New examples of non-abelian group codes

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

New examples of non-abelian group codes

Cristina García Pillado ^1, , Santos González ^1, , Victor Markov ^2, , Consuelo Martínez ^3, and Alexandr Nechaev ^2,

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

Abstract
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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.

Keywords: semisimplicity, permutation equivalence., Abelian group code, weight distribution, Group code.

Mathematics Subject Classification: Primary: 94B05, 94B60; Secondary: 20C0.

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

show all references

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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