$G$-Equivalence in Group Algebras and Minimal Abelian Codes
Raul Antônio Ferraz, Marinês Guerreiro, César Polcino Milies · IEEE Transactions on Information Theory · 2013
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.