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

## TL;DR

This paper explores the classification of minimal abelian codes in finite abelian groups, establishing equivalences between different subgroup notions and providing formulas for counting non-equivalent codes based on group structure.

## Contribution

It proves the equivalence of G-isomorphism and isomorphism on certain subgroups, and derives formulas for counting non-G-equivalent minimal abelian codes.

## Key findings

- Number of non-G-equivalent minimal abelian codes equals the number of divisors of the group's exponent under certain conditions.
- Established the equivalence between G-isomorphism and subgroup isomorphism for cyclic quotient conditions.
- Calculated the number of such codes for specific classes of abelian groups.

## Abstract

Let $G$ be a finite abelian group. Ferraz, Guerreiro and Polcino Milies prove 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 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.