When are all group codes of a~noncommutative group Abelian (a~computational approach)?
Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 2, pp. 75-85
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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.
@article{FPM_2012_17_2_a1,
author = {C. Garc{\'\i}a Pillado and S. Gonz\'alez and V. T. Markov and C. Mart{\'\i}nez and A. A. Nechaev},
title = {When are all group codes of a~noncommutative group {Abelian} (a~computational approach)?},
journal = {Fundamentalʹna\^a i prikladna\^a matematika},
pages = {75--85},
publisher = {mathdoc},
volume = {17},
number = {2},
year = {2012},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/FPM_2012_17_2_a1/}
}
TY - JOUR AU - C. García Pillado AU - S. González AU - V. T. Markov AU - C. Martínez AU - A. A. Nechaev TI - When are all group codes of a~noncommutative group Abelian (a~computational approach)? JO - Fundamentalʹnaâ i prikladnaâ matematika PY - 2012 SP - 75 EP - 85 VL - 17 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/FPM_2012_17_2_a1/ LA - ru ID - FPM_2012_17_2_a1 ER -
%0 Journal Article %A C. García Pillado %A S. González %A V. T. Markov %A C. Martínez %A A. A. Nechaev %T When are all group codes of a~noncommutative group Abelian (a~computational approach)? %J Fundamentalʹnaâ i prikladnaâ matematika %D 2012 %P 75-85 %V 17 %N 2 %I mathdoc %U http://geodesic.mathdoc.fr/item/FPM_2012_17_2_a1/ %G ru %F FPM_2012_17_2_a1
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)?. Fundamentalʹnaâ i prikladnaâ matematika, Tome 17 (2012) no. 2, pp. 75-85. http://geodesic.mathdoc.fr/item/FPM_2012_17_2_a1/