G-equivalence in group algebras and minimal abelian codes

TL;DR

This paper characterizes when minimal abelian codes in group algebras are G-equivalent, providing necessary and sufficient conditions and correcting previous results in the literature.

Contribution

It establishes a precise criterion for G-equivalence of minimal abelian codes and rectifies earlier inaccuracies in the field.

Findings

Provides a necessary and sufficient condition for G-equivalence.

Corrects previous results in the literature.

Enhances understanding of the structure of minimal abelian codes.

Abstract

Let G be a finite abelian group and F a field such that char(F) does not divide |G|. Denote by FG the group algebra of G over F. A (semisimple) abelian code is an ideal of FG. Two codes I and J of FG are G-equivalent if there exists an automorphism of G whose linear extension to FG maps I onto J In this paper we give a necessary and sufficient condition for minimal abelian codes to be G-equivalent and show how to correct some results in the literature.