# American Institute of Mathematical Sciences

February  2016, 10(1): 1-10. doi: 10.3934/amc.2016.10.1

## New examples of non-abelian group codes

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

Received  November 2014 Revised  June 2015 Published  March 2016

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.
Citation: Cristina García Pillado, Santos González, Victor Markov, Consuelo Martínez, Alexandr Nechaev. New examples of non-abelian group codes. Advances in Mathematics of Communications, 2016, 10 (1) : 1-10. doi: 10.3934/amc.2016.10.1
##### References:
 [1] Des. Codes Crypt., 51 (2009), 289-300. doi: 10.1007/s10623-008-9261-z.  Google Scholar [2] Discr. Math. Appl., 14 (2004), 163-172. doi: 10.1515/156939204872347.  Google Scholar [3] John Wiley & Sons, New York, 1962.  Google Scholar [4] in Proc. 3rd Int. Castle Meeting Coding Theory Appl. (eds. J. Borges and M. Villanueva), Servei de Publicacions, 2011, 123-127. Google Scholar [5] Fund. Appl. Math., 17 (2012), 75-85. doi: 10.1007/s10958-012-1006-x.  Google Scholar [6] J. Algebra Appl., 12 (2013). doi: 10.1142/S0219498813500370.  Google Scholar [7] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at , ().   Google Scholar [8] in Acta Math. Hungar., 118 (2008), 105-113. doi: 10.1007/s10474-007-6169-4.  Google Scholar [9] North Holland, Amsterdam, 1974. Google Scholar [10] John Wiley & Sons, New York, 1977.  Google Scholar

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##### References:
 [1] Des. Codes Crypt., 51 (2009), 289-300. doi: 10.1007/s10623-008-9261-z.  Google Scholar [2] Discr. Math. Appl., 14 (2004), 163-172. doi: 10.1515/156939204872347.  Google Scholar [3] John Wiley & Sons, New York, 1962.  Google Scholar [4] in Proc. 3rd Int. Castle Meeting Coding Theory Appl. (eds. J. Borges and M. Villanueva), Servei de Publicacions, 2011, 123-127. Google Scholar [5] Fund. Appl. Math., 17 (2012), 75-85. doi: 10.1007/s10958-012-1006-x.  Google Scholar [6] J. Algebra Appl., 12 (2013). doi: 10.1142/S0219498813500370.  Google Scholar [7] M. Grassl, Bounds on the minimum distance of linear codes and quantum codes,, available online at , ().   Google Scholar [8] in Acta Math. Hungar., 118 (2008), 105-113. doi: 10.1007/s10474-007-6169-4.  Google Scholar [9] North Holland, Amsterdam, 1974. Google Scholar [10] John Wiley & Sons, New York, 1977.  Google Scholar
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