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.

In this paper, the minimum weight and the dimension of all cyclic codes of length $p^{n}$ over a field $\BBF_{q}$ , are computed, when $p$ is an odd prime and $\BBF_{q}$ a finite field with $q$ elements, assuming that $\overline{q}$ generates the group of invertible elements of $\BBZ_{p^{n}}$ . Furthermore, the minimum weight and dimension of codes which are sum of two minimal codes in $\BBF_{q}(C_{p}\times C_{p})$ are also computed. Finally, the efficiency of cyclic codes and noncyclic abelian codes of length $p^{2}$ are compared.

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.