Bivariate bicycle (BB) codes are a prominent class of quantum LDPC codes constructed from group algebras. While the logical dimension and quantum distance of \emph{coprime} BB codes are known to be determined by a greatest common divisor polynomial $g(z)$, the properties governing their fault tolerance under noisy measurement have remained implicit. In this work, we prove that this same polynomial $g(z)$ dictates the code's stabilizer redundancy and the structure of the classical \emph{syndrome codes} required for single-shot decoding. We derive a strict equality between the quantum rate and the stabilizer redundancy density, and we provide BCH-like bounds on the achievable single-shot measurement error tolerance. Guided by this framework, we construct small coprime BB codes with significantly improved syndrome distance ($d_S$) and evaluate them using BP+OSD. Our analysis reveals a structural bottleneck: within the coprime BB ansatz, high quantum rate imposes an upper bound on syndrome distance, limiting single-shot performance. These results provide concrete algebraic design rules for next-generation 2BGA codes in measurement-limited architectures.

翻译：双变量自行车（BB）码是一类从群代数构造的量子LDPC码。虽然已知互质BB码的逻辑维度和量子距离由最大公约数多项式$g(z)$决定，但其在含噪测量下的容错特性一直未得到明确阐释。本文证明该多项式$g(z)$同样决定了码的稳定子冗余度以及单次解码所需经典校验码的结构。我们严格推导了量子码率与稳定子冗余密度之间的等式关系，并给出了类似BCH码界限的单次测量容错能力理论界。在此框架指导下，我们构造了具有显著提升的校验距离（$d_S$）的小规模互质BB码，并采用BP+OSD方法进行评估。分析揭示了一个结构性瓶颈：在互质BB码的构造框架内，高量子码率会对校验距离施加上限，从而限制单次解码性能。这些结果为面向测量受限架构的新一代2BGA码提供了具体的代数设计准则。