On abelian and cyclic group codes

Angelo Marotta

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arXiv · 2209.14213

On abelian and cyclic group codes

Angelo Marotta

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Abstract

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 $C=\bigoplus_{i=1}^{s}Rep_{t}(\mathbb{F}_{q})$ and we establish that such a code is an abelian group code, for every pair of integers $s,t\geq1$. Finally, in a similar fashion as for abelian group codes, we give an equivalent characterisation of cyclic group codes.

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Angelo Marotta. 2022-09-28. On abelian and cyclic group codes. https://arxiv.org/abs/2209.14213

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