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 Fundam. Prikl. Mat., 2012, Volume 17, Issue 2, Pages 75–85 (Mi fpm1401)

When are all group codes of a noncommutative group Abelian (a computational approach)?

C. García Pilladoa, S. Gonzáleza, V. T. Markovb, C. Martíneza, A. A. Nechaevb

b M. V. Lomonosov Moscow State University

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 $|G|\notin\{24,48,54,60,64,72,96,108,120\}$, but for $F=\mathbb F_5$ and $G=\mathrm S_4$ 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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English version:
Journal of Mathematical Sciences (New York), 2012, 186:4, 578–585

UDC: 519.725+512.552.7

Citation: C. García Pillado, S. González, V. T. Markov, C. Martínez, A. A. Nechaev, “When are all group codes of a noncommutative group Abelian (a computational approach)?”, Fundam. Prikl. Mat., 17:2 (2012), 75–85; J. Math. Sci., 186:4 (2012), 578–585

Citation in format AMSBIB
\Bibitem{GarGonMar12}
\by C.~Garc{\'\i}a Pillado, S.~Gonz\'alez, V.~T.~Markov, C.~Mart{\'\i}nez, A.~A.~Nechaev
\paper When are all group codes of a~noncommutative group Abelian (a~computational approach)?
\jour Fundam. Prikl. Mat.
\yr 2012
\vol 17
\issue 2
\pages 75--85
\mathnet{http://mi.mathnet.ru/fpm1401}
\transl
\jour J. Math. Sci.
\yr 2012
\vol 186
\issue 4
\pages 578--585
\crossref{https://doi.org/10.1007/s10958-012-1006-x}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84866507992}

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This publication is cited in the following articles:
1. Gabriele Nebe, Artur Schäfer, “A nilpotent non abelian group code”, Algebra Discrete Math., 18:2 (2014), 268–273
2. C. García Pillado, S. González, V. T. Markov, C. Martínez, “Non-Abelian group codes over an arbitrary finite field”, J. Math. Sci., 223:5 (2017), 504–507
3. C. Garcia Pillado, S. Gonzalez, V. Markov, C. Martinez, A. Nechaev, “New examples of non-Abelian group codes”, Adv. Math. Commun., 10:1, SI (2016), 1–10
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