# 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] J. J. Bernal, Á del Río and J. J. Simón, An intrinsical description of group codes,, Des. Codes Crypt., 51 (2009), 289.  doi: 10.1007/s10623-008-9261-z.  Google Scholar [2] E. Couselo, S. González, V. Markov and A. Nechaev, Loop codes,, Discr. Math. Appl., 14 (2004), 163.  doi: 10.1515/156939204872347.  Google Scholar [3] R. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras,, John Wiley & Sons, (1962).   Google Scholar [4] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, Group codes which are not abelian group codes,, in Proc. 3rd Int. Castle Meeting Coding Theory Appl. (eds. J. Borges and M. Villanueva), (2011), 123.   Google Scholar [5] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, When all group codes of a noncommutative group are abelian (a computational approach),, Fund. Appl. Math., 17 (2012), 75.  doi: 10.1007/s10958-012-1006-x.  Google Scholar [6] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, Group codes over non-abelian groups,, 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] M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of $FS_4$,, in Acta Math. Hungar., 118 (2008), 105.  doi: 10.1007/s10474-007-6169-4.  Google Scholar [9] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North Holland, (1974).   Google Scholar [10] D. S. Passman, The Algebraic Structure of Group Rings,, John Wiley & Sons, (1977).   Google Scholar

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##### References:
 [1] J. J. Bernal, Á del Río and J. J. Simón, An intrinsical description of group codes,, Des. Codes Crypt., 51 (2009), 289.  doi: 10.1007/s10623-008-9261-z.  Google Scholar [2] E. Couselo, S. González, V. Markov and A. Nechaev, Loop codes,, Discr. Math. Appl., 14 (2004), 163.  doi: 10.1515/156939204872347.  Google Scholar [3] R. Curtis and I. Reiner, Representation Theory of Finite Groups and Associative Algebras,, John Wiley & Sons, (1962).   Google Scholar [4] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, Group codes which are not abelian group codes,, in Proc. 3rd Int. Castle Meeting Coding Theory Appl. (eds. J. Borges and M. Villanueva), (2011), 123.   Google Scholar [5] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, When all group codes of a noncommutative group are abelian (a computational approach),, Fund. Appl. Math., 17 (2012), 75.  doi: 10.1007/s10958-012-1006-x.  Google Scholar [6] C. García Pillado, S. González, V. Markov, C. Martínez and A. Nechaev, Group codes over non-abelian groups,, 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] M. Khan, R. K. Sharma and J. B. Srivastava, The unit group of $FS_4$,, in Acta Math. Hungar., 118 (2008), 105.  doi: 10.1007/s10474-007-6169-4.  Google Scholar [9] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes,, North Holland, (1974).   Google Scholar [10] D. S. Passman, The Algebraic Structure of Group Rings,, John Wiley & Sons, (1977).   Google Scholar
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