In this paper, we develop a technique to extend any bound for the minimum distance of cyclic codes constructed from its defining sets (ds-bounds) to Abelian (or multivariate) codes through the notion of B-apparent distance. We also study conditions for an Abelian code to verify that its B-apparent distance reaches its (true) minimum distance. Then, we construct some codes as an application.

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.

We give counterexamples to show that some results regarding equivalence of abelian group codes, that have been in the literature for quite some time, are not correct. Also, we give examples of special families of abelian groups for which these results do hold.