A characterization of abelian group codes in terms of their parameters

Abstract

In 1979, Miller proved that for a group G of odd order, two minimal group codes in F(2)G are G-equivalent if and only if they have identical weight distribution. In 2014, Ferraz-Guerreiro-Polcino Milies disproved Miller's result by giving an example of two non- G-equivalent minimal codes with identical weight distribution. In this paper, we give a characterization of finite abelian groups so that over a specific set of group codes, equality of important parameters of two codes implies the G-equivalence of these two codes. As a corollary, we prove that two minimal codes with the same weight distribution are G-equivalent if and only if for each prime divisor p of |G|, the Sylow p-subgroup of G is homocyclic.