发表机构
The Department of Basic Science, Air Force Engineering University（空军工程大学基础部）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出权重为8校验的二元自行车码，通过代数结构理论和搜索流水线构造新码，在n=144和n=72时超越BB基准的距离或编码率，并量化了较重校验的阈值代价。
AI 中文摘要
Bravyi等人提出的二元自行车（BB）码是权重为6校验的量子低密度奇偶校验码，以[[144,12,12]]为例，其kd^2/n=12。我们发展了具有权重为8校验（权重为4的生成多项式）的BB型码的代数结构理论，并将其与经过精确验证的搜索流水线相结合，用于构造和认证新码。我们证明了精确的维度公式k=2dim R/(A,B)（迫使k为偶数）、生成元对上的4ℓm元素对称群、X/Z距离相等性d_X=d_Z，以及一族子群-陪集核向量，这些向量给出了严格的距离上界和高距离构造的设计规则；所有距离均由交叉验证的位掩码验证器穷举计算。在n=144时，该流水线返回53个码的普查结果，其中最强的成员超越了BB基准：[[144,6,d≥15]]超过了基准距离12（认证d≥15），[[144,10,12]]以权重为8的校验达到了该距离，而[[144,16,10]]在kd^2/n=11.11（比基准低7.4%）的情况下编码了多三分之一的逻辑量子比特，且解码性能不差。在n=72时，[[72,14,8]]达到了kd^2/n=12.44——是相同长度BB码的两倍多——并且解码更好；电路级内存实验将我们的权重为8的码置于约0.1%的伪阈值，而在相同模型下BB参考码约为0.4%，这量化了较重校验的阈值代价。所有结构陈述均在整体普查上进行了数值验证。
英文摘要
Bivariate bicycle (BB) codes of Bravyi \emph{et al.}~\cite{Bravyi2024} are quantum low-density parity-check codes with weight-$6$ checks, exemplified by $[[144,12,12]]$ with $kd^2/n=12$. We develop the algebraic structure theory of BB-type codes with weight-$8$ checks (weight-$4$ generator polynomials) and use it, together with an exactly validated search pipeline, to construct and certify new codes. We prove an exact dimension formula $k=2\dim R/(A,B)$ (forcing even $k$), a $4\ell m$-element symmetry group on generator pairs, an $X/Z$ distance equality $d_X=d_Z$, and a family of subgroup-coset kernel vectors giving rigorous distance upper bounds and a design rule for high-distance constructions; all distances are computed exhaustively by a cross-validated bit-mask verifier. At $n=144$ the pipeline returns a census of $53$ codes whose strongest members surpass the BB benchmark: $[[144,6,d\ge 15]]$ exceeds the benchmark distance $12$ (certified $d\ge 15$), $[[144,10,12]]$ reaches it with weight-$8$ checks, and $[[144,16,10]]$ encodes a third more logical qubits at $kd^2/n=11.11$ ($7.4\%$ below benchmark) while decoding no worse. At $n=72$, $[[72,14,8]]$ attains $kd^2/n=12.44$---more than twice the same-length BB code---and decodes better; a circuit-level memory experiment places our weight-$8$ codes at $\approx 0.1\%$ pseudo-threshold versus $\approx 0.4\%$ for the BB reference under an identical model, quantifying the threshold cost of the heavier checks. All structural statements are verified numerically on the whole census.
