Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2016 (2016), Article ID 9020173, 12 pages
http://dx.doi.org/10.1155/2016/9020173
Research Article

Self-Dual Abelian Codes in Some Nonprincipal Ideal Group Algebras

1Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok 10330, Thailand
2Department of Mathematics, Faculty of Science, Silpakorn University, Nakhon Pathom 73000, Thailand

Received 25 June 2016; Accepted 27 September 2016

Academic Editor: Kishin Sadarangani

Copyright © 2016 Parinyawat Choosuwan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. S. D. Berman, “On the theory of group codes,” Klbernetika, vol. 3, no. 1, pp. 31–39, 1967. View at Google Scholar
  2. S. D. Berman, “Semi-simple cyclic and abelian codes,” Kibernetika, vol. 3, pp. 21–30, 1967. View at Google Scholar
  3. J. J. Bernal and J. J. Simón, “Information sets from defining sets in abelian codes,” IEEE Transactions on Information Theory, vol. 57, no. 12, pp. 7990–7999, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. H. Chabanne, “Permutation decoding of abelian codes,” IEEE Transactions on Information Theory, vol. 38, no. 6, pp. 1826–1829, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. C. Ding, D. R. Kohel, and S. Ling, “Split group codes,” IEEE Transactions on Information Theory, vol. 46, no. 2, pp. 485–495, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. S. Jitman, S. Ling, H. Liu, and X. Xie, “Abelian codes in principal ideal group algebras,” IEEE Transactions on Information Theory, vol. 59, no. 5, pp. 3046–3058, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. B. S. Rajan and M. U. Siddiqi, “Transform domain characterization of abelian codes,” IEEE Transactions on Information Theory, vol. 38, no. 6, pp. 1817–1821, 1992. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. G. Nebe, E. M. Rains, and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Algorithms and Computation in Mathematics, vol. 17, Springer, Berlin, germany, 2006. View at MathSciNet
  9. S. Jitman, S. Ling, and P. Solé, “Hermitian self-dual abelian codes,” IEEE Transactions on Information Theory, vol. 60, no. 3, pp. 1496–1507, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. R. E. Sabin, “On determining all codes in semi-simple group rings,” in Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, G. Cohen, T. Mora, and O. Moreno, Eds., vol. 673 of Lecture Notes in Computer Science, pp. 279–290, Springer, New York, NY, USA, 1993. View at Publisher · View at Google Scholar
  11. X. Kai and S. Zhu, “On cyclic self-dual codes,” Applicable Algebra in Engineering, Communication and Computing, vol. 19, no. 6, pp. 509–525, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. Y. Jia, S. Ling, and C. Xing, “On self-dual cyclic codes over finite fields,” IEEE Transactions on Information Theory, vol. 57, no. 4, pp. 2243–2251, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. L. Fisher and S. K. Sehgal, “Principal ideal group rings,” Communications in Algebra, vol. 4, no. 4, pp. 319–325, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. Q. Dinh, “Constacyclic codes of length ps over Fpm+uFpm,” Journal of Algebra, vol. 324, no. 5, pp. 940–950, 2010. View at Publisher · View at Google Scholar
  15. H. Q. Dinh, “Constacyclic codes of length 2s over Galois extensions rings of F2+uF2,” IEEE Transactions on Information Theory, vol. 55, no. 4, pp. 1730–1740, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. H. Q. Dinh, B. T. Nguyen, and S. Sriboonchitta, “Skew constacyclic codes over finite fields and finite chain rings,” Mathematical Problems in Engineering, vol. 2016, Article ID 3965789, 17 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  17. S. Jitman, S. Ling, and P. Udomkavanich, “Skew constacyclic codes over finite chain rings,” Advances in Mathematics of Communications, vol. 6, no. 1, pp. 39–63, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. S. Benson, “Students ask the darnedest things: a result in elementary group theory,” Mathematics Magazine, vol. 70, no. 3, pp. 207–211, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  19. H. M. Kiah, K. H. Leung, and S. Ling, “Cyclic codes over GRp2, m of length pk,” Finite Fields and Their Applications, vol. 14, no. 3, pp. 834–846, 2008. View at Publisher · View at Google Scholar
  20. S. T. Dougherty and Y. H. Park, “On modular cyclic codes,” Finite Fields and Their Applications, vol. 13, no. 1, pp. 31–57, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. H. Q. Dinh and S. R. Lopez-Permouth, “Cyclic and negacyclic codes over finite chain rings,” IEEE Transactions on Information Theory, vol. 50, no. 8, pp. 1728–1744, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. H. M. Kiah, K. H. Leung, and S. Ling, “A note on cyclic codes over GR(p2, m) of length pk,” Designs, Codes and Cryptography, vol. 63, no. 1, pp. 105–112, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. S. Jitman, S. Ling, and E. Sangwisut, “On self-dual cyclic codes of length pa over GR(p2,s),” Advances in Mathematics of Communications, vol. 10, pp. 255–273, 2016. View at Google Scholar