发表机构
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
本文提出基于多项式环的代数预筛选枚举框架，通过除数驱动搜索构造出高$kd^2/n$的量子Bicycle LDPC码，突破群论搜索局限，得到多组优良参数的量子码并明确其构造边界。
AI 中文摘要
Bicycle（双块循环）量子低密度奇偶校验（LDPC）码包含部分已知最优的小型量子码，但其设计依赖群代数公式，其中码的维数和最小距离仅能通过矩阵计算获取。本文证明，在循环情形下，该构造可简化为多项式环$\text{GF}(2)[x]/(x^l-1)$：自正交性自动满足，量子维数可通过多项式最大公因式直接读取，最小距离可通过Calderbank对应（与$\text{GF}(4)$上加法码的对应）精确验证，从而将码搜索转化为经代数预筛选的枚举过程，可覆盖现有表格未充分涵盖的参数范围。基于该框架的计算机搜索得到了短码$[[42,12,4]]_2$和$[[62,12,4]]_2$，并生成了一系列具有优良品质因数$kd^2/n$的码，包括$kd^2/n=14.85$的$[[66,20,7]]_2$，其块长不到双变量Bicycle码$[[144,12,12]]_2$（$kd^2/n=12$）的一半，此外还得到了$[[46,2,8]]_2$、$[[66,2,9]]_2$、$[[66,4,8]]_2$、$[[66,6,8]]_2$，以及块长$n=90$时的$[[90,16,6]]_2$、$[[90,18,6]]_2$、$[[90,20,6]]_2$。对$n=48$的穷尽搜索明确了该框架的适用边界：我们从最小48元群中构造出$[[48,10,6]]_2$码（Aydin–Tamo–Barg实现需72元群），并证明距离5会导致稳定子秩损失，从而排除了$[[48,10,5]]_2$属于权8对称陪集族的可能。该框架为超越群论搜索能力的Bicycle型量子LDPC码开辟了系统的构造途径，并精确界定了真正陪集论现象的起始位置。
英文摘要
Bicycle (two-block circulant) quantum low-density parity-check (LDPC) codes include some of the best known small quantum codes, yet their design has relied on group-algebra formulations in which the dimension and distance are accessible only through matrix computation. We show that in the cyclic case the construction collapses into the polynomial ring $\F_2[x]/(x^{l}-1)$: self-orthogonality is automatic, the quantum dimension is read off from a polynomial gcd, and the minimum distance is certified exactly through the Calderbank correspondence to additive codes over $\F_4$, turning code search into an algebraically pre-filtered enumeration that reaches parameter regimes poorly covered by existing tables. A computer search based on this framework recovers the short codes $[[42,12,4]]_2$ and $[[62,12,4]]_2$ and produces a family of codes with competitive figure of merit $kd^2/n$, including $[[66,20,7]]_2$ with $kd^2/n=14.85$, above the bivariate bicycle code $[[144,12,12]]_2$ ($kd^2/n=12$) at less than half the block length, together with $[[46,2,8]]_2$, $[[66,2,9]]_2$, $[[66,4,8]]_2$, $[[66,6,8]]_2$ and, at $n=90$, $[[90,16,6]]_2$, $[[90,18,6]]_2$, $[[90,20,6]]_2$. An exhaustive census at $n=48$ delineates the boundary of this picture: we exhibit a $[[48,10,6]]_2$ code from a minimal $48$-element group (the Aydin--Tamo--Barg realization uses $72$ elements), and prove that distance $5$ forces a stabilizer-rank loss, which excludes $[[48,10,5]]_2$ from the weight-$8$ symmetric coset family. The framework thus opens a systematic route to bicycle-type quantum LDPC codes beyond the reach of group-theoretic searches, and identifies exactly where genuinely coset-theoretic phenomena begin.
Commentsv2: Added references on decoding comparison
