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]

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

show all references

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

[1]

Sascha Kurz. The $[46, 9, 20]_2$ code is unique. Advances in Mathematics of Communications, 2021, 15 (3) : 415-422. doi: 10.3934/amc.2020074
[2]

Dandan Cheng, Qian Hao, Zhiming Li. Scale pressure for amenable group actions. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1091-1102. doi: 10.3934/cpaa.2021008
[3]

Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637
[4]

Zhimin Chen, Kaihui Liu, Xiuxiang Liu. Evaluating vaccination effectiveness of group-specific fractional-dose strategies. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021062
[5]

Ricardo A. Podestá, Denis E. Videla. The weight distribution of irreducible cyclic codes associated with decomposable generalized Paley graphs. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021002
[6]

Joe Gildea, Adrian Korban, Abidin Kaya, Bahattin Yildiz. Constructing self-dual codes from group rings and reverse circulant matrices. Advances in Mathematics of Communications, 2021, 15 (3) : 471-485. doi: 10.3934/amc.2020077
[7]

V. Kumar Murty, Ying Zong. Splitting of abelian varieties. Advances in Mathematics of Communications, 2014, 8 (4) : 511-519. doi: 10.3934/amc.2014.8.511
[8]

Zemer Kosloff, Terry Soo. The orbital equivalence of Bernoulli actions and their Sinai factors. Journal of Modern Dynamics, 2021, 17: 145-182. doi: 10.3934/jmd.2021005
[9]

Andrey Kovtanyuk, Alexander Chebotarev, Nikolai Botkin, Varvara Turova, Irina Sidorenko, Renée Lampe. Modeling the pressure distribution in a spatially averaged cerebral capillary network. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021016
[10]

Ritu Agarwal, Kritika, Sunil Dutt Purohit, Devendra Kumar. Mathematical modelling of cytosolic calcium concentration distribution using non-local fractional operator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021017
[11]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058
[12]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011
[13]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221
[14]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133

2019 Impact Factor: 0.734

Tools

Metrics

  • PDF downloads (113)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences