Designs, Codes and Cryptography

, Volume 70, Issue 3, pp 347–358 | Cite as

On self-dual cyclic codes over finite chain rings

  • Aicha Batoul
  • Kenza Guenda
  • T. Aaron Gulliver


In this paper, we give necessary and sufficient conditions for the existence of non-trivial cyclic self-dual codes over finite chain rings. We prove that there are no free cyclic self-dual codes over finite chain rings with odd characteristic. It is also proven that a self-dual code over a finite chain ring cannot be the lift of a binary cyclic self-dual code. The number of cyclic self-dual codes over chain rings is also investigated as an extension of the number of cyclic self-dual codes over finite fields given recently by Jia et al.


Cyclic codes Self-dual codes Finite chain rings 

Mathematics Subject Classification

94B05 94B15 


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  1. 1.
    Abualrub T., Oehmke R.: On the generators of \({\mathbb {Z}_4}\) cyclic codes of length 2e. IEEE Trans. Inf. Theory 49(9), 2126–2133 (2003)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bannai E., Dougherty S.T., Harada M., Oura M.: Type II codes, even unimodular lattices and invariant rings. IEEE Trans. Inf. Theory 45(4), 1194–1205 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Blackford T.: Cyclic codes over \({\mathbb {Z}_4}\) of oddly even length. Appl. Discret. Math. 128, 27–46 (2003)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Blackford T.: Negacyclic codes over \({\mathbb {Z}_4}\) of even length. IEEE. Trans. Inf. Theory 49(6), 1417–1424 (2003)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bonnecaze A., Solé P., Calderbank A.R.: Quaternary quadratic residue codes and unimodular lattices. IEEE Trans. Inf. Theory 41(2), 366–377 (1995)CrossRefMATHGoogle Scholar
  6. 6.
    Calderbank A.R. , Sloane N.J.A.: Modular and p-adic cyclic codes. Des. Codes Cryptogr. 6(1), 21–35 (1996)Google Scholar
  7. 7.
    Demazure M.: Cours D’Algèbre: Primalité, Divisibilité, Codes. Cassini, Paris (1997)MATHGoogle Scholar
  8. 8.
    Dinh H., López-Permouth S.R.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans Inf. Theory 50(8), 1728–1744 (2004)CrossRefMATHGoogle Scholar
  9. 9.
    Dougherty S.T., Gulliver T.A., Wong J.N.C.: Self-dual codes over \({\mathbb {Z}_8}\) and \({\mathbb {Z}_9}\) . Des. Codes Cryptogr. 41(3), 235–249 (2006)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Dougherty S.T., Liu H., Park Y.H.: Lifted codes over finite chain rings. Math. J. Okayama Univ. 53, 39–53 (2010)MathSciNetGoogle Scholar
  11. 11.
    Dougherty S.T., Harada M., Solé P.: Self-dual codes over rings and the Chinese remainder theorem. Hokkaido Math. J. 28, 253–283 (1999)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Dougherty S.T., Kim J.L.: Construction of self-dual codes over chain rings. Int. J. Inf. Coding Theory 1(2), 171–190 (2010)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Guenda K.: New MDS self-dual codes over finite fields. Des. Codes Cryptogr. 62(1), 31–42 (2012)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Guenda K., Gulliver T.A.: MDS and self-dual codes over rings. Finite Fields Appl. (2011, submitted).Google Scholar
  15. 15.
    Hammons A.R. Jr., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The Z 4 linearity of Kerdock, Preparata, Goethals and related codes. IEEE Trans. Inf. Theory 40(2), 301–319 (1994)CrossRefMATHGoogle Scholar
  16. 16.
    Jia Y., Ling S., Xing C.: On self-dual cyclic codes over finite fields. IEEE Trans. Inf. Theory 57(4), 2243–2251 (2011)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Kanwar P., López-Permouth S.R.: Cyclic codes over the integers modulo p m. Finite Fields Appl. 3(4), 334–352 (1997)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    López-Permouth S.R., Szabo S.:Repeated root cyclic and negacyclic codes over Galois rings. In: Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. Springer Lecture Notes in Computer Science, vol. 5527, pp. 219–222 (2009).Google Scholar
  19. 19.
    Moree P., Solé P.: Around the Pellikán’s conjecture on very odd sequences. Manuscr. Math. 117, 219–238 (2005)CrossRefMATHGoogle Scholar
  20. 20.
    Norton G.H., Sălăgean A.: On the structure of linear and cyclic codes over a finite chain ring. Appl. Algebr. Eng. Commun. Comput. 10(6), 489–506 (2000)CrossRefMATHGoogle Scholar
  21. 21.
    Sloane N.J.A., Thompson J.G.: Cyclic self-dual codes. IEEE. Trans. Inf. Theory 29(3), 364–366 (1983)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Rains E., Sloane N.J.A.: Self-dual codes. In: Pless V.S., Huffman W.C. (eds) Handbook of Coding Theory, pp. 177–294. Elsevier, Amsterdam (1998).Google Scholar
  23. 23.
    Roman S.: Coding and Information Theory, Graduate Texts Mathematics, vol. 134. Springer, New York (1992)Google Scholar
  24. 24.
    Skersys G.: Calcul du group d’automorphismes des codes, PhD Thesis. Laco, Limoges (1999).Google Scholar
  25. 25.
    Yucas J.L., Mullin G.L.: Self-reciprocal irreducible polynomials over finite fields. Des. Codes Cryptogr. 33(3), 275–281 (2004)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Wolfmann J.: Negacyclic and cyclic codes over \({\mathbb {Z}_4}\) . IEEE Trans. Inf. Theory 45(7), 2522–2532 (1999)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Aicha Batoul
    • 1
  • Kenza Guenda
    • 1
  • T. Aaron Gulliver
    • 2
  1. 1.Faculty of Mathematics USTHBUniversity of Science and Technology of AlgiersAlgiersAlgeria
  2. 2.Department of Electrical and Computer EngineeringUniversity of VictoriaVictoriaCanada

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