arXiv:2507.21237 · IEEE Transactions on Information Theory

(2,2)-GB Codes: Classification and Comparison With Weight-4 Surface Codes

François Arnault, Philippe Gaborit, Nicolas Saussay

Published May 7, 2026• 0 citations

Abstract

Generalized Bicycle (GB) codes offer a compelling alternative to surface codes for quantum error correction. This paper focuses on (2,2)-Generalized Bicycle codes, constructed from pairs of binary circulant matrices with two non-zero elements per row. Leveraging a lower bound on their minimum distance, we construct three novel infinite families of optimal (2,2)-GB codes with parameters [[2n2, 2,n]], [[4r2, 2, 2r]], and [[(2t+1)2+1, 2, 2t+1]]. These families match the performance of Kitaev’s toric code and the best 2D weight-4 surface codes, reaching known theoretical limits. In particular, the second family breaks a long-held belief by providing optimal even-distance GB codes, previously deemed impossible. All are CSS codes derived from Cayley graphs. Recognizing that standard equivalence relations do not preserve their CSS structure, we introduce a CSS-preserving equivalence relation for rigorous comparison of Cayley graph-based CSS codes. Under this framework, the first two families are inequivalent to all previously known optimal weight-4 2D surface codes, while the third family is equivalent to the best-known odd-distance 2D surface code. Finally, we classify all extremal, non-equivalent (2, 2)-GB codes with length below 200 and present a comparison table with existing notable 2D weight-4 surface codes.

DOI: 10.1109/tit.2026.3691161

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arXiv:2507.21237 · IEEE Transactions on Information Theory

(2,2)-GB Codes: Classification and Comparison With Weight-4 Surface Codes

François Arnault, Philippe Gaborit, Nicolas Saussay

Published May 7, 2026• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.1109/tit.2026.3691161

Related papers

Distance Bounds for Generalized Bicycle Codes

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Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.Generalized bicycle (GB) codes is a class of quantum error-correcting codes constructed from a pair of binary circulant matrices. Unlike for other simple quantum code ansätze, unrestricted GB codes may have linear distance scaling. In addition, low-density parity-check GB codes have a naturally overcomplete set of low-weight stabilizer generators, which is expected to improve their performance in the presence of syndrome measurement errors. For such GB codes with a given maximum generator weight w, we constructed upper distance bounds by mapping them to codes local in D≤w−1 dimensions, and lower existence bounds which give d≥O(n1/2). We have also conducted an exhaustive enumeration of GB codes for certain prime circulant sizes in a family of two-qubit encoding codes with row weights 4, 6, and 8; the observed distance scaling is consistent with A(w)n1/2+B(w), where n is the code length and A(w) is increasing with w.

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Multivariate Multicycle Codes for Complete Single-Shot Decoding

Feroz Ahmed Mian, Owen Gwilliam, Stefan Krastanov

We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length- $\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. 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 $[[96, 12, 8]]$ , $[[144, 12, 12]]$ , $[[216, 12, 14]]$ , $[[288, 12, 16]]$ , $[[324, 12, 20]]$ , $[[432, 12, 27]]$ , $[[486, 24, 12]]$ , $[[630, 70, 9]]$ , and $[[648, 18, 23]]$ . Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.We introduce multivariate multicycle (MM) codes, a new family of quantum error-correcting codes (QECCs) that unifies bivariate bicycle, multivariate bicycle, abelian two-block group algebra, generalized bicycle, trivariate tricycle, and toric codes. MM codes are Calderbank-Shor-Steane (CSS) codes defined from length-$\textit{t+1}$ chain complexes with $\textit{$t \ge 4$}$. 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 $[[96, 12, 8]]$, $[[144, 12, 12]]$, $[[216, 12, 14]]$, $[[288, 12, 16]]$, $[[324, 12, 20]]$, $[[432, 12, 27]]$, $[[486, 24, 12]]$, $[[630, 70, 9]]$, and $[[648, 18, 23]]$. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.

quant-pharXiv:2601.18879Jan 28, 20260 citations

[[d^2+1,2,d]]

for odd

d \geq 3

and

[[d^2,2,d]]

for even

d \geq 4

, both having the property that each check qubit is connected to exactly four data qubits similar to surface codes. For the odd-distance family, we analyze the structure of low-weight logical Pauli operators and demonstrate the existence of a fault-tolerant logical CNOT gate between the two logical qubits, achievable through a simple relabeling of data qubits. We further construct a syndrome extraction pattern for both families that does not imply minimum distance reduction arising from extraction circuit faults that propagate from the check qubits to the data qubits. Finally, we numerically evaluate their logical error rates under a code capacity depolarizing noise model using the belief propagation ordered statistics decoding (BP-OSD) and minimum-weight perfect-matching (MWPM) decoders, yielding thresholds of approximately

14-16\%

for the odd and even families, very similar to those of rotated surface codes.

D_3

, a pair of weighted permutation shifts generates a regular

C_6\times C_{12}

action and recovers the known

[[144,12,12]]

and

[[144,14,14]]

bivariate bicycle codes. We also present a computer-verified

[[144,16,12]]

code over

A_4\times C_6

, with every check of weight eight. This example belongs to the established nonabelian two-block group-algebra family. Exhaustive low-weight enumeration and an intrinsic stabilizer-support invariant certify that it is inequivalent to any ordinary abelian BB or two-block group-algebra realization, even under qubit permutations combined with local Clifford operations. Finally, a nonuniform weighted-shift construction over

\mathbb{F}_2[C_3]

yields a

[[18,4,3]]

code with check weights six and eight. An exact enumeration of its minimum stabilizer supports rules out every ordinary binary 2BGA realization at that length, under the same equivalence. This establishes that the unrestricted weighted-shift formulation extends beyond the ordinary 2BGA class, even with an abelian seed group.

\textit{t+1}

chain complexes with $\textit{$t \ge 4$}$. 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]]

[[144, 12, 12]]

[[216, 12, 14]]

[[288, 12, 16]]

[[324, 12, 20]]

[[432, 12, 27]]

[[486, 24, 12]]

[[630, 70, 9]]

, and

[[648, 18, 23]]

. Notably, our codes achieve confinement profiles that surpass all known single-shot-decodable quantum CSS codes of practical blocksize. Our codes are also the first explicit instances of collapsed 5D through 9D higher dimensional QECCs, with check weights significantly lower than those of recent small instances of quantum Tanner codes.