Bivariate Bicycle Codes over Group Algebras
Abstract
We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D3, a pair of weighted permutation shifts generates a regular C6× C12 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 A4× C6, 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 𝔽2[C3] 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.
Related
- Spectral Theory of Semisimple Bivariate Bicycle Codes
- Logical Operator Decomposition for Distance Analysis of Bivariate Bicycle Codes
- Metachecks in Bivariate Bicycle Codes: Syndrome Distance, Measurement Faults, and Repair Limits
- Quantum bivariate bicycle codes with weight-8 checks surpassing the BB benchmark
- Large Language Model-Guided Discovery of Weight-Five Bivariate Bicycle Codes
BibTeX
@misc{liu2026bivariate,
title = {{Bivariate Bicycle Codes over Group Algebras}},
author = {Chaobin Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.36213},
eprint = {2609.36213},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.36213},
}
RIS
TY - UNPB
TI - Bivariate Bicycle Codes over Group Algebras
AU - Chaobin Liu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.36213
UR - https://arxiv.org/abs/2609.36213
AB - We study a weighted-shift formulation of bivariate bicycle group-algebra (BBGA) codes over finite group algebras. One binary shift acts by right multiplication and the other by left multiplication; associativity ensures that these shifts commute even for a nonabelian seed group. Bivariate polynomials in the shifts, including mixed monomials, therefore define CSS codes. We give explicit sparsity bounds and distinguish nonabelian seed groups from codes that are intrinsically inequivalent to abelian realizations. For D_3, a pair of weighted permutation shifts generates a regular C_6× 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× 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 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.
ER -