American Institute of Mathematical Sciences

Advanced
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

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

[1]

Jorge P. Arpasi. On the non-Abelian group code capacity of memoryless channels. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020058
[2]

Masaaki Harada, Ethan Novak, Vladimir D. Tonchev. The weight distribution of the self-dual $[128,64]$ polarity design code. Advances in Mathematics of Communications, 2016, 10 (3) : 643-648. doi: 10.3934/amc.2016032
[3]

Eldho K. Thomas, Nadya Markin, Frédérique Oggier. On Abelian group representability of finite groups. Advances in Mathematics of Communications, 2014, 8 (2) : 139-152. doi: 10.3934/amc.2014.8.139
[4]

Martino Borello, Francesca Dalla Volta, Gabriele Nebe. The automorphism group of a self-dual $[72,36,16]$ code does not contain $\mathcal S_3$, $\mathcal A_4$ or $D_8$. Advances in Mathematics of Communications, 2013, 7 (4) : 503-510. doi: 10.3934/amc.2013.7.503
[5]

Denis S. Krotov, Patric R. J.  Östergård, Olli Pottonen. Non-existence of a ternary constant weight $(16,5,15;2048)$ diameter perfect code. Advances in Mathematics of Communications, 2016, 10 (2) : 393-399. doi: 10.3934/amc.2016013
[6]

Laura Luzzi, Ghaya Rekaya-Ben Othman, Jean-Claude Belfiore. Algebraic reduction for the Golden Code. Advances in Mathematics of Communications, 2012, 6 (1) : 1-26. doi: 10.3934/amc.2012.6.1
[7]

Irene Márquez-Corbella, Edgar Martínez-Moro, Emilio Suárez-Canedo. On the ideal associated to a linear code. Advances in Mathematics of Communications, 2016, 10 (2) : 229-254. doi: 10.3934/amc.2016003
[8]

Serhii Dyshko. On extendability of additive code isometries. Advances in Mathematics of Communications, 2016, 10 (1) : 45-52. doi: 10.3934/amc.2016.10.45
[9]

John Franks, Michael Handel. Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface. Journal of Modern Dynamics, 2013, 7 (3) : 369-394. doi: 10.3934/jmd.2013.7.369
[10]

Olof Heden. The partial order of perfect codes associated to a perfect code. Advances in Mathematics of Communications, 2007, 1 (4) : 399-412. doi: 10.3934/amc.2007.1.399
[11]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889
[12]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2019, 0 (0) : 0-0. doi: 10.3934/amc.2020074
[13]

Alexander Barg, Arya Mazumdar, Gilles Zémor. Weight distribution and decoding of codes on hypergraphs. Advances in Mathematics of Communications, 2008, 2 (4) : 433-450. doi: 10.3934/amc.2008.2.433
[14]

M. Delgado Pineda, E. A. Galperin, P. Jiménez Guerra. MAPLE code of the cubic algorithm for multiobjective optimization with box constraints. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 407-424. doi: 10.3934/naco.2013.3.407
[15]

Andrew Klapper, Andrew Mertz. The two covering radius of the two error correcting BCH code. Advances in Mathematics of Communications, 2009, 3 (1) : 83-95. doi: 10.3934/amc.2009.3.83
[16]

Masaaki Harada, Takuji Nishimura. An extremal singly even self-dual code of length 88. Advances in Mathematics of Communications, 2007, 1 (2) : 261-267. doi: 10.3934/amc.2007.1.261
[17]

José Gómez-Torrecillas, F. J. Lobillo, Gabriel Navarro. Information--bit error rate and false positives in an MDS code. Advances in Mathematics of Communications, 2015, 9 (2) : 149-168. doi: 10.3934/amc.2015.9.149
[18]

Sergio Estrada, J. R. García-Rozas, Justo Peralta, E. Sánchez-García. Group convolutional codes. Advances in Mathematics of Communications, 2008, 2 (1) : 83-94. doi: 10.3934/amc.2008.2.83
[19]

Heping Liu, Yu Liu. Refinable functions on the Heisenberg group. Communications on Pure & Applied Analysis, 2007, 6 (3) : 775-787. doi: 10.3934/cpaa.2007.6.775
[20]

Stefan Haller, Tomasz Rybicki, Josef Teichmann. Smooth perfectness for the group of diffeomorphisms. Journal of Geometric Mechanics, 2013, 5 (3) : 281-294. doi: 10.3934/jgm.2013.5.281

2018 Impact Factor: 0.879

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences