基于移位界与遗传算法构造优良阿贝尔码
Constructing Good Abelian Codes via Shift Bounds and Genetic Algorithms
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中文总结 AI 辅助
该研究通过广义移位界推导最小距离下界，结合遗传算法搜索，在F₃、F₄上得到优于Grassl表格的优良阿贝尔码，还可通过Construction X构造更多优良线性码，为发现未知参数线性码提供可行方法。
中文摘要 AI 辅助
本文研究通过有限域上的阿贝尔码构造线性码。利用多元多项式商环的代数结构，我们采用广义移位界推导最小距离的下界，该方法扩展了循环码的经典van Lint-Wilson界。我们显式构造了多个无限族阿贝尔码，包括扩展了已知循环构造的二元和三元情形。为寻找更多参数优良的阿贝尔码，我们应用遗传算法，该算法在以分圆陪集的二进制染色体表示的定义集上进行搜索；适应度函数将计算得到的最小距离与已知最优线性码（BKLC）界进行比较。该搜索在F₃和F₄上得到多个打破纪录的码，优于Grassl表格中的结果。此外，这些码的嵌套结构可应用于Construction X，产生更多参数改进的线性码。结果表明，阿贝尔码结合启发式搜索是发现参数未知的线性码的可行途径。
英文摘要
This paper investigates the construction of linear codes via abelian codes over finite fields. By exploiting the algebraic structure of multivariate polynomial quotient rings, we derive lower bounds on the minimum distance using a generalized shift bound, which extends the classical van Lint-Wilson bound for cyclic codes. Several infinite families of abelian codes are explicitly constructed, including binary and ternary cases that extend previously known cyclic constructions. To find more abelian codes with good parameters, we apply a genetic algorithm that searches over defining sets represented as binary chromosomes of cyclotomic cosets; the fitness function compares the computed minimum distance against the best known linear code (BKLC) bounds. The search yields multiple record-breaking codes over F_3 and F_4, with improvements over Grassl's tables. Furthermore, the nested structure of these codes enables the application of Construction X, yielding additional linear codes with improved parameters. The results demonstrate that abelian codes, combined with heuristic search, form a viable way for discovering linear codes with unknown parameters.
