On abelian and cyclic group codes

We determine a condition on the minimum Hamming weight of some special abelian group codes and, as a consequence of this result, we establish that any such code is, up to permutational equivalence, a subspace of the direct sum of $s$ copies of the repetition code of length $t$, for some suitable positive integers $s$ and $t$. Moreover, we provide a complete characterisation of permutation automorphisms of the linear code $C={⨁}_{i=1}^{s}Re{p}_t({\mathbb{F}}_q)$ and we establish that such a code is an abelian group code, for every pair of integers $s,t\ge 1$. Finally, in a similar fashion as for abelian group codes, we give an equivalent characterisation of cyclic group codes.