Abstract
Let G be a finite group and F be a field. Any linear code over F that is permutation equivalent to some code defined by an ideal of the group ring FG will be called a G-code. The theory of these “abstract” group codes was developed in 2009. A code is called Abelian if it is an A-code for some Abelian group A. Some conditions were given that all G-codes for some group G are Abelian but no examples of non-Abelian group codes were known at that time. We use a computer algebra system GAP to show that all G-codes over any field are Abelian if |G| < 128 and |G| ∉ {24, 48, 54, 60, 64, 72, 96, 108, 120}, but for F = \( {\mathbb{F}_5} \) and G = S4 there exist non-Abelian G-codes over F. It is also shown that the existence of left non-Abelian group codes for a given group depends in general on the field of coefficients, while for (two-sided) group codes the corresponding question remains open.
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 17, No. 2, pp. 75–85, 2011/12.
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Pillado, C.G., González, S., Markov, V.T. et al. When are all group codes of a noncommutative group Abelian (a computational approach)?. J Math Sci 186, 578–585 (2012). https://doi.org/10.1007/s10958-012-1006-x
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DOI: https://doi.org/10.1007/s10958-012-1006-x