$G$ -Equivalence in Group Algebras and Minimal Abelian Codes

Let G be a finite Abelian group and BBF a field such that char(BBF ) does not divide |G|. Denote by BBF G the group algebra of G over BBF. A (semisimple) Abelian code is an ideal of BBF G. Two codes ℑ₁ and ℑ₂ of BBF G are G-equivalent if there exists an automorphism ψ of G whose linear extension to BBF G maps ℑ₁ onto ℑ₂. 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.