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This research advances quantum error correction by introducing weighted-shift formulations for bivariate bicycle group-algebra codes over nonabelian seed groups. The study demonstrates how these constructions generate CSS codes with specific sparsity bounds that are intrinsically inequivalent to standard abelian realizations, even under local Clifford operations and qubit permutations. By providing explicit examples like the [[144,16,12]] code, the work expands the known landscape of stabilizer codes beyond ordinary two-block group-algebra families.
Bivariate Bicycle Codes over Group Algebras
arXiv:2609.36213v1 Announce Type: new 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 $D_3$, 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.
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 $D_3$, 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.
arXiv:2609.36213 [quant-ph]
(or arXiv:2609.36213v1 [quant-ph] for this version)
https://doi.org/10.48550/arXiv.2609.36213 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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