Constructing Good Abelian Codes via Shift Bounds and Genetic Algorithms

Abstract: 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.