群代数上的双变量自行车码
Bivariate Bicycle Codes over Group Algebras
- Bowie State University（鲍伊州立大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出加权移位表述的双变量自行车群代数码，扩展至非阿贝尔种子群，给出稀疏性界限，并构造了不等价于阿贝尔实现的码，如$[[144,16,12]]$和$[[18,4,3]]$码，证明该表述超越普通2BGA类。
AI中文摘要：
我们研究了有限群代数上双变量自行车群代数（BBGA）码的加权移位表述。一个二元移位通过右乘作用，另一个通过左乘作用；结合性保证了即使对于非阿贝尔种子群，这些移位也交换。因此，移位中的双变量多项式（包括混合单项式）定义了CSS码。我们给出了显式的稀疏性界限，并区分了非阿贝尔种子群与本质上不等价于阿贝尔实现的码。对于$D_3$，一对加权置换移位生成正则$C_6\times C_{12}$作用，并恢复了已知的$[[144,12,12]]$和$[[144,14,14]]$双变量自行车码。我们还提出了一个计算机验证的$[[144,16,12]]$码，基于$A_4\times C_6$，每个校验权重为八。该示例属于已建立的非阿贝尔双块群代数族。穷举低权重枚举和内在稳定子支撑不变量证明它与任何普通阿贝尔BB或双块群代数实现不等价，即使在量子比特置换与局部Clifford操作组合下也是如此。最后，在$\mathbb{F}_2[C_3]$上的非均匀加权移位构造产生了一个$[[18,4,3]]$码，校验权重为六和八。对其最小稳定子支撑的精确枚举排除了在该长度下任何普通二元2BGA实现，在相同等价条件下。这确立了无限制加权移位表述扩展到了普通2BGA类之外，即使种子群是阿贝尔的。
英文摘要：
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.
