On abelian and cyclic group codes
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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^sRep_t(𝔽_q) and we establish that such a code is an abelian group code, for every pair of integers s,t≥1. Finally, in a similar fashion as for abelian group codes, we give an equivalent characterisation of cyclic group codes.
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