A characterization of abelian group codes in terms of their parameters

TL;DR

This paper characterizes when minimal abelian group codes are $G$-equivalent based on their parameters, refining previous results by linking code equivalence to Sylow subgroup structures.

Contribution

It provides a new characterization of abelian groups where code parameters determine $G$-equivalence, especially relating to Sylow $p$-subgroups.

Findings

Two minimal codes with the same weight distribution are $G$-equivalent iff Sylow $p$-subgroups are homocyclic.

Disproves Miller's 1979 result by providing counterexamples.

Establishes conditions under which code parameters imply $G$-equivalence.

Abstract

In 1979, Miller proved that for a group $G$ of odd order, two minimal group codes in $\mathbb{F}_2G$ are $G$-equivalent if and only they have identical weight distribution. In 2014, Ferraz-Guerreiro-Polcino Milies disprove 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.