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.
AKSU, FATMA ALTUNBULAK and TUVAY, İPEK
"On the number of non-$G$-equivalent minimal abelian codes,"
Turkish Journal of Mathematics: Vol. 45:
1, Article 26.
Available at: https://journals.tubitak.gov.tr/math/vol45/iss1/26