On the number of non-G-equivalent minimal abelian codes

Abstract

Let G be a finite abelian group. Ferraz, Guerreiro, and Polcino Milies (2014) proved that the number of G-equivalence classes of minimal abelian codes is equal to the number of G-isomorphism classes of subgroups for which corresponding quotients are cyclic. In this article, we prove that the notion of G-isomorphism is equivalent to the notion of isomorphism on the set of all subgroups H of G with the property that G/H is cyclic. As an application, we calculate the number of non-G-equivalent minimal abelian codes for some specific family of abelian groups. We also prove that the number of non-G-equivalent minimal abelian codes is equal to the number of divisors of the exponent of G if and only if for each prime p dividing the order of G, the Sylow p-subgroups of G are homocyclic.