半单二元双变量自行车码的谱理论
Spectral Theory of Semisimple Bivariate Bicycle Codes
AI总结：
该研究扩展二维循环码经典理论，针对双变量自行车码提出代数方法，推导逻辑维度公式与最小距离下界，构建码对称性系统理论，给出无需数值搜索的码生成示例，并将分析扩展至BCH基乘积构造。
AI中文摘要：
我们扩展了二维循环码的经典理论，针对双变量自行车码提出一种代数方法。利用弗罗贝尼乌斯幂等元，推导了逻辑维度公式并建立了最小距离的下界；构建了码对称性的系统理论，以构造坐标置换的结构化单项块子群。多个显式示例展示了如何从头生成这些码，无需依赖数值搜索；附录将分析扩展至基于BCH的乘积构造。
英文摘要：
Extending the classical theory of two-dimensional cyclic codes, we develop an algebraic approach to bivariate bicycle codes. Using Frobenius-orbit idempotents, formulas for logical dimensions are derived and lower bounds on minimum distances are established. A systematic theory of code symmetries is formulated to construct a structured block-monomial subgroup of coordinate permutations. Several explicit examples show how to generate these codes from first principles without relying on numerical searches. An appendix extends the analysis to BCH-based product constructions.
