Mathematical objects in this paper are group codes. In the first part of the paper we present a survey with some of the main results about group codes, mainly the existence of group codes that are not abelian group codes, the minimal length and the minimal dimension of such codes and the existence of a non-abelian group code that has better parameters than any abelian group code. In particular, in a previous paper , we have shown that the minimal dimension of a group code that is not abelian group code is 4. However, all known examples of group codes of dimension 4 that are non-abelian group codes are constructed using groups that are not p-groups.
We do not know if such codes exist for the case of p-groups, but in the second part of this paper we prove that, under some restrictions on the base field, all four-dimensional G-codes for an arbitrary finite p-group G are abelian.
KeywordsGroup codes Length Dimension Groups Non-abelian groups Abelian groups
Authors want specially remember Victor Markov and acknowledge his valuable contribution to this paper before he passed away in July 2019.
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