:amc}.
Our framework enables efficient exploration of MM code families via exhaustive polynomial search, as demonstrated in Table~\ref{tab:table2}.
}
%\textbf{Two general constructions of CSS quantum codes.} Left (AMC codes~\cite{lin2025abelianmulticyclecodessingleshot}): Defined over the abelian group algebra $\mathbb{F}_2[G]$, where $G$ is a finite abelian group. For any element $a \in \mathbb{F}_2[G]$, the $\ell \times \ell$ matrix $M(a)$ (with $\ell = |G|$) is defined via the $\mathbb{F}_2$-linear regular representation. Given a sequence $\{a_j\}_{j=1}^D \subset \mathbb{F}_2[G]$ of $D \ge 2$ elements, the matrices $A_j = M(a_j)$ pairwise commute and are assembled into a multi-block chain complex $\mathrm{MBC}(A_1,\dots,A_D)$ via iterative tensor products of 1-complexes, from which a CSS code is extracted. Right (MM codes, this work): We work with the polynomial ring $\mathbb{F}_2[x_1,\dots,x_D]$ and impose cyclic relations via the ideal $I = \langle x_1^{\ell_1}-1, \dots, x_D^{\ell_D}-1 \rangle$, giving the quotient ring $S = \mathbb{F}_2[x_1,\dots,x_D] / I$. Since every finite abelian group decomposes as 
%$G \cong C_{\ell_1} \times \cdots \times C_{\ell_D}$ by the fundamental 
%theorem of finitely generated abelian groups, this yields 
%$S \cong \mathbb{F}_2[G]$, covering the same class of groups as AMC codes. Polynomials in $S$ admit a natural circulant matrix representation, and given $\{F_j\}_{j=1}^t \subset S$, we form the Koszul complex $K_\bullet([F_1,\dots,F_t];\, S)$ and replace each polynomial with its circulant matrix to extract a CSS code. This framework enables efficient exploration of MM code families via exhaustive polynomial search, as demonstrated in Table~\ref{tab:table2}.}
}
\label{fig:MMcodesconstruction}
\end{figure*}

\subsection{Challenges in creating higher-length complexes}

The TT codes provide only partial single-shot decoding (they have metachecks only for the Z parity checks). 
\editcolor{A longer complex lets one provide metachecks for both the X and Z parity checks. The chief difficulty now is to figure out an "easy" technique to construct sufficiently long complexes. 
The technique of tensor products employed in the discovery of the TT codes can, in principle, be used to create such even longer complexes~\cite{lin2025abelianmulticyclecodessingleshot}, but explicitly describing the boundary maps (i.e.\ the parity check and metacheck matrices) of such complexes can grow complicated with those methods. Another alternative is the more general approach of AMC codes~\cite{lin2025abelianmulticyclecodessingleshot}, using a so-called "multiblock complex" (MBC) rule, also generating long enough complexes, at the cost of following a relatively complicated iterative procedure. Using the Koszul complex over a quotient ring, as we advocate in this paper, offers a simple, efficient way to describe these matrices that has already been in use for a long time in other fields of mathematics. In particular, it substantially expedites generating random samples of such codes, as we demonstrate through the many good codes we have discovered, listed in Table ~\ref{tab:table2}. For a comparison, see Fig.~\ref{fig:MMcodesconstruction}.}

%Moreover,  contrasts the two constructions: while TT codes are a special case of the recently introduced AMC codes~\cite{lin2025abelianmulticyclecodessingleshot}, which work with matrix representations of group algebra elements assembled via iterated tensor products, our construction works directly with polynomials in a multivariate quotient ring. Beyond the conceptual difference, this algebraic 
%perspective brings practical advantages: it enables the use of commutative-algebra tools, admits a simple description of boundary maps, and substantially expedites random sampling over the polynomial space --- leading to the new MM code families in Table~\ref{tab:table2}, which include codes over $C_{\ell_1} \times C_{\ell_2} \times C_{\ell_3} \times C_{\ell_4}$ not previously reported in the AMC framework.}

\section{Koszul Complex Formalism for Quantum Codes\label{sec:koszul_complex_formalism_for_quantum_codes}}

We establish that the abstract chain complexes (and, equivalently, their boundary maps which provide parity checks and potentially metachecks)
defining the recently introduced 2-block and 3-block group algebra codes (BB~\cite{Bravyi_2024}, TT codes~\cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic} and more),
are manifestations of a fundamental object in commutative algebra and algebraic geometry: the Koszul complex. This connection places these previous quantum error correcting code constructions within a well-studied mathematical framework, allowing us to systematically derive their properties, generalize them to higher dimensions (thus providing both X and Z metachecks), and perform efficient numerical searches for new codes.

\subsection{Koszul Complexes\label{sec:koszul_complexes}}

Koszul complexes can be defined in a few different ways. A definition (shown below) in terms of tensor products makes it easy to restate previous code constructions in terms of Koszul complexes. Another equivalent definition, in terms of exterior algebra, makes the derivation of boundary maps more explicit. Here we present both definitions and use the exterior algebra approach to restate the codes from Sec.~\ref{sec:previous_constructions} in new, simpler, more general terms, making them also much easier to construct through modern computer algebra systems like OSCAR~\cite{OSCAR,OSCAR-book}.

Let $R$ be a commutative Noetherian ring with identity, and let
\editcolor{$\underline{x} = (x_1,\dots,x_t)$} be a finite sequence of elements of $R$.
The associated Koszul complex is a standard $R$-complex admitting several equivalent constructions~\cite{atiyah2018introduction, bruns1998cohen, eisenbud2013commutative, sather2009homological, stacks-project}. We recall the tensor product and exterior algebra
descriptions, which will be used interchangeably.
\begin{definition}[Tensor product construction~\cite{sather2009homological}]
For $x\in R$, let
\begin{equation}
K_\bullet(x_1; R)=\bigl(0 \longleftarrow R \xleftarrow{x_1} R \longleftarrow 0\bigr)
\end{equation}
denote the $R$-complex concentrated in degrees $1$ and $0$.
For $\underline{\mathbf{x}}=(x_1,\dots,x_t)$, define
\begin{equation}
K_\bullet(\mathbf{\underline{x}}; R)
\cong
K_\bullet(x_1; R)\otimes_R \cdots \otimes_R K_\bullet(x_t; R)
\end{equation}
where the tensor product is taken in the category of $R$-complexes.
This $R$-complex is called the \emph{Koszul complex} on $\mathbf{\underline{x}}$.
\end{definition}
An equivalent description is obtained using wedge product from exterior algebra. The reader may consult~\cite{abraham2012manifolds, bruns1998cohen, eisenbud2013commutative, darling1994differential} for more details.
\begin{definition}[Exterior algebra construction~\cite{sather2009homological}]
Let $\underline{x}$ be a sequence of elements in $R$. Define the $R$-modules:

\editcolor{\begin{align}
K_0(\underline{x}; R) &= R \quad\text{ with basis }\{1\}    \\
K_1(\underline{x}; R) &= R^t \quad\text{with basis }\{e_1,\dots,e_t\}
\end{align}}
and for $1 < i \leq \editcolor{t}$, consider $K_i(\underline{x}) = R^{\binom{\editcolor{t}}{i}}$ with basis
\begin{align}
{\{e_{j_1} \wedge \cdots \wedge e_{j_i} \mid 1 \leq j_1 < \cdots < j_i \leq \editcolor{t}\}}  
\end{align}
The Koszul complex $K(\underline{x})$ is the $R$-complex

\editcolor{\begin{align}
0 \leftarrow K_0(\underline{x};R) \xleftarrow{\partial_1} K_1(\underline{x}; R) \xleftarrow{\partial_2} \cdots \xleftarrow{\partial_t} K_t(\underline{x}; R) \leftarrow 0
\end{align}}
where the differentials $\partial_i: K_i(\underline{x}) \to K_{i-1}(\underline{x})$ are defined on basis elements by
\begin{align}
\label{eq:explicit_formula}
\partial_i(e_{j_1} \wedge \cdots \wedge e_{j_i}) = \sum_{s=1}^i (-1)^{s+1} x_{j_s} \; e_{j_1} \wedge \cdots \wedge \widehat{e}_{j_s} \wedge \cdots \wedge e_{j_i}
\end{align}
where $\widehat{e}_{j_s}$ indicates that the factor $e_{j_s}$ is omitted from the wedge product.
Notice that the $K_i(\underline{x}; R)$ is a free $R$-module, so the homomorphism $\partial_i$ is completely determined by its values on the basis. In particular, $\partial_1: K_1(\underline{x};R) \to K_0(\underline{x}; R)$ is given by $\partial_1(e_j) = x_j$ for each $j = 1,\dots,\editcolor{t}$.
\end{definition}

If $R=\mathbb{F}_2[G]$ with $G$ being finite abelian group, then all sign factors vanish.

\subsection{Examples of Boundary Maps and Code Constructions\label{sec:boundary_maps_and_code_constructions}}

The exterior algebra description provides explicit formulas for the boundary maps, which correspond directly to the parity check and metacheck matrices in CSS code constructions. \editcolor{We create the "standard" length $t+1$ Koszul complex (over a polynomial quotient ring)}, which, as we saw above, has a simple explicit form for its boundary maps. To then generate a specific instance of a code from this class of complexes, the variables appearing in boundary maps are replaced by circulant matrix representations of polynomials from multivariate quotient rings. Thus deciding on a $t$ and on these polynomials completely defines a code instance (and if $t$ is large enough, we also have metachecks, see \editcolor{Eq.}~\eqref{eq:explicit_formula}).

We restate all of the previously established code constructions (see Sec.~\ref{sec:previous_constructions}) in our Koszul complex formalism:

\begin{example}[BB codes~\cite{Bravyi_2024} (\(t=2\))]
Consider $R = \mathbb{F}_2[x,y]$, and let $S = R/\langle x^\ell-1, y^m-1 \rangle 
\cong \mathbb{F}_2[\mathbb{Z}_\ell \times \mathbb{Z}_m]$. Let $\pi: R \rightarrow S$ be the natural projection mapping. Note that $\ker(\pi) = \langle x^\ell-1, y^m-1 \rangle$, since for any ring $R$ and ideal $I \subseteq R$, the kernel of $r \mapsto r + I$ from $R$ to $R/I$ is $I$ \cite[Theorem 15.4]{gallian2021contemporary}. Given bivariate polynomials $f(x,y)$, $g(x,y) \in R$, we define their images in $S$ as $F = \pi(f(x,y))$ and $G = \pi(g(x,y))$.
 
Form the Koszul complex $K_\bullet([F,G]; S)$:
\begin{equation}\label{eq:koszul-complex-sequence-t2}
0 \longleftarrow K_0 \xleftarrow{\partial_1} K_1 \xleftarrow{\partial_2} K_2 \longleftarrow 0
\end{equation}
which has the standard $S$-modules and bases (for $t=2$):
\begin{alignat}{3}
&K_0 \cong S &\text{ with basis }&\{1\} \\
&K_1 \cong S^2 &\text{ with basis }&\{e_1, e_2\} \\
&K_2 \cong S &\text{ with basis }&\{e_1 \wedge e_2\}
\end{alignat}
and which has the standard boundary maps:
\begin{align}\label{eq:boundary-maps-t2}
\partial_2 = \begin{bmatrix} F & G \end{bmatrix}, \quad 
\partial_1 = \begin{bmatrix} G \\ F \end{bmatrix}.
\end{align}
To construct the BB code, 
we now take the quotient by the relations $x^\ell = 1$ and $y^m = 1$ to get a chain complex of $S$-modules.
Here we rely on the standard isomorphism between the ring of $n \times n$ circulant matrices over a $\mathbb{F}_2$ and the group algebra $\mathbb{F}_2[C_n]$ of the cyclic group~\cite{hurley2006group}. 
This enables us to replace each polynomial $F$, $G$ with the circulant matrix representation \cite[Theorem~2.1]{albuquerque2011hopfalgebraicpropertiescirculant}. As a result, we obtain a CSS code with $P_Z = \partial_2$ and $P_X = \partial_1^\top$. 
\end{example}

\begin{example}[TT codes~\cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic} (\(t=3\))]
Consider $R = \mathbb{F}_2[x,y,z]$, and let 
$S = R/\langle x^\ell-1, y^m-1, z^p-1 \rangle 
\cong \mathbb{F}_2[\mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_p]$.
Let $\pi: R \rightarrow S$ be the natural projection mapping. Given trivariate polynomials $f(x,y,z)$, $g(x,y,z)$, $h(x,y,z) \in R$, we define their images in $S$ as $F = \pi(f(x,y,z))$, $G = \pi(g(x,y,z))$ and $H = \pi(h(x,y,z))$.

Form the Koszul complex $K_\bullet([F,G,H]; S)$:
\begin{equation}
0 \longleftarrow K_0 \xleftarrow{\partial_1} K_1 \xleftarrow{\partial_2} K_2 
\xleftarrow{\partial_3} K_3 \longleftarrow 0
\end{equation}
which has the standard $S$-modules and bases (for $t=3$):
\begin{alignat}{3}
&K_0 \cong S &\text{ with basis }&\{1\} \\
&K_1 \cong S^3 &\text{ with basis }&\{e_1, e_2, e_3\} \\
&K_2 \cong S^3 &\text{ with basis }&\{e_1 \wedge e_2,\ e_1 \wedge e_3,\ e_2 \wedge e_3\} \\
&K_3 \cong S &\text{ with basis }&\{e_1 \wedge e_2 \wedge e_3\}
\end{alignat}
and which has the standard boundary maps:
\begin{align}
\partial_3 &= \begin{bmatrix} F & G & H \end{bmatrix}, &
\partial_2 &= \begin{bmatrix} 
G & H & 0 \\ 
F & 0 & H \\ 
0 & F & G 
\end{bmatrix}, &
\partial_1 &= \begin{bmatrix} H \\ G \\ F \end{bmatrix}
\end{align}
To obtain the TT code, we impose the cyclic relations $x^\ell = 1$, $y^m = 1$\editcolor{,} and $z^p=1$\editcolor{, and} replace each polynomial \editcolor{$F(x,y,z)$, $G(x,y,z)$, $H(x,y,z)$} with \editcolor{its} circulant matrix representation \editcolor{under the} isomorphism~\cite[Theorem~2.1]{albuquerque2011hopfalgebraicpropertiescirculant}.
This produces a chain complex over $S$ that encodes a CSS code with $P_Z = \partial_2$, $P_X = \partial_1^\top$, 
and $Z$-metacheck $M_Z = \partial_3$.
\end{example}
\begin{example}[Our new Multivariate Multicycle (MM) codes at \(t=4\)]
Consider the polynomial ring $R = \mathbb{F}_2[w,x,y,z]$ and let
$S = R/\langle w^\ell-1, x^m-1, y^n-1, z^p-1 \rangle$, 
which is isomorphic to $\mathbb{F}_2[\mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_n \times \mathbb{Z}_p]$. Let $\pi: R \rightarrow S$ be the natural projection mapping. Given multivariate polynomials $f(w,x,y,z)$ ,$g(w,x,y,z)$, $h(w,x,y,z)$, $i(w,x,y,z) \in R$, we define their images in $S$ as $F = \pi(f(w,x,y,z))$, $G = \pi(g(w,x,y,z))$, $H = \pi(h(w,x,y,z))$ and $I = \pi(i(w,x,y,z))$.
Form the Koszul complex $K_\bullet([F,G,H,I]; S)$:
\begin{equation}
0 \longleftarrow K_0 \xleftarrow{\partial_1} K_1 \xleftarrow{\partial_2} K_2 
\xleftarrow{\partial_3} K_3 \xleftarrow{\partial_4} K_4 \longleftarrow 0
\end{equation}
which has the standard $S$-modules and bases (for $t=3$):
\begin{alignat}{3}
&K_0 \cong S &\text{ with basis }&\{1\} \\
&K_1 \cong S^4 &\text{ with basis }&\{e_1, e_2, e_3, e_4\} \\
&K_2 \cong S^6 &\text{ with basis }&\{e_1 \wedge e_2,\ e_1 \wedge e_3,\ e_1 \wedge e_4, \nonumber \\
&&&\ e_2 \wedge e_3,\ e_2 \wedge e_4,\ e_3 \wedge e_4\} \\
&K_3 \cong S^4 &\text{ with basis }&\{e_1 \wedge e_2 \wedge e_3,\ e_1 \wedge e_2 \wedge e_4, \nonumber \\
&&&\ e_1 \wedge e_3 \wedge e_4,\ e_2 \wedge e_3 \wedge e_4\} \\
&K_4 \cong S &\text{ with basis }&\{e_1 \wedge e_2 \wedge e_3 \wedge e_4\}
\end{alignat}
and which has the standard boundary maps:
\begin{align}
\label{eq:4d_koszul_complex}
\partial_4 &= \begin{bmatrix} F & G & H & I \end{bmatrix}, &
\partial_3 &= \begin{bmatrix} 
G & H & I & 0 & 0 & 0 \\ 
F & 0 & 0 & H & I & 0 \\ 
0 & F & 0 & G & 0 & I \\ 
0 & 0 & F & 0 & G & H 
\end{bmatrix} \\
\partial_2 &= \begin{bmatrix} 
H & I & 0 & 0 \\ 
G & 0 & I & 0 \\ 
0 & G & H & 0 \\ 
F & 0 & 0 & I \\ 
0 & F & 0 & H \\ 
0 & 0 & F & G 
\end{bmatrix}, &
\partial_1 &= \begin{bmatrix} I \\ H \\ G \\ F \end{bmatrix}
\end{align}
To obtain the MM code, following the process we've seen twice now, 
we replace each polynomial \editcolor{$F(w,x,y,z)$, $G(w,x,y,z)$, $H(w,x,y,z)$, $I(w,x,y,z)$} with the circulant matrix using the isomorphism 
\cite[Theorem~2.1]{albuquerque2011hopfalgebraicpropertiescirculant}. 
This produces a chain complex over $S$ that encodes a CSS code with $P_Z = \partial_3$, $P_X = \partial_2^\top$, 
and metachecks $M_Z = \partial_4$, $M_X = \partial_1^\top$.
\end{example}

\section{Multivariate Multicycle Codes\label{sec:multivariate_multicycle_codes}}

Finally, we show how the mechanistic procedure shown in Sec.~\ref{sec:koszul_complex_formalism_for_quantum_codes} can be used not only for representing many classes of known codes in simpler and more unified terms, but also for the creation of new codes, possessing otherwise unavailable valuable properties (like complete single-shot decoding capabilities, record-breaking confinement, and possibly non-Clifford fault-tolerant logical gates). In this section we simply elaborate upon the $t=4$ example above.

\subsection{Metacheck CSS Codes (mCSS Codes)}
We introduce the notion of a \emph{metacheck CSS code (mCSS code)} as a 5-term chain complex that extends the standard CSS code construction.

Recall that a CSS code involved two parity-check matrices $P_Z$ and $P_X$ satisfying $P_X P_Z^{\top} = 0$,
and hence we could rephrase it as a three-term chain complex, as in Eq.~\eqref{eqn: CSS as chain complex}.
Metachecks are additional maps $M_X$ and $M_Z$ that satisfy $M_X P_X = 0$ and $M_Z P_Z = 0$.
We can encode a CSS code with metachecks as a five-term chain complex.

\begin{definition}[mCSS code]
An \emph{mCSS code} is a 5-term chain complex of finite-dimensional vector spaces over $\mathbb{F}_2$:
\begin{align}
C_{-1} &\xleftarrow{\partial_0} C_0 \xleftarrow{\partial_1} C_1 \xleftarrow{\partial_2} C_2 \xleftarrow{\partial_3} C_3
\end{align}
whose inner 3-term complex
\begin{align}
C_0 \xleftarrow{\partial_1} C_1 \xleftarrow{\partial_2} C_2
\end{align}
defines a conventional CSS code with parity-check matrices $P_X = \partial_1$ and $P_Z = \partial_2^\top$, 
and whose remaining differentials determine metacheck matrices $M_X = \partial_0$ and $M_Z =\partial_4^\top$.
\end{definition} 

Any sufficiently long chain complex of vector spaces over $\mathbb{F}_2$ contains a 5-term subcomplex,
which thus provides an mCSS code.
For instance, any Koszul complex with length \editcolor{$t+1 \geq 5$} will contain a 5-term subcomplex that is nonzero in every term.

It can be convenient to ask for the following property.

\begin{definition}[Balanced mCSS Code]
An mCSS code
\begin{align}
C_{-1} &\xleftarrow{M_X} C_0 \xleftarrow{P_X} C_1 \xleftarrow{P_Z} C_2 \xleftarrow{M_Z} C_3
\end{align}
is {\em balanced} if $\dim C_{-1} = \dim C_3$ and $\dim C_0 = \dim C_2$.
\end{definition}

Balanced codes are convenient because the $X$‑ and $Z$‑stabilizer generator matrices have identical matrix dimensions up to transposition.

A simple method to obtain a balanced mCSS code is to extract the middle 5-term segment of a Koszul complex as follows.

\begin{proposition}
Let $S$ be a commutative algebra over $\mathbb{F}_2$ such that $s:=\dim_{\mathbb{F}_2}(S)$ is finite.
Let $t = 2q$ and $\underline{a} = (a_1, \ldots, a_t)$ be a sequence of elements of $S$.
The middle 5-term segment
\begin{align}
K_{q-2} \leftarrow K_{q-1} \leftarrow K_{q} 
\leftarrow K_{q+1} \leftarrow K_{q+2}  
\end{align}
of the Koszul complex $K_\bullet(\underline{a};S)$ is a balanced mCSS code.
\end{proposition}

More generally, unbalanced mCSS codes can be constructed by selecting any 5-term segment from such a Koszul complex or, more broadly, from any chain complex.
\begin{statement}
A useful fact is that for the Koszul complex $K(x_1, \dots, x_t)$, the rank of the free module in homological degree $k$ is
\begin{equation}
\rank K_k(x_1, \dots, x_t) = \binom{t}{k}
\end{equation}
\end{statement}
\subsection{The Multivariate Multicycle (MM) Construction\label{sec:main_construction}}

We now systematically generalize the BB and TT codes by applying the proposition to a special class of commutative algebras~$S$.

\begin{definition}
Let $t \geq 2$, and let $\ell_1, \dots, \ell_D \in \mathbb{Z}_{\geq 2}$ with $D \geq 2$.
The polynomial ring $R = \mathbb{F}_2[x_1, \dots, x_D]$ has a quotient ring 
\editcolor{$S = \mathbb{F}_2[x_1, \dots, x_D]/\langle x_1^{\ell_1} - 1, \dots, x_D^{\ell_D} - 1 \rangle \cong \mathbb{F}_2[\mathbb{Z}_{\ell_1} \times \cdots \times \mathbb{Z}_{\ell_D}]$}
for a finite abelian group.
Hence $S$ is a finite dimensional vector space over~$\mathbb{F}_2$.
Let $\pi: R \rightarrow S$ be the natural projection mapping.
Take polynomials $f_1, \dots, f_t \in R$, and define their images in $S$ as:
\begin{equation}
F_i = \pi(f_i(x_1,\dots,x_D)) \quad (i = 1, \dots, t).
\end{equation}

Form the Koszul complex \editcolor{$K_\bullet([F_i, \cdots, F_t]; S)$} over the quotient ring $S$ as follows

\begin{align}
0 \longleftarrow K_0 \xleftarrow{\partial_1} K_1 \xleftarrow{\partial_2} K_2 
\xleftarrow{\partial_3} \cdots \xleftarrow{\partial_t} K_t \longleftarrow 0
\end{align}
where $K_q = S^{\binom{t}{q}}$. Let $q = \lfloor t/2 \rfloor$.  
The MM code is the CSS code obtained by placing physical qubits on $K_q$ with parity-check matrices
\begin{align}
P_X &= \partial_q^{\!\top} : K_q \longrightarrow K_{q-1} \\
P_Z &= \partial_{q+1}: K_{q+1} \longrightarrow K_q
\end{align}

To construct the MM code, replace each polynomial $F_i$ in the boundary maps with the corresponding circulant matrix representation using the isomorphism 
\cite[Theorem~2.1]{albuquerque2011hopfalgebraicpropertiescirculant}.
\end{definition}

The number of physical qubits is \editcolor{$n = \dim_{\mathbb{F}_2} K_q = \binom{t}{q} \prod_{i=1}^{D} \ell_i$}. If $t \geq 4$, the code has an X-metacheck matrix
\begin{align}
M_X = \partial_{q-1}^{\!\top}: K_{q-1} \longrightarrow K_{q-2}
\end{align}
and if $t \geq 3$, it has a Z-metacheck matrix
\begin{align}
M_Z = \partial_{q+2}: K_{q+2} \longrightarrow K_{q+1}
\end{align}
These satisfy
\begin{align}
M_X P_X = 0, \quad M_Z P_Z = 0
\end{align}
providing the metachecks required for complete single-shot decoding.

For general parameters $t$ and $D$ where $t$ is the number of polynomials and $D$ is the "spatial" dimension, with block size \editcolor{$n = \prod_{i=1}^{D} \ell_i$}, the boundary maps $\partial_q$ is a matrix of size
\editcolor{
\begin{align}
\binom{t}{q} n \; \times \; \binom{t}{q-1} n.
\end{align}}
Thus the parity‑check matrices are the following
\editcolor{
\begin{align}
P_X = \partial_q^{\!\top} \text{\ \ of size\ \ } \binom{t}{q-1} n \times \binom{t}{q} n \\
P_Z = \partial_{q+1} \text{\ \ of size\ \ } \binom{t}{q} n \times \binom{t}{q+1} n. 
\end{align}}
When \editcolor{$t \ge 4$} the corresponding metacheck matrices with row and column sizes are
\editcolor{
\begin{align}
M_X = \partial_{q-1}^{\!\top} \text{\ \ of size\ \ } \binom{t}{q-2} n \times \binom{t}{q-1} n\\
M_Z = \partial_{q+2} \text{\ \ of size\ \ } \binom{t}{q+1} n \times \binom{t}{q+2} n. 
\end{align}}

The CSS orthogonality condition \(P_X P_Z^{\!\top} = 0\) and metacheck conditions \(M_X P_X = 0\), \(M_Z P_Z = 0\) follow from the fact that the Koszul complex over the multivariate polynomial ring satisfies \(\partial_i \partial_{i+1} = 0\) for all \(i\).
\begin{remark}
    The case \editcolor{$D=1$} (a single cyclic group) is equivalent to \editcolor{$D=2$} with $\ell_2=1$ via the isomorphism
$\mathbb{F}_2[x]/\langle x^{\ell_1}-1 \rangle \cong \mathbb{F}_2[x,y]/\langle x^{\ell_1}-1, y-1 \rangle$. Thus, requiring \editcolor{$D \geq 2$} does not entail a loss of generality.
\end{remark}

\begin{example}[\editcolor{MM code (\(t=4, D=4\))}]
A newly discovered \editcolor{\([[648, 60, (9,9)]]\)} MM code: Let $\ell_1=3,\ell_2=3,\ell_3=3,\ell_4=4$ and 
$R = \mathbb{F}_2[w,x,y,z]$ and $S= R/\langle w^3-1,x^3-1,y^3-1,z^4-1\rangle$. Let $\pi: R \rightarrow S$ be the natural projection mapping. Given multivariate polynomials $f(w,x,y,z),g(w,x,y,z), h(w,x,y,z), i(w,x,y,z) \in R$, we define their images in $S$ as $F = \pi(f(w,x,y,z))$, $G = \pi(g(w,x,y,z))$, $H = \pi(h(w,x,y,z))$ and $I = \pi(i(w,x,y,z))$. Consider the polynomials
\editcolor{\begin{align}
&F(w,x,y,z) = (1 + x)(1 + yz) \\
&G(w,x,y,z) = (1 + y)(1 + zw) \\
&H(w,x,y,z) = (1 + z)(1 + wx) \\
&I(w,x,y,z) = (1 + w)(1 + xy)
\end{align}}
The associated Koszul complex \editcolor{$K_\bullet([F,G,H,I]; S)$} has dimensions:
\begin{align}
\dim_{\mathbb{F}_2} C_0 &= \tbinom{4}{0}\prod_{i=1}^{4} \ell_i = 108 \\
\dim_{\mathbb{F}_2} C_1 &= \tbinom{4}{1}\prod_{i=1}^{4} \ell_i = 432 \\
\dim_{\mathbb{F}_2} C_2 &= \tbinom{4}{2}\prod_{i=1}^{4} \ell_i = 648 \\
\dim_{\mathbb{F}_2} C_3 &= \tbinom{4}{3}\prod_{i=1}^{4} \ell_i = 432 \\
\dim_{\mathbb{F}_2} C_4 &= \tbinom{4}{4}\prod_{i=1}^{4} \ell_i = 108
\end{align}
The differentials associated with $R$ complex have following matrix dimensions:
$\partial_1 = M_X^\top = 108 \times 432$, $\partial_2 = P_X^\top = 432 \times 648$, $\partial_3 = P_Z = 432 \times 648$, $\partial_4 = M_Z = 108 \times 432$.

With $q = \lfloor 4/2 \rfloor = 2$, physical qubits are placed in $C_2$, giving:  \editcolor{$n = \dim C_2 = 648$ and $k = 60$ logical qubits, and distance $d = 9$} computed using mixed-integer programming method of~\cite{landahl2011color} and the HiGHS linear programming solver~\cite{huangfu2018parallelizing}. This distance is cross-verified using the connected cluster algorithm from the \texttt{dist‑m4ri} program~\cite{Pryadko-2025-distm4ri}. The parity-check matrices are:
\begin{align}
\quad P_X &= \partial_2^{\!\top}: K_2 \to K_1 \\
\quad P_Z &= \partial_3: K_3 \to K_2 \quad
\end{align}
Metachecks (both exist for $t=4$) for complete single-shot decoding:
\begin{align}
\quad M_X &= \partial_1^{\top}: K_1 \to K_0
\\
\quad M_Z &= \partial_4: K_4 \to K_3
\end{align}
These satisfy $P_X P_Z^{\!\top}=0$, $M_X P_X=0$, and $M_Z P_Z =0$.
\end{example}
\begin{figure*}
    \centering
\includegraphics[width=0.6\linewidth]{subfamiliesofMM.pdf}
    \caption{\editcolor{\textbf{Multivariate multicycle (MM) codes generalize the explicit instances of Abelian multicycle (AMC) codes introduced in \cite{lin2025abelianmulticyclecodessingleshot}}. Important subfamilies of MM codes include the AMC (also referred to as AMC$_2$, as they are constructed from weight-2 group algebra elements), introduced in Ref.~\cite{lin2025abelianmulticyclecodessingleshot}; trivariate tricycle (TT) codes \cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic}; Abelian 2BGA codes \cite{lin2023quantumtwoblockgroupalgebra}; cyclic hypergraph product codes (C2, CxR) \cite{aydin2025cyclic}; bivariate bicycle (BB) codes \cite{Bravyi_2024}; multivariate bicycle (MB) codes \cite{voss2025multivariate}; generalized bicycle (GB) codes; and Haah's cubic codes on an $L \times L \times L$ cubic lattice \cite{panteleev2022asymptoticallygoodquantumlocally}.}}
    \label{fig:conceptual_diagram_of_subfamilies}
\end{figure*}
Here is another example to provide more clarity about code construction process:
\begin{example}
With $S = \mathbb{F}_2[x,y]/\langle x^{21}-1, y^{18}-1\rangle$ and polynomials \editcolor{$F(x,y) = x^3 + y^{10} + y^{17}$, $G(x,y) = x^{19} + x^3 + y^5$ in $S$}, we construct Koszul complex $\mathrm{K}([F,G]; S)$. The boundary maps are:
\begin{align}
\partial_2 = \begin{pmatrix} F & G \end{pmatrix}, \quad
\partial_1 = \begin{pmatrix} G \\ F \end{pmatrix}
\end{align}
with entries in $S$. Each polynomial $p \in \{F,G\}$ is substituted with a corresponding circulant matrix $\mathrm{CircMat(p)}$, which yields the following:
\begin{align}
\widetilde{\partial}_1 = \begin{pmatrix} \mathrm{CircMat(G)} \\ \mathrm{CircMat}(F) \end{pmatrix} \in \mathbb{F}_2^{756 \times 378} \\
\widetilde{\partial}_2 = \begin{pmatrix} \mathrm{CircMat}(F) & \mathrm{CircMat}(G) \end{pmatrix} \in \mathbb{F}_2^{378 \times 756}.
\end{align}
The CSS code matrices are then $P_X = \widetilde{\partial}_2^\top$ and $P_Z = \widetilde{\partial}_1$ and this leads to construction of $[[756, 16, \leq 34]]$ from Table 3 of~\cite{Bravyi_2024}.
\end{example}
\subsection{Subfamilies of MM codes}
\label{sec:subfamilies_of_mm_code}
Now that we have introduced a \editcolor{length-5} complex, we continue the restating of known codes in terms of our MM codes. This is a natural extension of Sec.~\ref{sec:boundary_maps_and_code_constructions}, but we wanted to keep it separate, because only now we have codes that admit a complete set of metachecks. \editcolor{To provide a broader perspective on how our construction fits within the larger family of QEC codes, we refer the reader to Fig.~\ref{fig:conceptual_diagram_of_subfamilies}}.

\begin{remark}[\editcolor{MM code (\(t=4, D=4\))}]
\label{mm:4dtoric}
Let $G = \mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_p \times \mathbb{Z}_r$ with $\ell = m = p = r$, and consider the group algebra $R = \mathbb{F}_2[G] \cong \mathbb{F}_2[w,x,y,z]/\langle w^\ell-1, x^m-1, y^p-1, z^r-1 \rangle$. The MM code defined by the polynomials $A(w,x,y,z) = 1+w$, $B(w,x,y,z) = 1+x$, $C(w,x,y,z) = 1+y$, $D(w,x,y,z) = 1+z$ in this quotient ring is equivalent to the 4D toric code on an $\ell \times m \times p \times r$ lattice.
\end{remark}
\begin{remark}[\editcolor{MM code (\(t=4, D=1\))}]
\label{mm:amc}
The Abelian Multicycle (AMC) codes of Ref.~\cite{lin2025abelianmulticyclecodessingleshot} are formulated in terms of the group algebra $\mathbb{F}_2[G]$ of a finite abelian group $G$. That provides significant generality, as there is great freedom in choosing what $G$.
All of the explicit examples in~\cite{lin2025abelianmulticyclecodessingleshot} use single cyclic groups $\mathbb{Z}_\ell$ and corresponding single-variable polynomials $a_i(x) \in \mathbb{F}_2[x]/(x^\ell-1)$. These explicit examples are a subclass of our MM codes where $G = \mathbb{Z}_\ell = \mathbb{Z}_\ell \times \mathbb{Z}_m \times \mathbb{Z}_p \times \mathbb{Z}_r$ (with $m=p=r=1$) is a direct product of four cyclic groups isomorphic to the quotient ring $\mathbb{F}_2[w,x,y,z]/\langle w^\ell-1, x^m-1, y^p-1, z^r-1\rangle$.
\end{remark}

\begin{remark}
\label{mm:bb}
Let $R = \mathbb{F}_2[G]$ with $G = \mathbb{Z}_{\ell_1} \times \cdots \times \mathbb{Z}_{\ell_t}$. For \editcolor{$t=2, D=2$}, our framework recovers BB codes~\cite{Bravyi_2024}; for \editcolor{$t=3, D=3$}, TT codes~\cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic}; and for $t\geq 4$, new MM codes.
\end{remark}

\begin{remark}[\editcolor{MM code (\(t=2, D=2\))}]
\label{mm:c2cxr}
The symmetric cyclic hypergraph product C2 code~\cite{aydin2025cyclic} with parameters 
$[[2n_c^{2}, 2k_c^2, d_c]]$, and check weight $2w$ is a 
special case of MM codes. It is obtained from two copies of a 
classical $[n_c,k_c,d_c]$ cyclic code with weight-$w$ polynomial $p(x)$ in the 
quotient ring $\mathbb{F}_2[x,y]/\langle x^{n_c}-1, y^{n_c}-1\rangle$. For example, 
the weight-6 $[[450,32,8]]$ C2 code from Table I of Ref.~\cite{aydin2025cyclic} can be obtained from univariate polynomials $p(x)=1+x+x^4$,  $p(y)=1+y+y^4$ in 
$\mathbb{F}_2[x,y]/\langle x^{15}-1,y^{15}-1\rangle$.

Similarly, the repeated cyclic hypergraph product CxR codes, introduced in Ref.~\cite{PRXQuantum.5.040328} are also special cases of MM codes. With $b = d_c$, the CxR code has parameters $[[2n_c d_c, k_c,  d_c]]$ and check weights $\omega=w+2$~\cite{aydin2025cyclic}. It is constructed  using the classical code polynomial $p(x)$ of weight $w$ and the repetition code $1+y$ in $\mathbb{F}_2[x,y]/\langle x^{n_c}-1, y^{d_c}-1\rangle$. For example, the weight-5 $[[240,8,8]]$ CxR code from Table I of Ref.~\cite{aydin2025cyclic} is obtained by using $p(x)=1+x+x^4$ and $p(y) = 1 + y$ in $\mathbb{F}_2[x,y]/\langle x^{15}-1,y^{8}-1\rangle$.
\end{remark}

\begin{remark}[\editcolor{MM code (\(t=2, D=2\))}]
\label{mm:lacross}
The $[[2n^2, 2k^2, d]]$ La-Cross codes~\cite{pecorari2025high} with periodic boundary conditions are a subfamily of MM code with $t=2$ and $G = \mathbb{Z}_n \times \mathbb{Z}_n$. 
For example, given a seed polynomial $h(a)=1+a+a^k$, setting $A = h(x)$ and $B = h(y)$ in the quotient ring $\mathbb{F}_2[x,y]/\langle x^n-1, y^n-1\rangle$ recovers the $[[98,18,4]]$ code with $n=7$ and $k=3$~\cite{pecorari2025high}.
\end{remark}

\begin{remark}[\editcolor{MM code (\(t=2, D=3\))}]
\label{mm:haah-mm}
Haah's cubic code~\cite{haah2011local} on an $L \times L \times L$ lattice corresponds to the case $t=2$, \editcolor{$D=3$} in our MM code framework. Taking the group $G = \mathbb{Z}_L \times \mathbb{Z}_L \times \mathbb{Z}_L$ and polynomials
$A(x, y, z) = 1 + x + y + z$ , $B(x, y, z) = 1 + xy + xz + yz$ in $\mathbb{F}_2[x,y,z]/\langle x^L-1, y^L-1, z^L-1\rangle$ our MM code reproduces Haah's cubic code. For example, with $L=8$ one obtains the weight 8-limited $[[1024, 30, d]]$ code ($13 \le \editcolor {d} \le 32$) studied in Appendix B of Ref.~\cite{panteleev2022asymptoticallygoodquantumlocally}.
\end{remark}

\begin{remark}[\editcolor{MM code (\(t=2, D=2\))}]
\label{mm:gb}
The $[[30, 8, 4]]$ GB code from Ref.~\cite{liang2025generalized} is recovered as special case of MM code with shift patterns $a_{\text{shifts}} = [0, 2, 8]$ and $b_{\text{shifts}} = [0, 1, 4]$ with $\ell = 15$. Moreover, all one-dimensional generalized bicycle codes from Tables~V through~VIII of Ref.~\cite{liang2025generalized} are systematically recoverable within this framework.
\end{remark}
\begin{remark}[\editcolor{MM code (\(t=2, D=2\))}]
\label{mm:2bga}
MM codes contain abelian 2BGA codes as a special case. When the group is a direct product of cyclic groups $G = \mathbb{Z}_m \times \mathbb{Z}_2$, the corresponding MM code construction uses the quotient ring $\mathbb{F}_2[s,x]/\langle s^2-1, x^m-1\rangle$, which is isomorphic to the group algebra $\mathbb{F}_2[\mathbb{Z}_m \times \mathbb{Z}_2]$. For instance, the $[[16, 2, 4 ]]$ code from Table~2 of Ref.~\cite{lin2023quantumtwoblockgroupalgebra} is generated by: $A(s,x) = 1 + x$, $B(s,x) = 1 + x + s + x^2 + s x + s x^3$ using the group algebra of the direct product of cyclic groups $\mathbb{Z}_4 \times \mathbb{Z}_2$.
\end{remark}

\begin{remark}[\editcolor{MM code (\(t=2, D=2\))}]
\label{mm:mb}
The MB~\cite{voss2025multivariate} code with group $G_r = \mathbb{Z}_{l_1} \times \mathbb{Z}_{l_2} \times \cdots \times \mathbb{Z}_{l_r}$ is a special case of the MM code construction. The weight-6 $[[48, 4, 6]]$ TB code from Table 2 of~\cite{voss2025multivariate} is realized as MM code with generating polynomials $A(x,y,z) =  x^3 + y^5$ and $B(x,y,z) = x + y^2 + y^5 + z^5$ with additional variable via $z = xy$ over $S = \mathbb{F}_2[x, y] / \langle x^4 - 1, y^6 - 1 \rangle$. Given this, all TB codes in~\cite{voss2025multivariate} can be realized as MM code by choosing two integers $l$, $m$ and two generating polynomials $A$ and $B$ in $\mathbb{F}_2[x, y] / \langle x^l - 1,\; y^m - 1 \rangle$ where the additional variable $z=xy$ fully defines TB code.
\end{remark}

The MM construction is inherently commutative, built from polynomial quotient rings isomorphic to abelian group algebras. Non-abelian 2BGA codes require different techniques such as the lifted product~\cite{panteleev2022asymptoticallygoodquantumlocally} and balanced product~\cite{Breuckmann_2021}. The former construction employs left and right regular representations of the non-commutative algebra, leveraging the associativity property to ensure the required orthogonality condition for CSS codes. In contemporary literature ~\cite{salescabrera2025codesalgebrasdirectproducts, salescabrera2025dualitiesdihedralgeneralisedquaternion, willenborg2025dihedralquantumcodes}, quantum codes from non-commutative group algebras are constructed by exploiting the Wedderburn–Artin decomposition of the semisimple group algebra into matrix rings over finite fields.

\section{Numerical Simulations\label{sec:numerical_simulations}}
The implementation of our MM code was carried out using the OSCAR computer algebra system~\cite{OSCAR, OSCAR-book} and integrated into the open source library QuantumClifford.jl~\cite{quantumclifford, stefan_krastanov_2025_17932783,kimlee2025quantumsavorywritesymbolicallyrun}. All explicit code instances discovered in this work are made available for direct use in the QuantumClifford.jl toolkit,
% Stefan -- I removed the direct github link -- use archival tools like zenodo to make doi links that are guaranteed to exist even after github is dead
which we hope will be valuable for future work in quantum error correction. Explicit matrix files are also provided in a git repository~\cite{feroz_ahmed_mian_2026_18396531}. Our implementation, also described in the Appendix, leverages OSCAR's comprehensive capabilities for computational group theory and homological algebra. 

\subsubsection{Code-capacity Model}

\begin{figure}[h!]
    \centering
    \includegraphics[width=8.5cm]{tesseract_LER_vs_p.png}
    \caption{\textbf{Tesseract decoder performance across quantum error correcting codes.} Including our new MM codes $[[96,12,4]]$, $[[96,12,8]]$, $[[96,44,4]]$, single-shot decodable 4D topological codes, and 2D topological codes. The legend includes the encoding rate \editcolor{$r = k/n$}. Our codes show high rate and high distance and are being single-shot decodable. That includes options with particularly high rates (close to half), at the cost of a lower pseudo-threshold.} \label{fig:tesseract_ler_vs_p}
\end{figure}

We evaluated several MM codes under code-capacity depolarizing noise using the Tesseract search-based decoder~\cite{beni2025tesseractsearchbaseddecoderquantum}. For each code, we constructed a Stim~\cite{gidney2021stim} circuit implementing the code-capacity noise model and compiled it to a detector error model~\cite{Derks_2025} (DEM) file. In the code-capacity noise model, each data qubit independently experiences a Pauli $X$ or $Z$ error with probability $p$ during a single error-correction step, while all stabilizer measurements are assumed to be perfect. This simplified model isolates the effect of qubit errors on the logical failure rate without introducing measurement noise. The decoder was configured with beam search parameters---a priority queue limit of 10,000 and a detector beam width of 5---and used 10 detector orderings generated from the DEM to optimize decoding performance.


Our MM codes of length $n = 96$—specifically $[[96,12,4]]$, $[[96,12,8]]$, and the higher-rate $[[96,44,4]]$—were compared against topological codes with similar parameters: a $[[96,6,4]]$ 4D toric code and a $[[33,1,4]]$ 4D surface code (both of which are rare examples of existing codes admitting metachecks), as well as a few 2D codes. The results are shown in Fig.~\ref{fig:tesseract_ler_vs_p}. Our MM codes demonstrate high rates, high distances, and good thresholds. Compared to the single-shot-decodable 4D $[[96,6,4]]$ toric code with same number of physical qubits we have higher rates and distances. Remarkably, the $[[96,44,4]]$ MM code also provides a very high rate at the same high distance (only compromising on a lower pseudo-threshold). Compared to the $[[33,1,4]]$ single-shot-decodable 4D surface code, $[[96,12,8]]$ MM code notably achieves a factor of four increase in rate and factor of two increase in distance.
\subsubsection{Phenomenological noise model}
\label{subsec:phenomenologicalnoisemodel}
\editcolor{We employ the phenomenological noise model extending the code-capacity model across $T = \{ 1, \; 2d\}$ rounds of syndrome extraction, where $d = \min\{d_X, d_Z\}$ is the code distance. This model is intentionally idealized to isolate the combined effect of data qubit errors and measurement errors across multiple rounds, without modeling gate-level operations explicitly; a full circuit-level analysis is presented in next section}.

% \editcolor{The syndrome extraction circuits follow the coloration circuit 
% construction of Ref.~\cite[see Algorithm 1]{tremblay2022constant}, which uses a single ancilla qubit per stabilizer generator with no flag qubits. CNOT gates are scheduled according to a minimum edge coloring of the Tanner graph: all CNOTs assigned the same color act on disjoint qubit pairs and are therefore applied simultaneously in a single time step. For $X$ stabilizers, each ancilla is prepared in  $|{+}\rangle$ and measured in the $X$ basis at the end of the round; for $Z$ stabilizers, each ancilla is prepared in $|{0}\rangle$ and measured in the $Z$ basis. The total depth of the resulting circuit is $\deg(\mathcal{T}_X) + 2$ for $X$ stabilizers and $\deg(\mathcal{T}_Z) + 2$ for $Z$ stabilizers, where $\deg(\mathcal{T}_X)$ and $\deg(\mathcal{T}_Z)$ denote the maximum degree of the $X$ and $Z$ Tanner graphs respectively  ~\cite[see proposition 2]{tremblay2022constant}}.

\editcolor{Under the phenomenological noise model, data qubits are subjected to independent single-qubit depolarizing errors with probability $p$ at the start of each round, and each stabilizer measurement outcome is independently flipped with probability $p$.
%Gate, reset, and idling errors are set to zero. The final round of syndrome extraction is noiseless, so that logical errors can be determined exactly from the final data qubit measurements.
For $Z$-basis memory experiments, logical qubits are initialized in $\ket{0}$ and measured in the $Z$ basis at the end of the circuit (same, but in the $X$ basis, for $X$-basis memory experiments).
Memory circuits were generated using the \texttt{make\_css\_code\_memory\_circuit} method from the LDPC package~\cite{roffe_decoding_2020} and sampled using Sinter~\cite{gidney2021stim} with the \texttt{TesseractSinterDecoder}.
%Logical error rates per round were estimated via Monte Carlo sampling with a minimum of $10^5$ shots per physical error rate point.
Fig.~\ref{fig:phenomenological_model_xbasis} and Fig.~\ref{fig:phenomenological_model_zbasis} present the results, showing very competitive behavior for our MM code instance.
}

\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{PhenomenologicalNoiseModelXBasis.png}
    \caption{\editcolor{\textbf{Performance comparison of near-$n$ codes under phenomenological noise in the $X$-basis memory experiment with the \texttt{TesseractSinterDecoder}.} Four codes are compared: the symmetric MM $[[144,12,(12, 12)]]$ code with  weight-9 stabilizers checks; the BB $[[144,12,(12,12)]]$ code with weight-6 stabilizers; the AMC $[[180,6,(12,12)]]$ code with weight-6 stabilizers; and the asymmetric TT $[[135,12,(14,6)]]$ code with weight-9 $X$-type and weight-6 $Z$-type stabilizers. In the $X$-basis memory experiment, the logical $X$ observable is tracked and the circuit is vulnerable to $Z$ errors. The TT code has $d_Z = 6$ in this experiment, which is considerably smaller than the $d_Z = 12$ of the MM and BB codes, placing it at a disadvantage at this block length.}}\label{fig:phenomenological_model_xbasis}
\end{figure}

%\editcolor{Fig.~\ref{fig:phenomenological_model_xbasis} illustrates that the weight-9 limited $[[144,12,12]]$ Multivariate Multicycle (MM) code exhibits consistent performance in both $X$- and $Z$-basis memory experiments under the phenomenological noise model. The observed pseudo-thresholds are competitive with those of weight-6 AMC and BB codes, despite the slightly row higher weight of stabilizer generators. Notably, the MM code significantly outperforms the $[[135,12,(14,6)]]$ TT codes of comparable block length in the X-basis memory experiment. In fact, as shown in the Appendix~\ref{section:more_phenomenological_noise_experiments}, all MM codes we considered outperform TT codes in the X-memory experiment}.

%\editcolor{Fig.~\ref{fig:phenomenological_model_zbasis} illustrates the TT code’s asymmetry between $X$- and $Z$-stabilizers leads to improved performance in the $Z$-memory experiment relative to $X$-memory. For a broader comparison of performance trade-offs between TT and MM codes at similar block lengths, see the Appendix}.

\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{PhenomenologicalNoiseModelZBasis.png}
    \editcolor{\caption{\textbf{Performance comparison of near-$n$ codes under phenomenological noise in the $Z$-basis memory experiment with the \texttt{TesseractSinterDecoder}.} The same codes as in Fig.~\ref{fig:phenomenological_model_xbasis}. The TT code has $d_X = 14$ in this experiment, which is larger than the $d_X = 12$ of the MM and BB codes, placing it at an advantage at this block length.}\label{fig:phenomenological_model_zbasis}}
\end{figure}

%\editcolor{Of note, the AMC and MM codes are \emph{symmetric} in the sense that their $X$ and $Z$ distances are equal, $d_X = d_Z = d_{min}$, so the advertised minimum distance provides equal protection in both bases. By contrast, the TT codes are, \emph{asymmetric}: their $X$ and $Z$ distances inherently differ, so the effective protection in a given basis memory experiment is governed by the smaller of the two distances rather than the nominal code distance $d_{min}$. As a consequence, at the relatively same block length $N$, a symmetric code such as weight-9 MM $[[144,12,(12,12)]]$ outperform an asymmetric code such as TT $[[135,12,(14,6)]]$ in a $X$-basis memory experiment simply because the TT code's $Z$ distance is smaller than its $X$ distance and vice versa. For the MM, BB, and AMC code families, we considered explicit instances with true minimum code distance of 12 to ensure a fair comparison.}

%\editcolor{In general, the  asymmetry in TT code X and Z-bases makes a fixed-$N$ comparison between symmetric codes ($d_X = d_Z$) and asymmetric TT codes inherently basis-dependent: the TT code is considerably stronger in one basis and weaker in the other, so neither basis alone gives a complete picture of its performance. By contrast, MM, AMC, and BB codes satisfy $d_X = d_Z$ by construction, providing equal protection in both bases and making their performance in either memory experiment directly representative of their overall fault-tolerance capability}.

\subsubsection{Circuit-level noise model}

\editcolor{To evaluate the impact of realistic gate-level noise on code performance, and in particular, the effect of potentially larger check weights, we simulate a full circuit-level noise model. We adopt the standard single-ancilla circuit construction of Ref.~\cite{tremblay2022constant}, as implemented in the \texttt{LDPC} package~\cite{roffe_decoding_2020}. In this approach, each stabilizer generator is associated with one ancilla qubit, and CNOT gates are scheduled via a minimum edge coloring of the Tanner graph. This ensures that gates of the same color act on disjoint qubits and can therefore be executed simultaneously within a single time step.}

\editcolor{Under the circuit-level noise model, every operation in the circuit is subject to depolarizing or flip errors at a uniform physical error rate $p$. Specifically: each CNOT gate is followed by a two-qubit depolarizing error of strength $p$; each data qubit undergoes a single-qubit depolarizing error of strength $p$ at the start of every round; each measurement outcome is flipped with probability $p$; each qubit reset is followed by a state-flip error of strength $p$ ($X$ error after $\ket{0}$ resets, $Z$ error after $\ket{+}$ resets); and each qubit idling during a CNOT layer suffers a single-qubit depolarizing error of strength $p$. Memory-experiment circuits were generated in Stim \cite{gidney2021stim} and decoded using the \texttt{TesseractSinterDecoder} via Sinter, configured with a detector beam width of $8$, beam climbing enabled, no-revisit and merge-errors flags active, a priority queue limit of $10{,}000$, and $10$ detector orderings generated via the \texttt{DetIndex} method \cite{beni2025tesseractsearchbaseddecoderquantum}}.
\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{CircuitLevelNoiseX.png}
\caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $X$-basis memory experiment decoded with \texttt{TesseractSinterDecoder}.} Four codes are compared: MM $[[144,12,(12,12)]]$ with weight-9 stabilizers, BB $[[144,12,(12,12)]]$ with weight-6 stabilizers, AMC $[[180,6,(12,12)]]$ with weight-6 stabilizers, and the asymmetric TT $[[135,12,(14,6)]]$ with weight-9 $X$-type and weight-6 $Z$-type stabilizers.
\label{fig:circuitlevel_model_xbasis}}}
\end{figure}
\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{CircuitLevelNoiseZ.png}
\caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $Z$-basis memory experiment decoded with \texttt{TesseractSinterDecoder}.} Four codes are compared: MM $[[144,12,(12,12)]]$ with weight-9 stabilizers, BB $[[144,12,(12,12)]]$ with weight-6 stabilizers, AMC $[[180,6,(12,12)]]$ with weight-6 stabilizers, and the asymmetric TT $[[135,12,(14,6)]]$ with weight-9 $X$-type and weight-6 $Z$-type stabilizers. Each code uses the single-ancilla coloration syndrome extraction circuit from the \texttt{LDPC} package. 
\label{fig:circuitlevel_model_zbasis}}}
\end{figure}

\editcolor{Fig.~\ref{fig:circuitlevel_model_xbasis} illustrates that our MM codes perform better than TT codes in the standard circuit-level noise model in the X-basis memory experiment. The TT $[[135,12,(14,6)]]$ code's small $d_Z = 6$ places it at a significant disadvantage relative to the other codes at comparable block length $N \approx 144$.
%, as its $X$-basis pseudothreshold is significantly lower than that of the symmetric MM and BB codes.
Fig.~\ref{fig:circuitlevel_model_zbasis} illustrates the $Z$-basis memory
%experiment under full circuit-level noise, where $X$ errors dominate and the effective protection is governed by $d_X$. Consistent with this, codes with larger $d_X$ achieve higher pseudothresholds.
The weight-9-limited $[[144,12,(12,12)]]$ MM code performs robustly, maintaining decent performance across both memory experiments due to its symmetric distances. In contrast, the TT $[[135,12,(14,6)]]$ code benefits in this setting from its larger $d_X = 14$, achieving the highest pseudothreshold.
%a reversal of its behavior in the $X$-basis experiment, where its smaller $d_Z = 6$ led to the weakest performance.
%This highlights the basis-dependent asymmetry of TT codes: enhanced protection in one basis comes at the cost of reduced protection in the other. By comparison, symmetric constructions such as MM and BB codes, with $d_X = d_Z = 12$, exhibit consistent behavior across bases. 
We further perform circuit-level noise experiments using state-of-the-art Left-Right circuits (LRC) \cite{strikis2026high} in Appendix~\ref{section:lrccircuitlevelnoisemodel}, where we demonstrate that LRC syndrome extraction yields pseudothreshold improvements for our weight-6  $[[144,6,(8,8)]]$ and $[[216,6,(12,12)]]$ MM codes.}
\subsubsection{Overlapping Window Decoding}
\label{subsection:owd_decoding}

\editcolor{\emph{Overlapping window decoding} (OWD) processes syndrome streams incrementally on bounded temporal windows, avoiding the need to store the full syndrome history~\cite{terhal2015quantum, skoric2023parallel}. Remarkably, Dennis et al.~\cite{Dennis_2002} recognized this principle as early as 2001:}

\begin{quote}
\textit{\editcolor{In our overlapping recovery scheme, we take action to remove only these long-lived defects, leaving those of more recent vintage to be dealt with in the next recovery step.}}
\end{quote}

\editcolor{Here, a defect refers to a syndrome change; persistent defects are more trustworthy than transient fluctuations from syndrome measurement noise.}

\editcolor{In practice, OWD examines a sliding window of $w$ consecutive syndrome rounds and proposes a correction using BP+OSD (belief propagation with ordered statistics decoding). Only the first $c \leq w$ rounds\textemdash the \emph{commit} region\textemdash are finalized, while the remaining $w - c$ rounds form a \emph{buffer} that is re-decoded in the subsequent window~\cite{skoric2023parallel}. The window advances by $c$ rounds, and when $w$ and $c$ remain constant as code distance increases, OWD achieves \emph{$(w,c)$-single-shot} decoding~\cite{scruby2026high}.}

% \subsubsection{Plateau experiments}
% \label{sec:plateauexperiments}

\editcolor{To assess the few-shot decoding performance of our codes under OWD, we perform plateau
experiments~\cite{lin2025abelianmulticyclecodessingleshot,
lin2025singleshottwoshotdecodinggeneralized,
jacob2025singleshotdecodingfaulttolerantgates}, in which the per-round logical error rate $P_L$ is measured as a function of $w$ at fixed $c$, total syndrome measurement rounds $T$, and physical error probability $p$. We conduct these experiments under both phenomenological and circuit-level noise models.}
\begin{figure*}
    \centering
    \includegraphics[width=1\linewidth]{plateau_xz_T15.png}
     \caption{\editcolor{\textbf{Plateau experiment in the $X$ and $Z$ memory experiments, phenomenological noise model.} Per-round logical error rate $P_L$ as a function of OWD window size $w$ under phenomenological noise at $p = 10^{-2}$ with $T=15$ syndrome extraction rounds and commit size $c=1$. The $X$ memory experiment is shown in (a) and $Z$ in (b). We test the MM codes [[144,12,(12,12)]] and [[486,24,(12,12)]]; the BB code [[144,12,12]]; the TT code [[135,12,(14,6)]]; and the AMC code [[180,6,(12,12)]]. The total rounds $T=15$ ensures sufficient syndrome history for the OWD decoder across window sizes up to $w=13$}.}
    \label{fig:plateau_XZ_T15phenom}
\end{figure*}

\paragraph{\editcolor{Phenomenological noise model.}}
\editcolor{Fig.~\ref{fig:plateau_XZ_T15phenom} shows the logical error rate per round as a function of OWD window size $w$ under phenomenological noise at $p = 0.01$ with $T=15$ syndrome extraction rounds. The MM $[[144,12,12]]$ code reaches plateau at $w = 2$ with $P_L \approx 10^{-4}$, while the MM $[[486,24,12]]$ code reaches plateau at $w = 3$ with lower $P_L \sim 5 \times 10^{-5}$. The plateau onset indicates that syndrome information within these windows is sufficient for accurate decoding. The BB $[[144,12,12]]$ code also plateaus at $w=2$ but with higher error rate, confirming that increased redundancy improves performance. The TT $[[135,12,6]]$ code shows continued improvement even at $w=13$, requiring longer syndrome history. The OWD decoder with commit size $c=1$, BP+OSD decoding (minimum sum, 1000 iterations, OSD order 0), was applied under phenomenological noise with equal error probability $p$ on data qubits errors and measurements errors.}

\paragraph{\editcolor{Circuit-level noise model.}}
\editcolor{Under full circuit-level noise with $T=10$ syndrome extraction rounds at $p = 10^{-3}$, Fig.~\ref{fig:plateau_XZ_T10circ} shows the MM $[[486,24,12]]$ code achieves plateau $P_L \approx 10^{-5}$, approximately one order of magnitude lower than both the BB $[[144,12,12]]$ and TT $[[135,12,6]]$ codes ($P_L \approx 10^{-4}$). The MM $[[486,24,12]]$ code demonstrates two-shot decoding at distance $d=12$.}

\begin{figure*}
    \centering
    \includegraphics[width=1.0\linewidth]{circ_plateau_xz_T10.png}
     \caption{\editcolor{\textbf{Plateau experiment in the $X$ and $Z$ memory experiments, circuit noise model} ($T = 10$, $p = 10^{-3}$, commit $c = 1$).}}
    \label{fig:plateau_XZ_T10circ}
\end{figure*}
% \editcolor{Fig.~\ref{fig:PlateauZCircuit-LevelNoise.png} shows the corresponding 
% $X$-basis plateau experiment under the same conditions. TT $[[72,6,(12,6)]]$ code fails to correct errors entirely, plateauing at $P_L \approx 10^{-1}$, well above the physical error probability $p = 10^{-3}$. This appears to be due to the TT code's small $d_Z = 6$: in the $X$-basis experiment $Z$ errors dominate, and the limited $Z$-distance provides insufficient protection under circuit-level noise at this error rate, regardless of the window size $w$.} 
% \begin{figure}
%     \centering
%     \includegraphics[width=1\linewidth]{PlateauXCircuit-LevelNoise.png}
%      \caption{\textbf{Plateau experiment for single-shot decoding in the $X$-basis memory experiment under circuit-level noise ($T = 10$ rounds, $p = 10^{-3}$, $50{,}000$ shots).} Logical error rate per round $P_L$ versus OWD window size $w$ for 
%     $[[144,12,(12,12)]]$ BB (orange), $[[486,24,(12,12)]]$ MM (red), and $[[72,6,(12,6)]]$ TT (purple).}
%     \label{fig:PlateauXCircuit-LevelNoise.png}
% \end{figure}

% \begin{figure}
%     \centering
%     \includegraphics[width=1\linewidth]{PlateauZCircuit-LevelNoise.png}
%     \caption{\textbf{Plateau experiment for single-shot decoding in the $Z$-basis memory experiment under circuit-level noise ($T = 10$ rounds, $p = 10^{-3}$, $50{,}000$ shots).} Logical error rate per round $P_L$ versus OWD window size $w$ for the same codes as Fig.~\ref{fig:PlateauXCircuit-LevelNoise.png}. TT successfully suppresses errors in this basis experiment due to its larger $d_X = 12$, plateauing near $P_L \approx 10^{-4}$, while MM again achieves the lowest plateau.}
%     \label{fig:PlateauZCircuit-LevelNoise.png}
% \end{figure}

\subsubsection{Confinement Profiles}
To evaluate robustness against measurement noise, we computed \emph{confinement profiles}~\cite{Quintavalle_2021} in Table~\ref{tab:confinement}—the minimum non-zero syndrome weights associated with irreducible errors of increasing weight—for each code using the connected cluster algorithm from \texttt{dist-m4ri} program~\cite{Pryadko-2025-distm4ri}. Confinement quantifies how the weight of the measured syndrome grows with the
reduced weight of a physical error
\cite[see Definition~3.44]{sriram2025diffusioncodesselfcorrectionsmallerset}.
In a quantum CSS code this relation is formalized via
boundary and coboundary confinement
\cite[see Definition~3.45]{sriram2025diffusioncodesselfcorrectionsmallerset}, which respectively state that $Z$-type ($X$-type) errors of small reduced weight must produce syndromes under the $X$-type ($Z$-type) checks whose weight grows at least linearly with the reduced error weight.
Higher confinement therefore guarantees that irreducible low-weight physical errors
necessarily flip many syndrome bits, ensuring that their syndromes remain
distinguishable even in the presence of measurement noise.  

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{1pt}
\begin{table*}[ht]
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\rowcolor{A}Code Instance & $[[n, k, (d_X, d_Z)]]$ & $r = \frac{k}{n}$ & $\frac{kd^2}{n}$ & $d_S$ & $\tilde{w_{X}}$ & $\tilde{w_{Z}}$ & $\overline{w_{X}}$& $\overline{w_{Z}}$ & Z-confinement & X-confinement\\
\hline
\rowcolor{B}2D Planar~\cite{liang2025planar} & $[[72, 8, (4, 4)]]$ & 0.11 & 1.78 & 1 & 5 & 5 & (5 & 5) & 1,1 & 1,1 \\
\hline
\rowcolor{B}2D Planar~\cite{liang2025planar} & $[[98, 8, (5, 5)]]$ & 0.08 & 2.04 & 1 & 6 & 6 & 6 & 6 & 1,1 & 1,1 \\
\hline
\rowcolor{B}2D Planar~\cite{liang2025planar} & $[[128, 8, (6, 6)]]$ & 0.06 & 2.25 & 1 & 6 & 6 & 6 & 6 & 1,1 & 1,1 \\
\hline
\rowcolor{B}2D Planar~\cite{liang2025planar} & $[[288, 8, (12, 12)]]$ & 0.02 & 4.00 & 1 & 6 & 6 & 6 & 6 &  1,2 & 1,1 \\
\hline
\rowcolor{C}2D Tile~\cite{steffan2025tile, eberhardt2024pruning, liang2025planar} & $[[288, 8, (14, 14)]]$ & 0.02 & 5.44 & 1 & 8 & 8 & 8 & 8 & 1,1 & 1,1,1\\
\hline
\rowcolor{C}2D Tile~\cite{steffan2025tile, eberhardt2024pruning, liang2025planar} & $[[288, 18, (13, 13)]]$ & 0.06 & 10.56 & 1 & 8 & 8 & 8 & 8 & 1,1 & 1,1 \\
\hline
\rowcolor{C}2D Tile~\cite{steffan2025tile, eberhardt2024pruning, liang2025planar} & $[[512, 18, (19, 19)]]$ & 0.03 & 12.69 & 1 & 8 & 8 & 8 & 8 & 1,1 & 1,1\\
\hline
\rowcolor{D}4D Surface~\cite{Berthusen_2024} & $[[33, 1, (4, 4)]]$ & 0.03 & 0.48 & 2 & 4 & 4 & 5 & 5 & 2,2,2,3 & 2,2,2,2 \\
\hline
\rowcolor{E}4D Toric~\cite{Dennis_2002} & $[[96, 6, (4, 4)]]$ & 0.06 & 1.00 & 4 & 6 & 6 & 6 & 6 & 4,4,4 & 4,4,4\\
\hline
\rowcolor{F}QuantumTanner~\cite{radebold2025explicit} & $[[36, 8, (3, 3)]]$ & 0.22 & 2.00 & 2 & 6 & 6 & 6 & 6 & 2,2,2, &  2,2,2, \\
\hline
\rowcolor{F}QuantumTanner~\cite{radebold2025explicit} & $[[54, 11, (4, 4)]]$ & 0.20 & 3.25 & 2 & 6 & 6 & 6 & 6 & 2,2,2,2, & 2,2,2,2, \\
\hline
\rowcolor{F}QuantumTanner~\cite{radebold2025explicit} & $[[72, 14, (4, 4)]]$ & 0.19 & 3.11 & 2 & 6 & 6 & 6 & 6 & 2,2,2,2, & 2,2,2,2, \\
\hline
\rowcolor{G}QuantumTanner~\cite{leverrier2025smallquantumtannercodes} & $[[84, 2, (10, 10)]]$ &0.02 & 2.38 & 1 & 6 & 6 & 6 & 6 & 2,2,2,2,2,1 & 2,2,2,2,2,1\\
\rowcolor{G}\hline
QuantumTanner~\cite{leverrier2025smallquantumtannercodes} & $[[144, 8, (12, 12)]]$ & 0.05 & 8.00 & 2 & 9 & 9 & 9 & 9 & 2,3,3,2,2,2 & 2,3,3,2,2,2 \\
\hline
\rowcolor{G}QuantumTanner~\cite{leverrier2025smallquantumtannercodes} & $[[144, 12, (11, 11)]]$ & 0.08 & 10.08 & 2 & 9 & 9 & 9 & 9 & 2,3,3,2,3,2 & 2,3,3,2,3,2 \\
\hline
\rowcolor{H}QuantumTanner~\cite{wang2026check} & $[[96, 24, (4, 4)]]$ & 0.25& 4.00 & 2 & 6 & 8 & 6 & 8 & 2,2,2,2 & 2,2,2,2 \\
\hline
\rowcolor{H}QuantumTanner~\cite{wang2026check} & $[[96, 30, (4, 4)]]$ & 0.31 & 5.00 & 2 & 8 & 8 & 8 & 8 & 2,2,2,2 & 2,2,2,2 \\
\hline
\rowcolor{H}QuantumTanner~\cite{wang2026check} & $[[128, 40, (4, 4)]]$ & 0.31 & 5.00 & 2 & 8 & 8 & 8 & 8 & 2,2,2,4 & 2,2,2,4 \\
\hline
\rowcolor{K}LiftedQuantumTanner~\cite{guemard2025lifts, guemard2025moderate} & $[[96, 2, (12, 12)]]$ & 0.02 & 3.00 & 2 & 6 & 6 & 8 & 8 & 2,2,2,2 & 2,2,2,2 \\
\hline
\rowcolor{K}LiftedQuantumTanner~\cite{guemard2025lifts, guemard2025moderate} & $[[160, 2, (16, 16)]]$ & 0.01 & 3.20 & 2 & 6 & 6 & 8 & 8 & 2,2,2,2, & 2,2,2,2, \\
\hline
\rowcolor{K}LiftedQuantumTanner~\cite{guemard2025lifts, guemard2025moderate} & $[[324, 4, (18, 18)]]$ & 0.01 & 4.00 & 2 & 5 & 5 & 5 & 5 & 2,2,2,2,2, & 2,2,2,2,3, \\
\hline
\rowcolor{L}Vanilla BivariateBicycle~\cite{Bravyi_2024} & $[[90, 8, (10, 10)]]$ & 0.08 & 8.89 & 2 & 6 & 6 & 6 & 6 & 3,4,3,2,3,2 & 3,4,3,2,3,2 \\
\hline
\rowcolor{L}Vanilla BivariateBicycle~\cite{Bravyi_2024} & $[[108, 8, (10, 10)]]$ & 0.07 & 7.40 & 2 & 6 & 6 & 6 & 6 & 3,4,3,2,3,2 & 3,4,3,2,3,2 \\
\hline
\rowcolor{L}Vanilla BivariateBicycle~\cite{Bravyi_2024} & $[[144, 12, (12, 12)]]$ & 0.08 & 12.00 & 3 & 6 & 6 & 6 & 6 & 3,4,3,4,3,4 & 3,4,3,4,3,4\\
\hline
\rowcolor{M}Self-Dual BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[56, 6, (8,8)]]$ & 0.10 & 6.85 & 2 & 8 & 8 & 8 & 8 & 4,4,4,4,4,2 & 4,4,4,4,4,2 \\
\hline
\rowcolor{M}Self-Dual BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[64, 8, (8,8)]]$ & 0.12 & 8.00 & 2 & 8 & 8 & 8 & 8 & 4,2,4,2,4,2 & 4,2,4,2,4,2\\
\hline
\rowcolor{M}Self-Dual BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[96, 12, (8,8)]]$ & 0.12 & 8.00 & 2 & 8 & 8 & 8 & 8 & 4,4,4,4,4,2 & 4,4,4,4,4,2 \\
\hline
\rowcolor{M}Self-Dual BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[98, 14, (6, 6)]]$ & 0.14 & 5.14 & 2 & 8 & 8 & 8 & 8 & 4,4,2,4,2,4 & 4,4,2,4,2,4 \\
\hline
\rowcolor{O}4-Cover BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[96, 10, (12, 12)]]$ & 0.10 & 15.00 & 4 & 8 & 8 & 8 & 8 & 4,6,6,4,4,6 & 4,6,6,4,4,6 \\
\hline
\rowcolor{O}3-Cover BivariateBicycle~\cite{liang2025selfdualbivariatebicyclecodes} & $[[96, 12, (10, 10)]]$ & 0.12 & 12.50 & 4 & 8 & 8 & 8 & 8 & 4,6,6,4,4,6 & 4,6,6,4,4,6\\
\hline
\rowcolor{O}7-Cover BivariateBicycle~\cite{symons2025sequences} & $[[98, 6, (12, 12)]]$ & 0.06 & 8.81 & 4 & 6 & 6 & 6 & 6 & 4,6,6,4,4,6 & 4,6,6,4,4,6 \\
\hline
\rowcolor{O}2-Cover BivariateBicycle~\cite{symons2025sequences} & $[[144, 14, (14,14)]]$ & 0.09 & 19.05 & 4 & 8 & 8 & 8 & 8 & 4,6,6,4,4,6 & 4,6,6,4,4,6 \\
\hline
\rowcolor{P}GeneralizedToric ~\cite{liang2025generalized} & $[[90, 8, (10, 10)]]$ & 0.08 & 8.89 & 2 & 6 & 6 & 6 & 6 & 3,4,3,2,3,2 & 3,4,3,2,3,4 \\
\hline
\rowcolor{P}GeneralizedToric ~\cite{liang2025generalized} & $[[96, 4, (12, 12)]]$ & 0.04 & 6.00 & 3 & 6 & 6 & 6 & 6 & 3,4,3,4,3,4 & 3,4,3,4,3,4\\
\hline
\rowcolor{P}GeneralizedToric ~\cite{liang2025generalized} & $[[98, 6, (12, 12)]]$ & 0.06 & 8.81 & 2 & 6 & 6 & 6 & 6 & 3,4,3,4,3,4 & 3,4,3,4,3,2 \\
\hline
\rowcolor{Q}ZSZ~\cite{Guo_2026} & $[[80,2,(8,8)]]$ & 0.02 & 1.60 & 1 & 6 & 6 & 6 & 6 & 3,2,3,2,1,4 & 3,2,3,2,1,2 \\
\hline
\rowcolor{Q}ZSZ~\cite{Guo_2026} & $[[108,2,(12,12)]]$ & 0.01 & 2.67 & 2 & 6 & 6 & 6 & 6 & 3,4,3,2,3,2 & 3,4,3,2,3,2 \\
\hline
\rowcolor{Q}ZSZ~\cite{Guo_2026} & $[[144,12,(8,8)]]$ & 0.08 & 5.33 & 3 & 6 & 6 & 6 & 6 & 3,4,3,4,3,4 & 3,4,5,4,3,4 \\
\hline
\rowcolor{Q}ZSZ~\cite{Guo_2026} & $[[288,12,(8,8)]]$ & 0.04 & 2.67 & 3 & 6 & 6 & 6 & 6 & 3,4,5,4,3,4 & 3,4,5,4,3,4 \\ 
\hline
\rowcolor{R}TrivariateTricycle~\cite{jacob2025singleshotdecodingfaulttolerantgates} & $[[72, 6, (12, 6)]]$ & 0.08 & 3.00 & 2 & 9 & 6 & 9 & 6 & 3,2,3,2,3,2 &
6,8,10,10,12,10 \\ 
\hline
\rowcolor{R}TrivariateTricycle~\cite{jacob2025singleshotdecodingfaulttolerantgates} & $[[81, 6, (12, 6)]]$ & 0.07 & 2.67 & 2 & 9 & 6 & 9 & 6 & 3,2,3,2, & 6,8,6,8,8,10 \\
\hline
\rowcolor{R}TrivariateTricycle~\cite{jacob2025singleshotdecodingfaulttolerantgates} & $[[90, 3, (15, 5)]]$ & 0.03 & 0.83 & 2 & 6 & 4 & 6 & 4 & 2,2,2,2, & 4,6,6,6,4,6\\
\hline
\rowcolor{S}AbelianMulticycle~\cite{lin2025abelianmulticyclecodessingleshot} & $[[84, 6, (7, 7)]]$ &  0.07 &3.50 & 4 & 6 & 6 & 6 & 6 & 4,6,8,8,4,6, & 4,6,8,8,4,6 \\
\hline
\rowcolor{S}AbelianMulticycle~\cite{lin2025abelianmulticyclecodessingleshot} & $[[96, 6, (8, 8)]]$ &  0.06 & 4.00 & 4 & 6 & 6 & 6 & 6 & 4,6,6,6,4,4, & 4,6,6,6,4,4\\
\hline
\rowcolor{S}AbelianMulticycle~\cite{lin2025abelianmulticyclecodessingleshot} & $[[108, 6, (9, 9)]]$ & 0.05 & 4.50 & 4 & 6 & 6 & 6 & 6 & 4,6,8,8,4,6 & 4,6,8,8,4,6\\
\hline
\rowcolor{T}MultivariateMulticycle & $[[96, 12, (4, 4)]]$ & 0.12 & 2.00 & 8 & 6 & 6 & 6 & 6 & 8,8,8,8 & 8,8,8,8\\
\hline
\rowcolor{T}MultivariateMulticycle & $[[96, 12, (8, 8)]]$ & 0.12 & 8.00 & 8 & 16 & 16 & 16 & 16 & 8,8,8,8,8,8 &  
8,8,8,8,8,8\\
\hline
\rowcolor{T}MultivariateMulticycle & $[[96, 44, (4, 4)]]$ & 0.45 & 7.33 & 8 & 12 & 12 & 12 & 12 & 8,8,8 & 8,8,8 \\
\hline
\rowcolor{T}MultivariateMulticycle & $[[144, 40, (4, 4)]]$ & 0.27 & 4.44 & 8 & 12 & 12 & 12 & 12 & 8,8,8,12 & 8,8,8,12 \\
\hline
\rowcolor{T}MultivariateMulticycle & $[[216, 12, (12, 12)]]$ & 0.05 & 8.00 & 4 & 9 & 9 & 10 & 10 & 4,6,8,8,10,8 & 4,6,8,8,10,8 \\
\hline
\rowcolor{T}MultivariateMulticycle & $[[486, 24, (12, 12)]]$ & 0.04 & 7.11 & 6 & 9 & 9 & 9 & 9 & 6,10,12,14,18,18 & 6,10,12,14,18,18 \\
\hline
\rowcolor{T}MultivariateMulticycle & $[[486, 66, (9, 9)]]$ & 0.13 & 11.00 & 8 & 12 & 12 & 12 & 12 & 8,12,12,16,16,12 & 8,12,12,16,16,12 \\
\hline
\rowcolor{T}MultivariateMulticycle & $[[576, 64, (6, 6)]]$ & 0.11 & 4.00 & 8 & 12 & 12 & 12 & 12 & 8,12,12,12,8,12 & 8,12,12,12,8,12 \\
\hline
\end{tabular}
\centering
\caption{\textbf{Code parameters \editcolor{$[[n,k,d]]$} and confinement profiles.} \editcolor{$n$} is the number of physical qubits, \editcolor{$k$} the number of logical qubits, and the true code distance \editcolor{$d = \min(d_X, d_Z)$} is the minimum weight of a non-trivial logical Pauli operator. The code distances were cross-verified using the connected cluster algorithm from the \texttt{dist‑m4ri} program~\cite{Pryadko-2025-distm4ri}. The encoding rate is $r = K/N$, the median and maximum row weight of the X- and Z-type stabilizer checks are ($w_{X}^{\mathrm{med}}$, $w_{Z}^{\mathrm{med}}$) and  ($w_{X}^{\mathrm{max}}$, $w_{Z}^{\mathrm{max}}$) respectively. Confinement profiles showcases the minimum syndrome weights for irreducible errors of weight $w = 1, \dots, 6$ -- our codes have record-breaking confinement profiles, showcasing their capabilities in single-shot decoding. The smallest entry in the profile, i.e.~the minimum syndrome weight among all tested errors, is given by $d_S$.}
\label{tab:confinement}
\end{table*}
Our codes, particularly $[[96, 12, 4]]$, $[[96, 12, 8]]$, and $[[96, 44, 4]]$ achieve confinement $8,8,8,8$, meaning that irreducible errors of weight $1$–$4$ each flip at least eight syndrome bits, surpassing the $4,4,4,4$ of the 4D toric code and all AMC codes in Table~1 of~\cite{lin2025abelianmulticyclecodessingleshot}. To the best of our knowledge, the confinement profile for our MM codes is the highest recorded. We note that degree of confinement directly affects single-shot properties: codes with low confinement, such as certain GB codes, exhibit limited single-shot capability~\cite{lin2025singleshottwoshotdecodinggeneralized, sriram2025diffusioncodesselfcorrectionsmallerset}, because high-weight errors can produce small-weight syndromes that are indistinguishable from measurement noise. In contrast, the high confinement of our MM codes for both $X$ and $Z$ errors ensures that even low-weight errors generate large, distinct syndromes that remain robust to imperfect syndrome measurements, enabling reliable single-shot decoding. 

Single-shot decoding is often colloquially attributed to one of two mechanisms: either the presence of metachecks or the expansion property of the underlying quantum expander code~\cite{Campbell_2019, Quintavalle_2021}. While this approach is widespread in the literature~\cite{Campbell_2019, Quintavalle_2021, Higgott_2023, ostrev2024classical, lin2025singleshottwoshotdecodinggeneralized, lin2025abelianmulticyclecodessingleshot}, it does not fully capture the mechanism underlying single-shot decoding. As a concrete example, the 4D toric code, whose decoder is typically analyzed in this setting, employs a parallelized local flip decoding that does not explicitly use metachecks~\cite{Dennis_2002}. Instead, its performance relies on a confinement property: small irreducible $Z$- or $X$-type errors produce syndromes whose weight grows proportionally with their reduced weight and satisfy the conditions of confinement \cite[see Definition 3.45]{sriram2025diffusioncodesselfcorrectionsmallerset}. In this sense, neither expansion nor explicit metachecks are required; what appears essential is the presence of confinement. Moreover, it is unclear whether introducing nontrivial stabilizer redundancy alone can produce single-shot capability without simultaneously inducing some form of confinement, raising the question of whether redundancy per se should be regarded as the fundamental resource enabling single-shot decoding. A similar situation arises for asymptotically good codes, which possess substantial global redundancy even though practical single-shot decoders~\cite{Gu_2024} do not explicitly exploit the use of metachecks.

Of note is a stronger measure known as "good linear confinement" \cite[see Definition 2.10]{tan2025singleshotuniversalityquantumldpc} which ensures that for each check matrix in an infinite family of codes of length $N$, sufficiently small irreducible errors measured by their reduced weight, produce syndromes whose weight grows at least linearly with the error weight. This property is sufficient to guarantee that a quantum code is single-shot against both adversarial and stochastic noise~\cite{Quintavalle_2021, tan2025singleshotuniversalityquantumldpc}.

\section{Discussion and Conclusions\label{sec:discussion}}
Our Multivariate Multicycle (MM) code framework provides a unification of previous quantum error correcting codes, namely BB~\cite{Bravyi_2024}, MB\cite{voss2025multivariate}, TT~\cite{jacob2025singleshotdecodingfaulttolerantgates, menon2025magictricyclesefficientmagic}, GB~\cite{PhysRevA.88.012311}, and abelian 2BGA~\cite{lin2023quantumtwoblockgroupalgebra} codes. It provides metachecks for complete single-shot decoding, with codes supporting partial single-shot Z-decoding for $t \geq 3$ and both X and Z single-shot decoding for $t \geq 4$. This improves upon TT codes, which only support single-shot Z-decoding. We report a number of low-weight, high-rate, high-distance codes with record-breaking confinement.

% Our framework is the first use of the Koszul complex over a polynomial ring $R$ and quotient ring $R/I$ for construction of quantum error correcting codes.

Some of the MM codes we provide in Appendix Table~\ref{tab:table2} are relatively high-weight. Nonetheless they are high-rate and high-distance, thus they will be valuable with weight-reduction techniques~\cite{hastings2016weight, hastings2021quantum, PRXQuantum.5.040302} which have shown to preserve or even improve the effective fault-tolerant distance while lowering stabilizer weights~\cite{Tan_2025}.
Notably, hook errors - which can reduce the effective distance of a code under noisy syndrome extraction~\cite{Gidney_2023} are believed to be absent in higher-dimensional codes under any single-ancilla measurement circuit \cite[Theorem 5]{Tan_2025}. We expect investigating weight-reduction techniques with MM codes to be a productive future project.

Similarly exciting future research direction is the possibility for transversal fault-tolerant non-Clifford gates for the MM codes. Recent work~\cite{barkeshli2023codimension,lin2024transversalnoncliffordgatesquantum, zhu2025transversalnoncliffordgatesqldpc, hsin2025classifyinglogicalgatesquantum, Hsin_2025, zhu2025noncliffordparallelizablefaulttolerantlogical, zhu2025topologicaltheoryqldpcnonclifford, golowich2024quantumldpccodestransversal, breuckmann2024cups, li2025poincar, zhu2026nonabelianqldpctqftformalism} has shown that the cochain complex that defines the metachecks in higher dimensional codes can be endowed with cohomological invariants, such as the cup product. In this framework for a CSS code, the cup product is the algebraic version of a topological intersection number, via Poincaré duality~\cite{MR1867354}. It is a multilinear operation on cocycles whose output is invariant under the addition of coboundaries. This cohomological invariance enables a constant-depth transversal logical gate, which motivates the pursuit of higher-dimensional codes to achieve both single-shot decoding and native, fault-tolerant non-Clifford logical gates. Building on this perspective, the cups-and-gates (CG) framework~\cite{breuckmann2024cups, Hsin_2025} provides a method for constructing transversal logical gates using cohomology invariants.
%Functions on cochains that depend only on cohomology classes assign phases to logical states, resulting in transversal unitaries that preserve the codespace; when local, these unitaries can be implemented by finite-depth circuits.
%Cup products, as well as other multilinear operations such as Steenrod squares, and Pontryagin squares, are examples of cohomological invariants have been utilized~\cite{breuckmann2024cupsgatesicohomology, Hsin_2025, li2025poincaredualitymultiplicativestructures}. For codes that do not admit a product satisfying the Leibniz rule at the chain level, CG introduces pre-orientations together with a weak DGA structure. Under this structure, the integrated Leibniz rule ensures that the cup product descends to a well-defined operation on cohomology, yielding cohomology invariants that correspond to transversal logical gates, even for non-topological CSS codes.
This framework naturally raises the question of whether MM codes can support well-defined cohomological invariants such as cup products, that give rise to non-Clifford fault-tolerant logical gates. Thus, in future work we plan to explore the existence of a transversal fault tolerant CCZ gate or gates higher up in the Clifford hierarchy for the MM codes.

\begin{acknowledgments}
FAM conceived of this project, executed the numerical experiments, and wrote the manuscript with input from the other authors. SK suggested various benchmarks for code performance, directed manuscript outline, and manages the open source framework in which much of the development happened. OG conducted a thorough review of the presentation of the mathematics, providing insightful comments and contributing to key discussions about Koszul complexes. We thank Tommy Hofmann for contributions to our open source stack and for referring us to Matthias Zach of the OSCAR team, and we thank Matthias Zach for insightful discussions about Koszul complexes. This work was supported by NSF grants 1941583, 2346089, 2402861, 2522101.
\end{acknowledgments}
\bibliography{bibliography}
\appendix

\section{Generating Algorithm and explicit instances of MM codes}
In this appendix, we present Algorithm~\ref{alg:mm-css-quo} for constructing Multivariate Multicycle (MM) codes via Koszul complexes over quotient rings, along with specific code instances discovered through numerical search. The algorithm takes polynomials $F_1,\dots,F_t$ in the quotient ring $S = \mathbb{F}_2[x_1, \dots, x_D] / \langle x_i^{\ell_i} - 1, \cdots x_D^{\ell_D} \rangle$ and constructs the Koszul complex $K_\bullet([F_1,\dots,F_t]; S)$ over $S$. Each boundary map $\partial_k$ is a matrix with entries in $S$, and each entry is converted to its circulant matrix representation using the standard isomorphism between group algebra of cyclic group and algebra of circulant square matrices.  Algorithm~\ref{alg:mm-css} presents a "symbolic" approach that first constructs the Koszul complex over the polynomial ring $\mathbb{F}_2[y_1,\dots,y_t]$ with the standard boundary maps having entries in $\{0,y_1,\dots,y_t\}$, then substitutes each $y_i$ with the circulant matrix of $F_i \in S$. Both methods rely on Algorithm~\ref{alg:polynomial-to-circulant} for converting polynomials to circulant matrices. After conversion, we verify that the matrices commute pairwise—a property inherent to circulant matrices. The algorithms are implemented in the open-source QuantumClifford.jl library, utilizing the OSCAR computer algebra system~\cite{OSCAR, OSCAR-book} and the Julia~\cite{bezanson2015juliafreshapproachnumerical} programming language.

\begin{algorithm}[H]
    \DontPrintSemicolon
    \SetAlgoLined
    \KwInput{$\ell_1, \dots, \ell_D$, $F_1, \dots, F_t \in S = \mathbb{F}_2[x_1, \dots, x_D] / \langle x_i^{\ell_i} - 1, \cdots x_D^{\ell_D} \rangle$}
    \KwOutput{$P_X$, $P_Z$, $M_X$, $M_Z$}
    $n \gets \prod_{i=1}^D \ell_i, t \gets \text{length}(F_1, \dots, F_t)$\;
    \For{$i \gets 1$ \KwTo $t$}{
        $C_i \gets \text{PolynomialToCirculant}(F_i, \ell_1, \dots, \ell_D)$
    }
    \tcp{Check pairwise commutativity}
    \For{$i \gets 1$ \KwTo $t$}{
        \For{$j \gets i+1$ \KwTo $t$}{
            \textbf{assert} $A_i A_j \equiv A_j A_i \pmod{2}$
        }
    }
    \tcp{Construct Koszul complex over S}
    $K \gets \text{KoszulComplex}([F_1, \dots, F_t], S)$
    $\partial \gets \text{Array}(t)$\;
    \For{$k \gets 1$ \KwTo $t$}{
        $\partial_k \gets \text{map}(K, k)$
        $B_k \gets \text{matrix}(\partial_k)$
        $(r,c) \gets \text{size}(B_k)$\;
        $M_k \gets \mathbf{0}_{r n \times c n}(\mathbb{Z})$\;
        \For{$i \gets 1$ \KwTo $r$}{
            \For{$j \gets 1$ \KwTo $c$}{
                $f \gets B_k[i,j]$\;
                \If{$f \neq 0$}{
                    $C \gets \text{PolynomialToCirculant}(f, \ell_1, \dots, \ell_D)$\;
                    $M_k[(i-1)n+1:in, (j-1)n+1:jn] \gets C$\;
                }
            }
        }
        $\partial[k] \gets M_k \bmod 2$\;
    }
    \For{$k \gets 2$ \KwTo $t$}{
        \textbf{assert} $\partial[k] \cdot \partial[k-1] \equiv 0 \pmod{2}$
    }
    \eIf{$t = 2$}{
        $P_X \gets \partial[1]^\top$, $P_Z \gets \partial[2]$\;
        $M_X \gets \varnothing$, $M_Z \gets \varnothing$\;
    }{
        $q \gets \lfloor t/2 \rfloor$\;
        $P_X \gets \partial[q]^\top$, $P_Z \gets \partial[q+1]$\;
        \If{$t \geq 4$}{$M_X \gets \partial[q-1]^\top$}
        \If{$t \geq 3$}{$M_Z \gets \partial[q+2]$}
    }    
    \textbf{assert} $P_X P_Z^\top \equiv 0 \pmod{2}$\;
    \If{$M_X \neq \varnothing$}{\textbf{assert} $M_X P_X \equiv 0 \pmod{2}$}
    \If{$M_Z \neq \varnothing$}{\textbf{assert} $M_Z P_Z \equiv 0 \pmod{2}$}
    \KwRet{$(P_X, P_Z, M_X, M_Z)$}
    \caption{MM code vis Koszul complex over quotient ring}
    \label{alg:mm-css-quo}
\end{algorithm}

\begin{algorithm}
    \DontPrintSemicolon
    \SetAlgoLined
    \SetKwComment{Comment}{// }{}
    \KwInput{$\ell_1, \dots, \ell_D$, $F_1, \dots, F_t \in R = \mathbb{F}_2[x_1, \dots, x_D] / \langle x_i^{\ell_i} - 1, \cdots x_D^{\ell_D} \rangle$}
    \KwOutput{$P_X$, $P_Z$, $M_X$, $M_Z$}
    \BlankLine
    \For{$i \gets 1$ \KwTo $t$}{
        $C_i \gets \text{PolynomialToCirculant}(F_i, \ell_1, \dots, \ell_D)$\;
    }
    \BlankLine
    \Comment*[l]{Check pairwise commutativity}
    \For{$i \gets 1$ \KwTo $t$}{
        \For{$j \gets i+1$ \KwTo $t$}{
            \textbf{assert} $A_i A_j \equiv A_j A_i \pmod{2}$
        }
    }
    \BlankLine
    \Comment*[l]{Construct Koszul complex boundary maps}
    $R, y \gets \mathbb{F}_2[y_1, \dots, y_t]$\;
    $K \gets \text{KoszulComplex}(y)$\;
    $\partial \gets \text{Array}(t)$\;
    \For{$k \gets 1$ \KwTo $t$}{
        $B_k \gets \text{matrix}(\text{map}(K, k))$ \tcp*{$\binom{t}{k} \times \binom{t}{k-1}$ matrix}
        $(r,c) \gets \text{size}(B_k)$\;
        $\partial_k \gets \mathbf{0}_{rn \times cn}(\mathbb{Z})$\;
        \For{$i \gets 1$ \KwTo $r$}{
            \For{$j \gets 1$ \KwTo $c$}{
                \If{$B_k[i,j] = y_m$}{
                    $\partial_k[(i-1)n+1:in, (j-1)n+1:jn] \gets A_m$\;
                }
            }
        }
        $\partial[k] \gets \partial_k \bmod 2$\;
    }
    \For{$k \gets 2$ \KwTo $t$}{
        \textbf{assert} $\partial[k] \cdot \partial[k-1] \equiv 0 \pmod{2}$
    }
    \BlankLine
    \eIf{$t = 2$}{
        $P_X \gets \partial[1]^\top$, $P_Z \gets \partial[2]$\;
        $M_X \gets \varnothing$, $M_Z \gets \varnothing$\;
    }{
        $q \gets \lfloor t/2 \rfloor$\;
        $P_X \gets \partial[q]^\top$, $P_Z \gets \partial[q+1]$\;
        \If{$t \geq 4$}{$M_X \gets \partial[q-1]^\top$}
        \If{$t \geq 3$}{$M_Z \gets \partial[q+2]$}
    }    
    \textbf{assert} $P_X P_Z^\top \equiv 0 \pmod{2}$\;
    \If{$M_X \neq \varnothing$}{\textbf{assert} $M_X P_X \equiv 0 \pmod{2}$}
    \If{$M_Z \neq \varnothing$}{\textbf{assert} $M_Z P_Z \equiv 0 \pmod{2}$}
    \KwRet{$(P_X, P_Z, M_X, M_Z)$}
    \caption{MM code via Koszul complex over polynomial ring}
    \label{alg:mm-css}
\end{algorithm}

\begin{algorithm}
    \DontPrintSemicolon
    \SetAlgoLined
    \KwInput{Polynomial $F$, orders $\ell_1, \dots, \ell_D$}
    \KwOutput{Circulant matrix $C \in \mathbb{F}_2^{n \times n}$ s.t. $n = \prod_{i=1}^D \ell_i$}
    \BlankLine
    $n \gets \prod_{i=1}^D \ell_i$\;
    $C \gets \mathbf{0}_{n \times n}(\mathbb{F}_2)$\;
    $f \gets \text{lift}(F)$\;
    \For{$j \gets 0$ \KwTo $n-1$}{
        $tmp \gets j$\;
        \For{$k \gets D$ \KwDownTo $1$}{
            $idxs[k] \gets tmp \bmod \ell_k$\;
            $tmp \gets \lfloor tmp / \ell_k \rfloor$\;
        }
        \For{$\tau \in \text{terms}(f)$}{
            $c \gets \text{coeff}(\tau,1)$\;
            \If{$c = 0$}{continue}
            $e_k \gets \text{degree}(\tau,k)$ for $k = 1, \dots, D$\;
            $row \gets 0$\;
            \For{$k \gets 1$ \KwTo $D$}{
                $row \gets row \cdot \ell_k + ((idxs[k] + e_k) \bmod \ell_k)$\;
            }
            $C[row+1,j+1] \gets C[row+1,j+1] + c$\;
        }
    }
    \KwRet{$C$}
    \caption{PolynomialToCirculant}
    \label{alg:polynomial-to-circulant}
\end{algorithm}
\section{More phenomenological noise experiments}
\label{section:more_phenomenological_noise_experiments}
\editcolor{We find the MM codes consistently perform better than TT codes over blocklengths of practical sizes. We ran the \texttt{SinterTesseractDecoder} with 50,000 shots to calculate the logical error rate per physical error rate}.
\begin{figure}[H]
    \centering
    \includegraphics[width=1\linewidth]{AppendixA.png}
    \caption{\textbf{MM codes consistently perform better than TT codes in the X memory experiment.} Seven codes are compared: MM $[[96,12,(8,8)]]$, MM $[[144,12,(12,12)]]$, MM $[[486,24,(12,12)]]$, MM $[[648,60,(9,9)]]$, TT $[[72,6,(12,6)]]$, TT $[[135,12,(14,6)]]$ and TT $[[180,12,(20,8)]]$. All MM codes have weight-12 stabilizer checks.}
    \label{fig:appendixfig1}
\end{figure}
\begin{figure}[H]
    \centering
    \includegraphics[width=1\linewidth]{AppendixB.png}
    \caption{\textbf{MM codes consistently perform better than TT codes in the Z memory experiment.} Seven codes are compared: MM $[[96,12,(8,8)]]$, MM $[[144,12,(12,12)]]$, MM $[[486,24,(12,12)]]$, MM $[[648,60,(9,9)]]$, TT $[[72,6,(12,6)]]$, TT $[[135,12,(14,6)]]$ and TT $[[180,12,(20,8)]]$. All MM codes have weight-12 stabilizer checks.}
    \label{fig:appendixfig1}
\end{figure}

% \section{More Plateau experiments under circuit-level noise}
% \editcolor{Circuit-level noise model simulations are computationally demanding; as noted in Section~\ref{sec:plateauexperiments}, the main comparison is therefore restricted to BB, MM, and TT codes. Here we present an additional simulation including the AMC~$[[180,6,(12,12)]]$ code under the same conditions ($T = 10$ syndrome extraction rounds, $p = 10^{-3}$, $50{,}000$ shots per window size). The result is shown in Fig.~\ref{fig:PlateauCircuitAMC}.}
% \label{section:plateau_circuit_amc}
% \begin{figure}[H]
%     \centering
%     \includegraphics[width=1\linewidth]{CircuitLevelXPlateauExperiments.png}
%     \caption{\textbf{Plateau experiment for single-shot decoding in the $X$-basis memory experiment under circuit-level noise, including AMC~$[[180,6,(12,12)]]$ ($T = 10$ rounds, $p = 10^{-3}$, $50{,}000$ shots).} Logical error rate per round $P_L$ versus OWD 
%     window size $w$ for $[[180,6,(12,12)]]$ AMC (cyan), $[[144,12,(12,12)]]$ BB (orange), $[[486,24,(12,12)]]$ MM (red), and $[[72,6,(12,6)]]$ TT (purple). AMC achieves the lowest plateau, consistent with its 
%     phenomenological noise performance Fig.~\ref{fig:plateau_X_T10phenom}). TT again fails to suppress errors in this basis due to its small $d_Z = 6$.}
%     \label{fig:PlateauCircuitAMC}
% \end{figure}
% \section{Complete Single-shot decodability of [[486, 24, (12,12)]] code}
% \editcolor{We demonstrate the performance of our $[[486, 24, 12]]$ MM code—which possesses the highest known confinement among many single-shot-decodable codes (see Table~\ref{tab:confinement})—achieves superior performance in both X and Z memory experiments. This suggests that high confinement is indicative of better single-shot decoding performance. Remarkably, this code also excels in the X-memory experiment, thereby demonstrating the \textit{complete-single-shot} capability of our codes under phenomenological noise model.}
% \begin{figure}[H]
%     \centering
%     \includegraphics[width=1\linewidth]{OWDPLOTBX.png}
%     \caption{Single-shot decoding using the OWD BP+OSD decoder in the X-memory experiment under the phenomenological noise model. Logical error rates per round for weight-12 MM $[[486,24,12]]$ (solid) and TT $[[135,12,6]]$ (dotted) codes with constant decoder parameters $(w,c)=(2,1)$ across varying numbers of syndrome extraction rounds $T \in \{1,2,6\}$. The TT code has weight-9 X-type and weight-6 Z-type stabilizer checks.}
%     \label{fig:owdplotAXBasisl4862412}
% \end{figure}
% \begin{figure}[H]
%     \centering
%     \includegraphics[width=1\linewidth]{OWDPLOTBZ.png}
%     \caption{Single-shot decoding using the OWD BP+OSD decoder in the Z-memory experiment under the phenomenological noise model. Logical error rates per round for weight-12 MM $[[486,24,12]]$ (solid) and TT $[[135,12,6]]$ (dotted) codes with constant decoder parameters $(w,c)=(2,1)$ across varying numbers of syndrome extraction rounds $T \in \{1,2,6\}$. The TT code has weight-9 X-type and weight-6 Z-type stabilizer checks.}
%     \label{fig:owdplotAZBasis4862412}
% \end{figure}

% \begin{figure}[H]
% \centering
% \begin{tikzpicture}[scale=0.7, transform shape]
% \owdpanel{0}{(0.2, 3.5)}
% \node[font=\Large\bfseries, text=frameC, anchor=east] at (-0.1, 7.2) {(a)};
% \node[font=\small, text=axisC, anchor=east] at (-0.1, 6.4) {$t{=}1$};
% \begin{scope}[shift={(0.2, 3.5)}, x=8.5mm, y=7.8mm]
%   \draw[decorate, decoration={calligraphic brace, amplitude=5pt},
%         line width=0.8pt, color=umassLight!70!black]
%     ($(p0-0-5.north west)+(-0.05,1.0)$) --
%     ($(p0-1-5.north east)+(0.05,1.0)$)
%     node[midway, above=4pt, font=\normalsize\bfseries, text=umassLight!55!black] {$w$};
%   \draw[decorate, decoration={calligraphic brace, amplitude=3.5pt},
%         line width=0.7pt, color=umassMed!70!black]
%     ($(p0-0-5.north west)+(-0.05,0.2)$) --
%     ($(p0-0-5.north east)+(0.05,0.2)$)
%     node[midway, above=4pt, font=\normalsize\bfseries, text=umassMed!55!black] {$c$};
%   \draw[decorate, decoration={calligraphic brace, amplitude=3pt},
%         line width=0.6pt, color=bufferC!60!black]
%     ($(p0-1-5.north west)+(-0.05,0.3)$) --
%     ($(p0-1-5.north east)+(0.05,0.3)$)
%     node[midway, above=3pt, font=\normalsize\bfseries, text=bufferC!60!black] {$w{-}c$};
% \end{scope}
% \draw[-{Stealth[length=3mm,width=2.5mm]}, line width=1.0pt, color=umassMed,
%       shorten >=2pt, shorten <=2pt]
%   (6.7, 3.0) -- (6.7, 0.2);
% \node[
%   font=\normalsize\itshape,
%   text=umassMed,
%   anchor=west,
%   fill=white,
%   inner sep=1pt
% ]  at (6.8, 1.6) {slide $c$};
% \owdpanel{1}{(0.2, -1.8)}
% \node[font=\Large\bfseries, text=frameC, anchor=east] at (-0.1, 1.9) {(b)};
% \node[font=\small, text=axisC, anchor=east] at (-0.1, 1.1) {$t{=}2$};
% \draw[-{Stealth[length=3mm]}, line width=0.7pt, color=axisC]
%   (0.2, -2.8) -- (7.5, -2.8);
% \node[
%   font=\normalsize\itshape,
%   text=axisC,
%   anchor=north
% ] at (3.85, -2.95) {time};
% \begin{scope}[shift={(0.2, -3.8)}]
%   \node[commit, minimum width=5mm, minimum height=5mm] (c) at (0,0) {};
%   \node[leglabel, anchor=west] at ($(c.east)+(0.12,0)$) {commit $c$};
%   \node[buffer, minimum width=5mm, minimum height=5mm] (b) at (2.4,0) {};
%   \node[leglabel, anchor=west] at ($(b.east)+(0.12,0)$) {buffer $w{-}c$};
%   \node[past, minimum width=5mm, minimum height=5mm] (p) at (5.2,0) {};
%   \node[leglabel, anchor=west] at ($(p.east)+(0.12,0)$) {past};
% \end{scope}
% \end{tikzpicture}
% \caption{\editcolor{\textbf{Overlapping window decoding with $(w,c)=(2,1)$.}
% (a) shows the decoding window at $t=1$, where round~1 forms the commit region ($c$) and round~2 remains in the overlap region ($w-c$). (b) shows the next decoding cycle at $t=2$, after the window advances by $c$: round~2 now becomes committed while a new round enters the overlap region. Choosing $c<w$ preserves temporal overlap between consecutive decoding windows, allowing tentative corrections in the overlap region to be revised in later cycles. This reduces the likelihood that error chains originating in the commit region extend beyond the active decoding window and are therefore mis-corrected~\cite{skoric2023parallel}.}}
% \label{fig:owddecoding}
% \end{figure}
\editcolor{\section{Circuit-Level Noise Modelling via Left-Right Circuits}}
\label{section:lrccircuitlevelnoisemodel}
\editcolor{We consider a new approach that uses the Left-Right Circuit (LRC) framework~\cite{strikis2026high}, which addresses a key challenge in syndrome circuit design known as \emph{hook errors}. Hook errors arise when a weight-1 error during the circuit propagates through the CNOT network to produce a high-weight residual error on the data qubits. If this residual error overlaps a minimum-weight logical operator, it effectively reduces the circuit distance below the code distance $d$, degrading fault-tolerance performance. Rather than mitigating hook errors through flag qubits~\cite{chamberland2018flag, chao2018quantum, lingling2020faulttolerant} or Shor-style syndrome extraction~\cite{shor1996fault}---which incur additional circuit depth and ancilla overhead---the LRC framework instead identifies CNOT schedules for which hook errors do not propagate onto the support of low-weight logical operators.}

\editcolor{\textit{}{A natural question is whether the benefit of LRC scheduling depends on stabilizer weight}. To investigate this, we evaluate our MM codes under both weight-6 and weight-9 $X$-stabilizer checks. For the weight-6 variants --- specifically MM~[[144,6,(8,8)]] and MM~[[216,6,(12,12)]] --- the LRC approach yields a clear improvement in the $Z$-basis memory experiment, as shown in Fig.~\ref{fig:circuitlevel_model_xbasis_w6} and Fig.~\ref{fig:circuitlevel_model_zbasis_w6}.}

\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{LRCXBasisweight6circuitlevel.png}
    \caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $X$-basis memory experiment for weight-6 MM codes with 
    \texttt{TesseractSinterDecoder}.} MM~[[144,6,(8,8)]] and MM~[[216,6,(12,12)]] are compared against BB~[[144,12,(12,12)]] under standard LDPC coloration circuits (default, dotted) and LRC circuits~\cite{strikis2026high} (LRC, solid). Both weight-6 MM codes exhibit a clear pseudo-threshold improvement under LRC scheduling, in contrast to the weight-9 MM code results shown in Figs.~\ref{fig:circuitlevel_model_xbasis} and~\ref{fig:circuitlevel_model_zbasis}.}}
    \label{fig:circuitlevel_model_xbasis_w}
\end{figure}


\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{LRCZBasisweight6circuitlevel.png}
    \caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $Z$-basis memory experiment for weight-6 MM codes with 
    \texttt{TesseractSinterDecoder}.} MM~[[144,6,(8,8)]] and MM~[[216,6,(12,12)]] are compared against BB~[[144,12,(12,12)]] under standard LDPC coloration circuits (default, dotted) and LRC circuits~\cite{strikis2026high} (LRC, solid). Both weight-6 MM codes exhibit a clear pseudo-threshold improvement under LRC scheduling, in contrast to the weight-9 MM code results shown in Figs.~\ref{fig:circuitlevel_model_xbasis} and~\ref{fig:circuitlevel_model_zbasis}.}}
    \label{fig:circuitlevel_model_zbasis_w6}
\end{figure}

\editcolor{We next compare the four main codes --- BB~[[144,12,(12,12)]], MM~[[144,12,(12,12)]], AMC~[[180,6,(12,12)]], and TT~[[135,12,6]] --- across both X- and Z- memory experiments in Figs.~\ref{fig:circuitlevel_model_xbasis_again} and~\ref{fig:circuitlevel_model_zbasis_again}. In the $X$-basis experiment (Fig.~\ref{fig:circuitlevel_model_xbasis}), the BB and AMC codes show noticeable threshold improvements under LRC scheduling. By contrast, the MM~[[144,12,(12,12)]] and TT~[[135,12,(14,6)]] codes --- both with weight-9 $X$-stabilizer checks --- show negligible improvements. Together with the weight-6 results in Fig.~\ref{fig:circuitlevel_model_zbasis_w6}, this suggests that the LRC benefit is closely tied to stabilizer weight: codes with lower-weight checks are more amenable to LRC-optimized approach for circuit-level simulations.}

\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{circuitlevelnoiselrcldpcXexp.png}
    \caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $X$-basis memory experiment with \texttt{TesseractSinterDecoder}.} Four codes are compared under standard LDPC coloration circuits (default, dotted) and LRC circuits~\cite{strikis2026high} (LRC, solid). BB~[[144,12,(12,12)]] and AMC~[[180,6,(12,12)]] show noticeable threshold improvements under LRC scheduling. In contrast, MM~[[144,12,12]] and TT~[[135,12,(14,6)]] --- both featuring weight-9 $X$-stabilizer checks --- show negligible improvement.}}
    \label{fig:circuitlevel_model_xbasis_again}
\end{figure}

\begin{figure}[h!]
    \centering
    \includegraphics[width=1\linewidth]{circuitlevelnoiselrcldpcZexp.png}
    \caption{\editcolor{\textbf{Performance comparison under circuit-level noise in the $Z$-basis memory experiment with \texttt{TesseractSinterDecoder}.} The same four codes and two syndrome extraction approaches as in Fig.~\ref{fig:circuitlevel_model_xbasis} are shown. The pattern is consistent with the $X$-basis results: BB and AMC benefit from LRC scheduling while the weight-9 MM and TT codes do not.}}
    \label{fig:circuitlevel_model_zbasis_again}
\end{figure}

\editcolor{For the LRC preprocessing pipeline, residual distances are estimated using min-sum BP+OSD with $200$ maximum iterations and OSD order~$3$, over $100$ attempts per residual error with a batch size of $10$, parallelised across $8$ workers. Up to $1{,}000$ colour permutations are evaluated and the top $5$ CNOT orderings are retained. The extended-code distance re-ranking step applies $1{,}000$ BP-OSD attempts with $500$ maximum iterations and OSD order~$7$. For the standard coloration circuits generated via the \texttt{LDPC} package~\cite{roffe_decoding_2020}, no CNOT ordering optimization is performed beyond the minimum edge coloring. Logical error rates are estimated via Monte Carlo sampling with up to $50{,}000$ shots and a minimum of $500$ errors per physical error rate point $p$ in the range $[10^{-4}, 10^{-1}]$, with $T = 2d$ syndrome extraction rounds where $d$ is the code distance.}

\editcolor{\section{Appendix: Search Procedure for Multivariate Multicycle Codes}}
\label{sec:appendix_search}
\editcolor{We describe the computational procedure used to search for high-rate and high distance MM codes. The search is implemented in Julia programming language,
relying on the \texttt{QECCore.jl}, \texttt{QuantumClifford.jl}, \texttt{Oscar.jl},
and \texttt{Nemo.jl} libraries for code construction,and on \texttt{Gurobi.jl} via \texttt{JuMP.jl} for distance computation.
We fix a target block length $N$ and factorize it as $n = 6 \cdot \ell_1
\cdot \ell_2 \cdot \ell_3 \cdot \ell_4$, enumerating all tuples $(\ell_1, \ell_2, \ell_3, \ell_4)$ with each $\ell_i \ge 2$ satisfying $\ell_1 \ell_2 \ell_3 \ell_4 = N/6$. For each valid tuple, we construct the quotient ring}
\editcolor{\begin{align}
    S = \mathbb{F}_2[w, x, y, z] \,\Big/\, \bigl\langle w^{\ell_1}-1,\; x^{\ell_2}-1,\; y^{\ell_3}-1,\; z^{\ell_4}-1 \bigr\rangle,
\end{align}}
\editcolor{whose monomials $\{w^a x^b y^c z^d\}$ with $0 \le a < \ell_1$, $0 \le b < \ell_2$, $0 \le c < \ell_3$, $0 \le d < \ell_4$ form a basis of size $\ell_1 \ell_2 \ell_3 \ell_4$. For each tuple we draw $n_{\mathrm{samples}}$ independent random instances, each consisting of four polynomials $F(w,x,y,z),\;G(w,x,y,z),\; H(w,x,y,z), \;I(w,x,y,z) \in S$ sampled uniformly at random as sums of exactly $t$ distinct monomials chosen without replacement from the full monomial basis. The corresponding MM code $\mathcal{Q} = \mathrm{MM}([\ell_1,\ell_2,\ell_3,\ell_4], [F,G,H,I])$ is then constructed, and instances yielding $k = 0$ logical qubits are discarded.
For the remaining codes, the $X$- and $Z$-distances are computed
independently via a mixed-integer program (MIP), using QuantumClifford.jl's \texttt{DistanceMIPAlgorithm} with Gurobi as the solver and a time limit of $T_{\max}$ seconds per instance. To characterize sparsity, we also record the median, and maximum row weights of both parity-check matrices $H_X$ and 
$H_Z$, defined for a matrix $H \in \mathbb{F}_2^{r \times n}$.}

\editcolor{\section{Recovering Subfamilies of MM codes}}
\editcolor{To support our remarks of Sec.~\ref{sec:subfamilies_of_mm_code} and to demonstrate the unifying nature of the MM code construction, we provide explicit numerical verification using an open-source implementation. All examples below are realized within the \texttt{QuantumClifford.jl}. The computations rely on \texttt{Oscar.jl} for computational group theory, and on \texttt{Gurobi.jl} for solving the distance via MIP}.
\subsection{Abelian Multicycle codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[96, 6, (8,8)]] AMC code
julia> using Oscar; using QuantumClifford.ECC;

julia> l, m, p, r = 16, 1, 1, 1;

julia> R, (w, x, y, z) = polynomial_ring(GF(2), [:w, :x, :y, :z]); # polynomial ring R = F_2[x,y]

julia> I = ideal(R, [w^l - 1, x^m - 1, y^p - 1, z^r - 1]); # Ideal I = (x^l - 1, y^m - 1)

julia> S, _ = quo(R, I);  # quotient ring S = R/I

julia> A = S(1 + w); # A(x,y) in S

julia> B = S(1 + w^3); # B(x,y) in S

julia> C = S(1 + w^5); # C(x,y) in S

julia> D = S(1 + w^7); # D(x,y) in S

julia> c = MultivariateMulticycle([l, m, p, r], [A, B, C, D]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=Gurobi))
(96, 6, 8)
\end{lstlisting}

\subsection{Bivariate Bicycle codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[144, 12, (12,12)]] BB code

julia> using Oscar; using QuantumClifford.ECC;

julia> l=12; m=6;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^l-1, y^m-1]);

julia> S, _ = quo(R, I);

julia> A = S(x^3 + y + y^2);

julia> B = S(y^3 + x + x2^);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=Gurobi))
(144, 12, 12)
\end{lstlisting}

\subsection{Trivariate Tricycle codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[72, 6, (12,6)]] BB code

julia> using Oscar; using QuantumClifford.ECC;

julia> l, m, p = 4, 3, 2;

julia> R, (x, y, z) = polynomial_ring(GF(2), [:x, :y, :z]);

julia> I = ideal(R, [x^l - 1, y^m - 1, z^p - 1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + y + x*y^2);

julia> B = S(1 + y*z + x*2y^2);

julia> C = S(1 + x*y^2*z + x^2*y);

julia> c = MultivariateMulticycle([l,m, p], [A, B, C]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=Gurobi))
(72, 6, 6)
\end{lstlisting}

\subsection{4D Toric codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[96, 6, (4,4)]] 4D Toric code

julia> using Oscar; using QuantumClifford.ECC;

julia> l, m, p, r = 2, 2, 2, 2;

julia> R, (w, x, y, z) = polynomial_ring(GF(2), [:w, :x, :y, :z]);

julia> I = ideal(R, [w^l - 1, x^m - 1, y^p - 1, z^r - 1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + w);

julia> B = S(1 + x);

julia> C = S(1 + y);

julia> D = S(1 + z);

julia> c = MultivariateMulticycle([l, m, p, r], [A, B, C, D]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=Gurobi))
(96, 6, 4)
\end{lstlisting}

\subsection{La-Cross codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[98, 18, (4,4)]] La-Cross code

julia> using Oscar; using QuantumClifford.ECC;

julia> n = 7;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^n-1, y^n-1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + x + x^3);

julia> B = S(1 + y + y^3);

julia> c = MultivariateMulticycle([n,n], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS, time_limit=900))
(98, 18, 4)
\end{lstlisting}

\subsection{Multivariate Bicycle codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[48, 4, (6,6)]] MB code
julia> using Oscar; using QuantumClifford.ECC;

julia> l=4; m=6;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^l-1, y^m-1]);

julia> S, _ = quo(R, I);

julia> z = x*y;

julia> A = S(x^3 + y^5);

julia> B = S(x + z^5 + y^5 + y^2);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS, time_limit=900))
(48, 4, 6)
\end{lstlisting}


\subsection{Generalized Bicycle codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[30, 8, (4,4)]] GB code
julia> using Oscar; using QuantumClifford.ECC;

julia> l=15; m=1;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^l-1, y^m-1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + x^2 + x^8);

julia> B = S(1 + x + x^4);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS, time_limit=900))
(30, 8, 4)
\end{lstlisting}

\subsection{Abelian two-block group algebra codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[16, 2, (4,4)]] 2BGA code
julia> using Oscar; using QuantumClifford.ECC;

julia> l=2; m=4;

julia> R, (s, x) = polynomial_ring(GF(2), [:s, :x]);

julia> I = ideal(R, [s^l-1, x^m-1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + x);

julia> B = S(1 + x + s + x^2 + s*x + s*x^3);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS, time_limit=900))
(16, 2, 4)
\end{lstlisting}

\subsection{Cyclic hypergraph product codes}
\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[450 32, (8,8)]] C2 code

julia> using Oscar; using QuantumClifford.ECC;

julia> l=15; m=15;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^l-1, y^m-1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + x + x^4);

julia> B = S(1 + y + y^4);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=HiGHS, time_limit=900))
(450, 32, 8)
\end{lstlisting}

\begin{lstlisting}[language=JuliaREPL]
# Code Example: [[240 8, (8,8)]] CxR code

julia> using Oscar; using QuantumClifford.ECC;

julia> l=15; m=8;

julia> R, (x, y) = polynomial_ring(GF(2), [:x, :y]);

julia> I = ideal(R, [x^l-1, y^m-1]);

julia> S, _ = quo(R, I);

julia> A = S(1 + x + x^4);

julia> B = S(1 + y);

julia> c = MultivariateMulticycle([l,m], [A,B]);

julia> import Gurobi;

julia> code_n(c), code_k(c), distance(c, DistanceMIPAlgorithm(solver=Gurobi, time_limit=900))
(240, 8, 8)
\end{lstlisting}
\subsubsection{Explicit low-weight instances of MM codes}

\editcolor{We discover several instances of weight-6 MM codes for practical quantum hardware platforms in Appendix~\ref{tab:mmcodetable3}. Although MM codes employ geometrically non-local checks, recent work \cite{xu2024constant} suggests such codes can be implemented efficiently on reconfigurable neutral atom arrays due to the product structure inherent in many qLDPC codes, including hypergraph product and lifted product codes, which can be exploited to implement non-local syndrome extraction circuits via parallel atom rearrangement with crossed acousto-optic deflectors (AODs), achieving an effectively constant overhead. Since MM codes share this product structure, the same compilation toolkit applies directly, and we expect that the high confinement of MM codes makes them an attractive candidate for this architecture.}

% \begin{figure*}
%     \centering
%     \includegraphics[width=1\linewidth]{weights.png}
%     \editcolor{\caption{Check weight comparison of 5D MM codes against AMC, TT, and Quantum Tanner codes. (a)~X-type check weight ($w_X$) versus code rate ($r = k/n$), with Z-type check weight ($w_Z$) encoded in the colorbar. (b)~Z-type check weight ($w_Z$) versus code rate, with X-type check weight ($w_X$). All codes are restricted to block length $n \in [60,200]$. The 5D MM code family achieves constant check weights $(w_X, w_Z) = (8,6)$ across varying block lengths. For comparison, we include: AMC codes, TT codes, and recent small instances of quantum Tanner codes from Table II of Ref.\cite{wang2026checkweightconstrainedquantumcodesbounds}. The 5D MM codes demonstrates low $w_X$ and $w_Z$ compared to quantum Tanner codes. These low, constant check weights make MM codes particularly suitable for practical fault-tolerant QEC.}}
%     \label{fig:placeholder}
% \end{figure*}

% \begin{figure*}
%     \centering
%     \includegraphics[width=1\linewidth]{distanceplot.png}
%     \editcolor{\caption{}}
%     \label{fig:placeholder}
% \end{figure*}
\section{Explicit instances on higher dimensional codes}
\renewcommand{\arraystretch}{1.6}
\begin{table*}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$}& \textbf{$\ell$, m, p, r} & \textbf{$\frac{kd^2}{n}$} & \textbf{$\tilde{w}$} & \textbf{$\overline{w}$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} \\
\hline
\rowcolor{A} $[[96, 12, (4,4)]]$ & 2, 2, 2, 2 & 2.0 & 6 & 6 &  $1 + wx$ & $1 + xy$ & $1 + yz $ & $1 + wz$ \\
\hline
\rowcolor{B} $[[96, 12, (4, 4)]]$ & $2, 2, 2, 2$ & 2.0 & 12& 12 & $1 + w + xy + zx$  & $1 + x + zy + zw$ & $1 + y + zx + zw$ & $1 + z + yx + yw$ \\
\hline
\rowcolor{D} $[[96, 44, (4,4)]]$ & 2, 2, 2, 2 & 7.3 & 12& 12 & $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
% \rowcolor{G} $[[144, 6, (4,4)]]$ & 2, 2, 2, 3 & \makecell{0.66} & 6 & 6 &  $1 + wx$ & $1 + xy$ & $1 + yz $ & $1 + wz$ \\
\rowcolor{A} $[[144, 12, (8,8)]]$ & 2, 2, 2, 3 & 5.3 & 9 & 10 &  $wxy + xyz$ & $y + zx + yx + zw$ & $x + zyxw$ & $z + y + x + zyx$ \\
\hline
\rowcolor{H}$[[192, 30, (8,8)]]$ & (2, 2, 2, 4) & 10.0 & 16 &  $18$ & $zx + zw + zyw + xw$ & $x + zx + yw + zyw$ & $x + yx + w + zyw$ & $z + y + zx + zyx$\\
\rowcolor{H}& & & & & $ + zxw + yxw$ & & $ + xw + zxw$ & $+ zyw + yxw$\\
\hline
\rowcolor{A} $[[216, 12, (12,12)]]$ & 2, 2, 3, 3 & 8.0 & 9 & 10 &  $wxy + xyz$ & $y + zx + yx + zw$ & $x + zyxw$ & $z + y + x + zyx$ \\
\hline
\rowcolor{B} $[[240, 12, (8,8)]]$ & 2, 2, 2, 5 & 3.2 & 13 & 16 &  $1 + y + zy + yx$ & $1 + y + zx + w$ & $y + zy + zyw + zxw$ & $zyx + yxw$ \\
\rowcolor{B} & & & & & $+ w + zw$ & $+ zyw + xw$ & &\\
\hline
% \rowcolor{G} $[[288, 6, (6,6)]]$ & 2, 2, 3, 4 & \makecell{} & 6 & 6 &  $1 + wx$ & $1 + xy$ & $1 + yz $ & $1 + wz$ \\
\rowcolor{D} $[[288, 52, (4,4)]]$ & 2, 2, 3, 4 & 2.9 & 12 & 12 & $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
\rowcolor{E} $[[324, 18, (9,9)]]$ & 2, 3, 3, 3 & 4.5 & 14 & 18 & $zyx + yw + zyw + xw$ & $xz + x$& $zy + x + zw + xw$  &$1 + zy + x + zyx$\\
\rowcolor{E} & & & & & $+ yxw + zyxw$ & & $ + zxw + zyxw$& $+ xw + zyxw$ \\
\hline
\rowcolor{A}$[[360, 30, (6,6)]]$ & 2, 2, 3, 5 & 3.0 & 14 & 16 &  $zy + zxw$ & $zy + w + zw + yw$ & $y + x + zxw + zyxw$ & $1 + z + zy + yx$ \\
\rowcolor{A} & & & & & & & & $ + w + yw + zyw + zxw$\\
\hline
\rowcolor{B}$[[384,80,(4,4)]]$ & 2, 2, 2, 8 & 3.5 & 12 & 12 &  $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
\rowcolor{C}$[[486, 18, (9,9)]]$& 3, 3, 3, 3 & 3.0 & 12 & 12 & $1 + w + xy + zx$  & $1 + x + zy + zw$ & $1 + y + zx + zw$ & $1 + z + yx + yw$ \\
\hline
\rowcolor{D}$[[486, 24, (12,12)]]$ & 3, 3, 3, 3 & 7.11 & 9 & 9 &  $1 + wx + x^2y$ & $1 + xy + y^2z$ & $1 + yz + wz^2$ & $1 + wz + w^2x$\\
\hline
\rowcolor{E}$[[486, 66, (9,9)]]$ & 3, 3, 3, 3 & 11.0 & 12 & 12 & $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
\rowcolor{A} $[[576, 64, (6,6)]]$ & 2, 3, 4, 4 & 4.0 & 12 & 12 & $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
\rowcolor{B} $[[648, 60, (9,9)]]$ & 3, 3, 3, 4 & 7.5 & 12 & 12 &  $(1 + x)(1 + yz)$ & $(1 + y)(1 + zw)$ & $(1 + z)(1 + wx)$ & $(1 + w)(1 + xy)$ \\
\hline
\rowcolor{C} $[[768, 12, (12, 12)]]$ & 2, 4, 4, 4 & 2.25 & 6 & 6 & $1 + wx$ & $1 + xy$ & $1 + yz $ & $1 + wz$ \\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Parameters and explicit instances of MM codes supporting complete single-shot decoding.} Here, $n$ is the number of physical qubits, $k$ the number of logical qubits, and $(d_X, d_Z)$ are $X$-type and $Z$-type distances. The efficiency rate is $\frac{kd^2}{n}$, the median and maximum row weight of the stabilizer checks are $\tilde{w}$ and $\overline{w}$, respectively. The multivariate quotient ring polynomials $F(w,x,y,z)$, $G(w,x,y,z)$, $H(w,x,y,z)$, and $I(w,x,y,z)$ are defined over $\mathbb{F}_2[w, x, y, z]/\langle w^\ell-1, x^m-1, y^p-1, z^r-1 \rangle$. Code distances were computed using the mixed-integer programming method of~\cite{landahl2011color} and the HiGHS linear programming solver~\cite{huangfu2018parallelizing}. The code distances were cross-verified using the connected cluster algorithm from the \texttt{dist‑m4ri} program~\cite{Pryadko-2025-distm4ri}. Unlike the 3D surface code with its asymmetric X- and Z-bases~\cite{Berthusen_2024}, both the 4D toric code and our 4D MM codes yield $d_X = d_Z$.}
\label{tab:table2}}
\end{table*}

% \begin{figure*}[!h]
%     \centering
% \includegraphics[width=1.0\linewidth]{tesseractdecodervsbposddecoder.png}
%      \caption{\textbf{Logical error rate per round versus physical error rate for three MM codes under phenomenological noise model, sampled using \texttt{sinter} package, which enables fast Monte Carlo simulation of quantum error-correction circuits. The Tesseract decoder with beam search (\texttt{$det\_beam=10$}), and  (\texttt{$pqlimit=1000$},
%     \texttt{num\_det\_orders=5}) was compared against the BPOSD decoder with OSD-CS method (\texttt{$osd\_order=9$}).}}
%     \label{fig:bposdtesseract}
% \end{figure*}
\begin{table*}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & ($\ell$, m, p, q) & $\frac{kd^2}{n}$ & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} \\
\hline
\rowcolor{B} $[[96, 12, (4,4)]]$ & (2,2,2,2) & 2.0 & $wz + xyz$ & $w + xyz$ & $wyz + xy$ & $xyz + xz$\\
\hline
\rowcolor{E} $[[144, 6, (6,6)]]$ & (2,2,3,2) & 1.5 & $wxz + wz$ & $xy^2z + yz$ & $wy^2 + xyz$ & $xz + y^2$\\ 
\hline
\rowcolor{F} $[[144, 6, (8,8)]]$ & (2,2,3,2) & 2.6 & $wxy + z$ & $wx + wyz$ & $wx + xy^2$ & $xyz + y^2z$\\
\hline
\rowcolor{G} $[[144, 12, (5,5)]]$ & (2,2,3,2) & 2.1 & $wxz + xy$ & $wxy^2 + wyz$ & $wy^2z + xyz$ & $wxy^2 + xyz$\\
\hline
\rowcolor{H} $[[192, 6, (8,8)]]$ & (4,2,2,2) & 2.0  & $w^2xz + z$ & $w^2yz + wyz$ & $w^3x + xyz$ & $w^3xyz + w^2z$\\ 
\hline
\rowcolor{K} $[[216, 6, (12,12)]]$ & (2,2,3,3) & 4.0 & $wxy + xy^2$ & $wy + xyz^2$ & $xy^2 + y^2z$ & $wxyz^2 + xz$\\
\hline
\rowcolor{L} $[[240, 24, (5,5)]]$ & (2,2,2,5) & 2.5 & $wxyz^4 + wz^3$ & $xyz^3 + 1$ & $wyz^3 + wyz^2$ & $wxz^2 + wx$\\
\hline
\rowcolor{M} $[[240, 12, (7,7)]]$ & (5,2,2,2) & 2.5 & $w^4y + w^3x$ & $w^4x + w^3$ & $w^4y + wx$ & $w^4z + w^2yz$\\
\hline
\rowcolor{N} $[[240, 6, (12,12)]]$ & (5,2,2,2) & 3.6 & $w^4xy + xz$ & $w^3 + w^2z$ & $w^3xyz + wy$ & $w^2y + w$\\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Small instances of multivariate multicycle (MM) codes with stabilizer weight $w = 6$.} Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}.}}
\label{tab:mmcodetable3}
\end{table*}

\begin{table*}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$, m, p, q} & $\frac{kd^2}{n}$ & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} \\
\hline
\rowcolor{A} $[[144, 12, (8,8)]]$ &  3,2,2,2 & 5.3 & $w^2xyz + wy + yz$ & $w^2y + wxz + x$ & $w^2x + wz + xy$ & $w^2yz + w + 1$\\
\hline
\rowcolor{B} $[[144, 12, (12,12)]]$ &  3,2,2,2 & 12.0  & $w^2xy + w + xyz$ & $w^2x + wxy + z$ & $w^2xz + wx + xyz$ & $w^2yz + wxy + xyz$\\
\hline
\rowcolor{C} $[[144, 24, (6,6)]]$ &  3,2,2,2 & 6.0 & $w^2yz + wz + 1$ & $w^2x + wz + z$ & $w^2z + wxz + yz$ & $w^2x + wxyz + xz$\\
\hline
\rowcolor{D} $[[216, 12, (12, 12)]]$ &  3,2,2,2 & 8.0 & $wx^2z + wx + x^2y$ & $w^2xy + wx^2z + yz$ & $w^2xyz + x + 1$ & $w^2x^2z + wx^2y + x^2y$\\
\hline
\rowcolor{E} $[[216, 12, (14, ?)]]$ &  2,3,3,2 & 10.9 & $wz + xy^2 + y$ & $wxz + wy + xy^2z$ & $wy^2 + wz + xy$ & $x^2y + x^2z + xy^2$\\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Small instances of multivariate multicycle (MM) codes with stabilizer weight $w = 9$.} All code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. Notably, the weight-9 MM $[[144,12,12]]$ code achieves higher confinement than the weight-9 $[[144,12,11]]$ quantum Tanner code of~\cite{leverrier2025smallquantumtannercodes} at the same block length and stabilizer weight.}}
\label{tab:mmcodetable4}
\end{table*}

\begin{table*}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[N, K, (d_X, d_Z) ]]$}& \textbf{$\ell$, m, p, q}  & $\frac{kd^2}{n}$ & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} \\
\hline
\rowcolor{C} $[[96, 12, (6,6)]]$ & 2,2,2,2 & 4.5 & $wy + xz + yz + z$ & $wxyz + wxz + w + xyz$ & $xyz + xz + yz + 1$ & $wxy + wy + wz + y$\\
\hline
\rowcolor{D} $[[96, 12, (8,8)]]$ & 2,2,2,2 & 8.0 & $wy + wz + w + 1$ & $wxz + wy + wz + w$ & $wxyz + w + xyz + y$ & $wxyz + w + yz + 1$\\
\hline
\rowcolor{B} $[[144, 6, (6,6)]]$ & 2,3,2,2 & 1.5 & $w + x^2yz + x^2z + yz$ & $x^2y + x + yz + y$ & $wxy + wxz + w + x^2$ & $wx^2y + x^2y + xy + yz$\\
\hline
\rowcolor{C} $[[144, 6, (8,8)]]$ & 3,2,2,2  & 2.7 & $wxyz + wxy + wy + yz$ & $w^2z + xyz + xy + yz$ & $w^2x + w^2yz + w^2 + 1$ & $w^2z + wy + w + xz$\\ 
\hline
\rowcolor{D} $[[144, 6, (10,10)]]$ & 3,2,2,2  & 4.2 & $w^2xz + w^2yz + wx + z$ & $w^2yz + wxz + w + xy$ & $wxy + wy + x + y$ & $wxz + xy + x + z$\\  
\hline
\rowcolor{E} $[[144, 6, (12,12)]]$  & 3,2,2,2  & 6.0 & $w^2 + wyz + wz + w$ & $w^2z + wxy + yz + z$ & $w^2y + wz + w + 1$ & $w^2z + wx + wy + 1$\\ 
\hline
\rowcolor{F} $[[144, 12, (6,6)]]$ & 3,2,2,2  & 3.0 & $w^2z + wyz + w + y$ & $w^2x + w^2 + wxyz + wyz$ & $w^2x + wz + x + y$ & $w^2z + wxy + wz + y$\\ %
\hline
\rowcolor{G} $[[144, 12, (8,8)]]$ & 2,3,2,2 & 5.3 & $x^2z + x^2 + yz + z$ & $wx^2y + w + x^2 + yz$ & $wxy + w + x^2y + z$ & $wxyz + wxy + x^2y + xz$\\ %
\hline
\rowcolor{H} $[[144, 12, (10,10)]]$ & 3,2,2,2 & 8.3 & $w^2x + xyz + x + y$ & $w^2y + wxy + wx + z$ & $w^2yz + w^2 + yz + z$ & $w^2x + wxyz + wyz + 1$\\ %
\hline
\rowcolor{I} $[[144, 12, (12,12)]]$  & 3,2,2,2 & 12.0 & $w^2xz + w^2yz + wy + xy$ & $wxy + wyz + wz + xy$ & $w^2xy + w^2 + wyz + 1$ & $w^2xz + w^2z + wxz + xy$\\ %
\hline
\rowcolor{J} $[[144, 18, (6,6)]]$  & 3,2,2,2 & 4.5 & $w^2xy + w^2yz + w^2 + z$ & $w^2xy + w^2yz + w^2y + yz$ & $w^2xy + wy + w + x$ & $w^2yz + wz + w + xy$\\ %
\hline
\end{tabular}
\editcolor{\caption{\textbf{Multivariate multicycle (MM) codes with block length $n = 144$ and stabilizer weight $w = 12$.} Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}.}}
\label{tab:mmcodetable5}
\end{table*}

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table*}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$, m, p, q} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} \\
\hline
\rowcolor{A} $[[192, 6, (8,8)]]$ & 4,2,2,2  & $w^3xyz + w^3x + wxyz + xy$ & $w^3y + w^3 + w^2 + wyz$ & $w^3z + w^3 + wx + wy$ & $w^3xyz + w^3yz + xz + x$\\ 
\hline
\rowcolor{C} $[[192, 12, (8,8)]]$ & 4,2,2,2 & $w^3xyz + w^2 + wx + yz$ & $w^2xyz + wxyz + wyz + wz$ & $w^3xy + w^3y + w + z$ & $w^2y + wxyz + xz + yz$\\ 
\hline
\rowcolor{D} $[[192, 18, (8,8)]]$ & 2,4,2,2 & $wx^2 + wxz + w + x^3$ & $wx^3z + wxyz + x^3yz + 1$ & $wxy + w + x^2y + z$ & $wx^2y + wxy + xz + x$\\
\hline
\rowcolor{E} $[[192, 18, (12,12)]]$ & 2,2,2,4  & $wxz + wyz^3 + w + xyz^3$ & $wz^2 + wz + x + yz^2$ & $wxz^2 + xyz + xz^2 + z$ & $wxyz^2 + wxyz + wyz^2 + xz^3$\\ 
\hline
\rowcolor{F}$[[192, 24, (6,6)]]$ & 2,2,2,4 & $wxyz^3 + wxy + xyz^3 + yz$ & $wxyz^2 + wyz + wz^2 + z$ & $wxyz + wxz + w + yz^3$ & $wyz^3 + xyz + xy + yz^3$\\
\hline
\rowcolor{G} $[[192, 24, (8,8)]]$ & 4,2,2,2 & $w^3xy + w^2xy + w^2yz + x$ & $w^2x + xy + x + y$ & $w^3xyz + w^3x + wxy + xy$ & $w^3 + wxyz + wy + xy$\\ 
\hline
\end{tabular}
\editcolor{\caption{\textbf{Multivariate multicycle (MM) codes with block length $n = 192$ and stabilizer weight $w = 12$.} Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}.}}
\label{tab:mmcodetable6}
\end{table*}

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} & \textbf{J}\\
\hline
\rowcolor{A} $[[60, 10, (6,3)]]$ & 6  & $1+v$ & $1+v^2$ & $1+v^3$ & $1+v^4$ & $1+v^5$\\ 
\hline
\rowcolor{B} $[[70, 10, (6,4)]]$ & 7  & $1+v$ & $1+v^2$ & $1+v^4$ & $1+v^5$ & $1+v^6$\\ 
\hline
\rowcolor{C} $[[80, 10, (6,4)]]$ & 8  & $1+v$ & $1+v^2$ & $1+v^3$ & $1+v^4$ & $1+v^7$\\ 
\hline
\rowcolor{D} $[[80, 10, (7,4)]]$ & 8  & $1+v$ & $1+v^3$ & $1+v^5$ & $1+v^6$ & $1+v^7$\\ 
\hline
\rowcolor{E} $[[90, 10, (6,4)]]$ & 9  & $1+v$ & $1+v^2$ & $1+v^5$ & $1+v^7$ & $1+v^8$\\ 
\hline
\rowcolor{F} $[[90, 10, (7,4)]]$ & 9  & $1+v$ & $1+v^2$ & $1+v^3$ & $1+v^4$ & $1+v^5$\\ 
\hline
\rowcolor{G} $[[100, 10, (8,4)]]$ & 10  & $1+v^3$ & $1+v^5$ & $1+v^6$ & $1+v^8$ & $1+v^9$\\ 
\hline
\rowcolor{H} $[[110, 10, (10,6)]]$ & 11  & $1+v^2$ & $1+v^6$ & $1+v^7$ & $1+v^8$ & $1+v^{10}$\\ 
\hline
\rowcolor{I} $[[120, 10, (9,6)]]$ & 12  & $1+v^4$ & $1+v^5$ & $1+v^9$ & $1+v^{10}$ & $1+v^{11}$\\ 
\hline
\rowcolor{K} $[[130, 10, (10,6)]]$ & 13  & $1+v^2$ & $1+v^5$ & $1+v^7$ & $1+v^9$ & $1+v^{10}$\\ 
\hline
\rowcolor{K} $[[140, 10, (10,6)]]$ & 14  & $1+v^4$ & $1+v^5$ & $1+v^8$ & $1+v^{11}$ & $1+v^{13}$\\ 
\hline
\rowcolor{L} $[[150, 10, (10,6)]]$ & 15  & $1+v^2$ & $1+v^4$ & $1+v^8$ & $1+v^{9}$ & $1+v^{14}$\\ 
\hline
\rowcolor{M} $[[150, 10, (12,6)]]$ & 15  & $1+v^5$ & $1+v^8$ & $1+v^{11}$ & $1+v^{13}$ & $1+v^{14}$\\ 
\hline
\rowcolor{N} $[[160, 10, (10,6)]]$ & 16  & $1+v^2$ & $1+v^4$ & $1+v^{7}$ & $1+v^{13}$ & $1+v^{15}$\\ 
\hline
\rowcolor{O} $[[160, 10, (11,6)]]$ & 16 & $1+v$ & $1+v^6$ & $1+v^{9}$ & $1+v^{13}$ & $1+v^{14}$\\ 
\hline
\rowcolor{P} $[[170, 10, (10,5)]]$ & 17 & $1+v^4$ & $1+v^7$ & $1+v^{12}$ & $1+v^{13}$ & $1+v^{15}$\\ 
\hline
\rowcolor{Q} $[[170, 10, (12,6)]]$ & 17  & $1+v$ & $1+v^4$ & $1+v^{7}$ & $1+v^{12}$ & $1+v^{15}$\\ 
\hline
\rowcolor{R} $[[180, 10, (12,7)]]$ & 18  & $1+v$ & $1+v^5$ & $1+v^{10}$ & $1+v^{11}$ & $1+v^{14}$\\ 
\hline
\rowcolor{S} $[[180, 10, (12,7)]]$ & 18 & $1+v$ & $1+v^5$ & $1+v^{10}$ & $1+v^{11}$ & $1+v^{14}$\\ 
\hline
\rowcolor{T} $[[190, 10, (12,7)]]$ & 19  & $1+v$ & $1+v^4$ & $1+v^{5}$ & $1+v^{11}$ & $1+v^{17}$\\ 
\hline
\rowcolor{U} $[[190, 10, (13,7)]]$ & 19  & $1+v^4$ & $1+v^{10}$ & $1+v^{12}$ & $1+v^{16}$ & $1+v^{18}$\\ 
\hline
\rowcolor{V} $[[200, 10, (12,7)]]$ & 20  & $1+v$ & $1+v^2$ & $1+v^{9}$ & $1+v^{14}$ & $1+v^{15}$\\ 
\hline
\rowcolor{W} $[[200, 10, (13,6)]]$ & 20  & $1+v^2$ & $1+v^3$ & $1+v^{7}$ & $1+v^{11}$ & $1+v^{12}$\\ 
\hline
\rowcolor{X} $[[210, 10, (13,6)]]$ & 21  & $1+v^5$ & $1+v^{12}$ & $1+v^{13}$ & $1+v^{18}$ & $1+v^{20}$\\ 
\hline
\rowcolor{Y} $[[210, 10, (14,7)]]$ & 21  & $1+v^3$ & $1+v^4$ & $1+v^{8}$ & $1+v^{10}$ & $1+v^{15}$\\ 
\hline
\rowcolor{Z} $[[220, 10, (13,6)]]$ & 22  & $1+v^{1}$ & $1+v^{2}$ & $1+v^{9}$ & $1+v^{14}$ & $1+v^{19}$\\ 
\hline
\rowcolor{A} $[[220, 10, (14,7)]]$ & 22  & $1+v^{4}$ & $1+v^{7}$ & $1+v^{10}$ & $1+v^{19}$ & $1+v^{21}$\\ 
\hline
\rowcolor{B} $[[230, 10, (13,7)]]$ & 23  & $1+v^{3}$ & $1+v^{5}$ & $1+v^{12}$ & $1+v^{16}$ & $1+v^{19}$\\ 
\hline
\rowcolor{C} $[[230, 10, (14,6)]]$ & 23  & $1+v^{5}$ & $1+v^{6}$ & $1+v^{9}$ & $1+v^{11}$ & $1+v^{21}$\\ 
\hline
\rowcolor{D} $[[240, 10, (13,7)]]$ & 24 & $1+v^{4}$ & $1+v^{7}$ & $1+v^{13}$ & $1+v^{21}$ & $1+v^{23}$\\
\hline
\rowcolor{E} $[[240, 10, (14,7)]]$ & 24  & $1+v$ & $1+v^{9}$ & $1+v^{10}$ & $1+v^{17}$ & $1+v^{20}$\\
\hline
\rowcolor{F} $[[250, 10, (13,7)]]$ & 25 & $1+v^8$ & $1+v^{11}$ & $1+v^{12}$ & $1+v^{15}$ & $1+v^{23}$\\ 
\hline
\rowcolor{G} $[[250, 10, (14,7)]]$ & 25 & $1+v^4$ & $1+v^5$ & $1+v^{6}$ & $1+v^{9}$ & $1+v^{14}$\\ 
\hline
\rowcolor{H} $[[250, 10, (15,8)]]$ & 25  & $1+v$ & $1+v^4$ & $1+v^{9}$ & $1+v^{15}$ & $1+v^{22}$\\ 
\hline
\rowcolor{I} $[[250, 10, (16,8)]]$ & 25  & $1+v^4$ & $1+v^9$ & $1+v^{13}$ & $1+v^{14}$ & $1+v^{19}$\\ 
\hline
\rowcolor{J} $[[260, 10, (13,7)]]$ & 26 & $1+v^5$ & $1+v^6$ & $1+v^{9}$ & $1+v^{12}$ & $1+v^{24}$\\ 
\hline
\rowcolor{K} $[[260, 10, (14,7)]]$ & 26 & $1+v^8$ & $1+v^{15}$ & $1+v^{20}$ & $1+v^{23}$ & $1+v^{25}$\\ 
\hline
\rowcolor{L} $[[260, 10, (15,8)]]$ & 26  & $1+v^3$ & $1+v^5$ & $1+v^{6}$ & $1+v^{7}$ & $1+v^{11}$\\ 
\hline
\rowcolor{M} $[[270, 10, (13,7)]]$ & 27  & $1+v^7$ & $1+v^{10}$ & $1+v^{11}$ & $1+v^{12}$ & $1+v^{23}$\\ 
\hline
\rowcolor{N} $[[270, 10, (15,6)]]$ & 27  & $1+v^5$ & $1+v^{8}$ & $1+v^{18}$ & $1+v^{24}$ & $1+v^{26}$\\ 
\hline
\rowcolor{O} $[[270, 10, (16,7)]]$ & 27 & $1+v^5$ & $1+v^6$ & $1+v^{17}$ & $1+v^{18}$ & $1+v^{20}$\\ 
\hline
\rowcolor{P} $[[280, 10, (14,7)]]$ & 28  & $1+v^5$ & $1+v^{11}$ & $1+v^{16}$ & $1+v^{19}$ & $1+v^{22}$\\ 
\hline
\rowcolor{Q} $[[280, 10, (15,7)]]$ & 28  & $1+v^3$ & $1+v^{5}$ & $1+v^{15}$ & $1+v^{16}$ & $1+v^{26}$\\ 
\hline
\rowcolor{R} $[[290, 10, (15,8)]]$ & 29  & $1+v$ & $1+v^{3}$ & $1+v^{5}$ & $1+v^{6}$ & $1+v^{15}$\\ 
\hline
\end{tabular}
\editcolor{\caption{\textbf{Collapsed 5-dimensional (5D) multivariate multicycle (MM) codes.} The multivariate quotient ring polynomials $F(v,w,x,y,z)$, $G(v,w,x,y,z)$, $H(v,w,x,y,z)$, $I(v,w,x,y,z)$ and $J(v,w,x,y,z)$ are defined over $\mathbb{F}_2[v,w, x, y, z]/\langle v^\ell-1, w-1, x-1, y-1, z-1 \rangle$. Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. Notably, all these 5D MM code have weight-8 X stabilizer checks and weight-6 Z stabilizer checks.}}
\label{tab:mmcodetable7}
\end{table}


\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} & \textbf{J} &  \textbf{K}\\
\hline
\rowcolor{A} $[[140, 20, (6,6)]]$ & 7  & $1+u$ & $1+u^2$ & $1+u^3$ & $1+u^4$ & $1+u^5$ & $1+u^6$ \\ 
\hline
\rowcolor{B} $[[160, 20, (6,6)]]$ & 8  & $1+u$ & $1+u^3$ & $1+u^4$ & $1+u^5$ & $1+u^6$ & $1+u^7$ \\ 
\hline
\rowcolor{C} $[[180, 20, (6,6)]]$ & 9  & $1+u$ & $1+u^3$ & $1+u^4$ & $1+u^6$ & $1+u^7$ & $1+u^8$ \\ 
\hline
\rowcolor{D} $[[200, 20, (7,7)]]$ & 10  & $1+u$ & $1+u^3$ & $1+u^6$ & $1+u^7$ & $1+u^8$ & $1+u^9$ \\ 
\hline
\rowcolor{E} $[[220, 20, (8,8)]]$ & 11  & $1+u$ & $1+u^3$ & $1+u^5$ & $1+u^7$ & $1+u^8$ & $1+u^{10}$ \\ 
\hline
\rowcolor{F} $[[240, 20, (8,8)]]$ & 12  & $1+u$ & $1+u^2$ & $1+u^3$ & $1+u^5$ & $1+u^9$ & $1+u^{10}$ \\ 
\hline
\rowcolor{G} $[[260, 20, (10,10)]]$ & 13  & $1+u^5$ & $1+u^7$ & $1+u^9$ & $1+u^{10}$ & $1+u^{11}$ & $1+u^{12}$ \\ 
\hline
\rowcolor{H} $[[280, 20, (9,9)]]$ & 14  & $1+u$ & $1+u^2$ & $1+u^3$ & $1+u^4$ & $1+u^{5}$ & $1+u^{11}$ \\
\hline
\rowcolor{I} $[[300, 20, (9,9)]]$ & 15  & $1+u$ & $1+u^3$ & $1+u^5$ & $1+u^6$ & $1+u^{8}$ & $1+u^{11}$ \\
\hline
\rowcolor{J} $[[320, 20, (9,9)]]$ & 16  & $1+u$ & $1+u^3$ & $1+u^5$ & $1+u^7$ & $1+u^{10}$ & $1+u^{15}$ \\
\hline
\rowcolor{K} $[[340, 20, (10,10)]]$ & 17  & $1+u^5$ & $1+u^7$ & $1+u^9$ & $1+u^{11}$ & $1+u^{14}$ & $1+u^{15}$ \\
\hline
\rowcolor{L} $[[360, 20, (10,10)]]$ & 18  & $1+u^3$ & $1+u^7$ & $1+u^{11}$ & $1+u^{13}$ & $1+u^{16}$ & $1+u^{17}$ \\
\hline
\rowcolor{M} $[[380, 20, (10,10)]]$ & 19  & $1+u^2$ & $1+u^3$ & $1+u^5$ & $1+u^{6}$ & $1+u^{11}$ & $1+u^{12}$ \\
\hline
\rowcolor{N} $[[380, 20, (11,11)]]$ & 19  & $1+u^9$ & $1+u^{13}$ & $1+u^{14}$ & $1+u^{15}$ & $1+u^{16}$ & $1+u^{17}$ \\
\hline
\rowcolor{P} $[[400, 20,(10,10)]]$ & 20  & $1+u^{8}$ & $1+u^{11}$ & $1+u^{14}$ & $1+u^{15}$ & $1+u^{17}$ & $1+u^{18}$ \\
\hline
\rowcolor{Q} $[[400, 20,(11,11)]]$ & 20  & $1+u^{3}$ & $1+u^{7}$ & $1+u^{11}$ & $1+u^{14}$ & $1+u^{16}$ & $1+u^{19}$ \\
\hline
\rowcolor{R} $[[400, 20,(12,12)]]$ & 20 & $1+u$ & $1+u^{2}$ & $1+u^{3}$ & $1+u^{6}$ & $1+u^{9}$ & $1+u^{15}$ \\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Collapsed 6-dimensional (6D) multivariate multicycle (MM) codes.} The multivariate quotient ring polynomials $F(u,v,w,x,y,z)$, $G(u,v,w,x,y,z)$, $H(u,v,w,x,y,z)$, $I(u,v,w,x,y,z)$, $J(u,v,w,x,y,z)$ and $K(u,v,w,x,y,z)$ are defined over $\mathbb{F}_2[u,v,w, x, y, z]/\langle u^\ell-1,v-1, w-1, x-1, y-1, z-1 \rangle$. Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. All these 6D MM code have weight-8 X stabilizer checks and weight-8 Z stabilizer checks.}}
\label{tab:mmcodetable8}
\end{table}

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table}
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} & \textbf{J} &  \textbf{K} &  \textbf{L}\\
\hline
\rowcolor{A} $[[280, 35, (8,4)]]$ & 8  & $1+t$ & $1+t^2$ & $1+t^3$ & $1+t^4$ & $1+t^5$ & $1+t^6$ & $1+t^7$\\ 
\hline
\rowcolor{B} $[[315, 35, (9,6)]]$ & 9  & $1+t$ & $1+t^2$ & $1+t^3$ & $1+t^4$ & $1+t^5$ & $1+t^7$ & $1+t^8$\\ 
\hline
\rowcolor{C} $[[350, 35, (10,6)]]$ & 10  & $1+t$ & $1+t^2$ & $1+t^3$ & $1+t^4$ & $1+t^5$ & $1+t^7$ & $1+t^9$\\ 
\hline
\rowcolor{D} $[[385, 35, (10,7)]]$ & 11  & $1+t$ & $1+t^2$ & $1+t^3$ & $1+t^4$ & $1+t^6$ & $1+t^8$ & $1+t^{10}$\\
\hline
\rowcolor{E} $[[420, 35, (10,6)]]$ & 12  & $1+t$ & $1+t^3$ & $1+t^6$ & $1+t^7$ & $1+t^8$ & $1+t^{10}$ & $1+t^{11}$\\
\hline
\rowcolor{F} $[[455, 35, (10,7)]]$ & 13  & $1+t^2$ & $1+t^6$ & $1+t^7$ & $1+t^8$ & $1+t^9$ & $1+t^{10}$ & $1+t^{11}$\\
\hline
\rowcolor{G} $[[455, 35, (11,7)]]$ & 13  & $1+t$ & $1+t^2$ & $1+t^4$ & $1+t^5$ & $1+t^6$ & $1+t^{10}$ & $1+t^{12}$\\
\hline
\rowcolor{I} $[[490, 35, (11,7)]]$ & 14  & $1+t$ & $1+t^3$ & $1+t^7$ & $1+t^8$ & $1+t^9$ & $1+t^{11}$ & $1+t^{12}$\\
\hline
\rowcolor{J} $[[490, 35, (12,7)]]$ & 14  & $1+t^3$ & $1+t^4$ & $1+t^6$ & $1+t^7$ & $1+t^9$ & $1+t^{12}$ & $1+t^{13}$\\
\hline
\rowcolor{K} $[[525, 35, (12,8)]]$ & 15  & $1+t^2$ & $1+t^3$ & $1+t^6$ & $1+t^7$ & $1+t^8$ & $1+t^{10}$ & $1+t^{11}$\\
\hline
\rowcolor{L} $[[490, 35, (15,9)]]$ & 15  & $1+t$ & $1+t^2$ & $1+t^3$ & $1+t^4$ & $1+t^5$ & $1+t^{8}$ & $1+t^{9}$\\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Collapsed 7-dimensional (7D) multivariate multicycle (MM) codes.} The multivariate quotient ring polynomials $F(t,u,v,w,x,y,z)$, $G(t,u,v,w,x,y,z)$, $H(t,u,v,w,x,y,z)$, $I(t,u,v,w,x,y,z)$, $J(t,u,v,w,x,y,z)$ and $K(t,u,v,w,x,y,z)$ are defined over $\mathbb{F}_2[t,u,v,w, x, y, z]/\langle t^\ell-1,u-1,v-1, w-1, x-1, y-1, z-1 \rangle$. Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. All these 5D MM code have weight-10 X stabilizer checks and weight-8 Z stabilizer checks.}}
\label{tab:mmcodetable9}
\end{table}

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table*}[h!]
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} & \textbf{J} &  \textbf{K} &  \textbf{L} & \textbf{M}\\
\hline
\rowcolor{A} $[[630, 70, (9,9)]]$ & 9  & $1+s$ & $1+s^2$ & $1+s^3$ & $1+s^4$ & $1+s^5$ & $1+s^6$ & $1+s^7$ & $1+s^8$\\ 
\hline
\rowcolor{B} $[[700, 70, (8,8)]]$ & 10  & $1+s$ & $1+s^2$ & $1+s^3$ & $1+s^4$ & $1+s^5$ & $1+s^6$ & $1+s^7$ & $1+s^9$\\ 
\hline
\rowcolor{C} $[[770, 70, (9,9)]]$ & 11  & $1+s$ & $1+s^2$ & $1+s^4$ & $1+s^6$ & $1+s^7$ & $1+s^8$ & $1+s^{9}$ & $1+s^{10}$\\ 
\hline
\rowcolor{D} $[[840, 70, (6,6)]]$ & 12  & $1+s$ & $1+s^2$ & $1+s^3$ & $1+s^4$ & $1+s^5$ & $1+s^7$ & $1+s^{8}$ & $1+s^{10}$\\
\hline
\end{tabular}
\editcolor{\caption{\textbf{Collapsed 8-dimensional (8D) multivariate multicycle (MM) codes.} The multivariate quotient ring polynomials $F(s,t,u,v,w,x,y,z)$, $G(s,t,u,v,w,x,y,z)$, $H(s,t,u,v,w,x,y,z)$, $I(s,t,u,v,w,x,y,z)$, $J(s,t,u,v,w,x,y,z)$, $K(s,t,u,v,w,x,y,z)$, $L(s,t,u,v,w,x,y,z)$ and $M(s,t,u,v,w,x,y,z)$ are defined over $\mathbb{F}_2[s,t,u,v,w, x, y, z]/\langle s^\ell-1, t-1,u-1,v-1, w-1, x-1, y-1, z-1 \rangle$. Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. All these 5D MM code have weight-10 X stabilizer checks and weight-8 Z stabilizer checks.}}
\label{tab:mmcodetable10}
\end{table*}

\renewcommand{\arraystretch}{1}
\setlength{\tabcolsep}{0.005pt}
\begin{table*}[h!]
\centering
\definecolor{A}{RGB}{248, 248, 255}
\definecolor{B}{RGB}{245, 255, 250}
\definecolor{C}{RGB}{255, 250, 240}
\definecolor{D}{RGB}{240, 248, 255}
\definecolor{E}{RGB}{255, 245, 238}
\definecolor{F}{RGB}{240, 255, 240}
\definecolor{G}{RGB}{255, 250, 250}
\definecolor{H}{RGB}{248, 248, 247}
\definecolor{I}{RGB}{245, 245, 245}
\definecolor{J}{RGB}{250, 250, 250}
\definecolor{K}{RGB}{245, 248, 255}
\definecolor{L}{RGB}{255, 248, 240}
\definecolor{M}{RGB}{240, 252, 245}
\definecolor{N}{RGB}{250, 245, 255}
\definecolor{O}{RGB}{245, 250, 255}
\definecolor{P}{RGB}{255, 252, 245}
\definecolor{Q}{RGB}{248, 255, 248}
\definecolor{R}{RGB}{255, 248, 248}
\definecolor{S}{RGB}{245, 250, 245}
\definecolor{T}{RGB}{250, 248, 255}
\definecolor{U}{RGB}{252, 252, 250}
\definecolor{V}{RGB}{248, 250, 255}
\definecolor{W}{RGB}{255, 252, 250}
\definecolor{X}{RGB}{250, 252, 248}
\definecolor{Y}{RGB}{252, 250, 248}
\definecolor{Z}{RGB}{248, 252, 250}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\textbf{$[[n, k, (d_X, d_Z) ]]$} & \textbf{$\ell$} & \textbf{F} & \textbf{G} & \textbf{H} & \textbf{I} & \textbf{J} &  \textbf{K} &  \textbf{L} & \textbf{M}  & \textbf{M}\\
\hline
\rowcolor{A} $[[1260, 126, (10,5)]]$ & 9  & $1+u$ & $1+u^2$ & $1+u^3$ & $1+u^4$ & $1+u^5$ & $1+u^6$ & $1+u^7$ & $1+u^8$ & $1+u^9$\\ 
\hline
\end{tabular}
\editcolor{\caption{\textbf{Collapsed 9-dimensional (9D) multivariate multicycle (MM) codes.} Code distances were computed using the \texttt{Gurobi} optimizer~\cite{optimization2020gurobi} via the mixed-integer 
programming method of Ref.~\cite{landahl2011color} as implemented in \texttt{QuantumClifford.jl}. All these 5D MM code have weight-10 X stabilizer checks and weight-8 Z stabilizer checks.}}
\label{tab:mmcodetable11}
\end{table*}
\end{document}
