Advertisement

Cyclic codes over \({\mathbb {F}}_{2^m}[u]/\langle u^k\rangle \) of oddly even length

  • Yonglin Cao
  • Yuan Cao
  • Fang-Wei Fu
Original Paper

Abstract

Let \({\mathbb {F}}_{2^m}\) be a finite field of characteristic 2 and \(R={\mathbb {F}}_{2^m}[u]/\langle u^k\rangle ={\mathbb {F}}_{2^m} +u{\mathbb {F}}_{2^m}+\ldots +u^{k-1}{\mathbb {F}}_{2^m}\) (\(u^k=0\)) where \(k\in {\mathbb {Z}}^{+}\) satisfies \(k\ge 2\). For any odd positive integer n, it is known that cyclic codes over R of length 2n are identified with ideals of the ring \(R[x]/\langle x^{2n}-1\rangle \). In this paper, an explicit representation for each cyclic code over R of length 2n is provided and a formula to count the number of codewords in each code is given. Then a formula to calculate the number of cyclic codes over R of length 2n is obtained. Moreover, the dual code of each cyclic code and self-dual cyclic codes over R of length 2n are investigated.

Keywords

Cyclic code Finite chain ring Non-principal ideal ring Dual code Self-dual code 

Mathematics Subject Classification

94B05 94B15 11T71 

Notes

Acknowledgments

Part of this work was done when Yonglin Cao was visiting Chern Institute of Mathematics, Nankai University, Tianjin, China. Yonglin Cao would like to thank the institution for the kind hospitality. This research is supported in part by the National Key Basic Research Program of China (Grant No. 2013CB834204) and the National Natural Science Foundation of China (Grant No. 11471255).

References

  1. 1.
    Abualrub, T., Siap, I.: Cyclic codes over the ring \({\mathbb{Z}}_2+u{\mathbb{Z}}_2\) and \({\mathbb{Z}}_2+u{\mathbb{Z}}_2+u^2{\mathbb{Z}}_2\). Des. Codes Cryptogr. 42, 273–287 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Al-Ashker, M., Hamoudeh, M.: Cyclic codes over \(Z_2+uZ_2+u^2Z_2+\ldots +u^{k-1}Z_2\). Tur. J. Math. 35(4), 737–749 (2011)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonnecaze, A., Udaya, P.: Cyclic codes and self-dual codes over \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inform. Theory 45, 1250–1255 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dinh, H.Q.: Constacyclic codes of length \(2^s\) over Galois extension rings of \({\mathbb{F}}_2+u{\mathbb{F}}_2\). IEEE Trans. Inform. Theory 55, 1730–1740 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Dinh, H.Q.: Constacyclic codes of length \(p^s\) over \({\mathbb{F}}_{p^m}+u {\mathbb{F}}_{p^m}\). J. Algebra 324, 940–950 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dougherty, S.T., Kim, J.-L., Kulosman, H., Liu, H.: Self-dual codes over commutative Frobenius rings. Finite Fields Appl. 16, 14–26 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Norton, G., Sălăgean-Mandache, A.: On the structure of linear and cyclic codes over finite chain rings. Appl. Algebra in Engrg. Comm. Comput. 10, 489–506 (2000)CrossRefzbMATHGoogle Scholar
  8. 8.
    Singh, A.K., Kewat, P.K.: On cyclic codes over the ring \({\mathbb{Z}}_p[u]/\langle u^k\rangle \). Des. Codes Cryptogr. 72, 1–13 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of SciencesShandong University of TechnologyZiboChina
  2. 2.College of Mathematics and EconometricsHunan UniversityChangshaChina
  3. 3.Chern Institute of Mathematics and LPMCNankai UniversityTianjinChina

Personalised recommendations