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

Let
$G$
be a finite Abelian group and
$ {\BBF }$
a field such that
$ \mathop {\rm char}({\BBF }) $
does not divide
$ \vert G\vert $
. Denote by
$ {\BBF } G$
the group algebra of
$G$
over
$ {\BBF }$
. A (semisimple) Abelian code is an ideal of
$ {\BBF } G$
. Two codes
${\cal I}_{1}$
and
${\cal I}_{2}$
of
$ {\BBF } G$
are
$G$
-equivalent if there exists an automorphism
$\psi $
of
$G$
whose linear extension to
$ {\BBF } G$
maps
${\cal I}_{1}$
onto
${\cal I}_{2}$
. 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.