New examples of non-abelian group codes

Cristina García  Pillado; Santos  González; Victor  Markov; Consuelo Martínez; Alexandr  Nechaev

doi:10.3934/amc.2016.10.1

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2016, Volume 10, Issue 1: 1-10. Doi: 10.3934/amc.2016.10.1

This issue Previous Article Editorial Next Article An approach to the performance of SPC product codes on the erasure channel

New examples of non-abelian group codes

1.
Department of Mathematics, University of Oviedo, Calvo Sotelo, s/n, 33007 Oviedo
2.
Department of Mechanics and Mathematics, Moscow State University
3.
Departamento de Matemáticas, Universidad de Oviedo, C/ Calvo Sotelo s/n, 33007 Oviedo

Received: October 31, 2014

Revised: May 31, 2015

Published: January 2016

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Abstract

In previous papers [4,5,6] we gave the first example of a non-abelian group code using the group ring $F_5S_4$. It is natural to ask if it is really relevant that the group ring is semisimple. What happens if the field has characteristic $2$ or $3$? We have addressed this question, with computer help, proving that there are also examples of non-abelian group codes in the non-semisimple case. The results show some interesting differences between the cases of characteristic $2$ and $3$. Furthermore, using the group $SL(2,F_3)$, we construct a non-abelian group code over $F_2$ of length $24$, dimension $6$ and minimal weight $10$. This code is optimal in the following sense: every linear code over $F_2$ with length $24$ and dimension $6$ has minimum distance less than or equal to $10$. In the case of abelian group codes over $F_2$ the above value for the minimum distance cannot be achieved, since the minimum distance of a binary abelian group code with the given length and dimension 6 is at most 8.

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Mathematics Subject Classification: Primary: 94B05, 94B60; Secondary: 20C05.

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