Advertisement

A class of constacyclic codes over \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \)

  • Habibul Islam
  • Tushar Bag
  • Om Prakash
Original Research
  • 106 Downloads

Abstract

In this paper, we study \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes over the ring \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) where \(u^{k}=0 \) with \(\lambda =(1+2u^{k-1})\) and \((3+2u^{k-1})\). It is shown that the Gray images of \(\lambda \)-constacyclic and skew \(\lambda \)-constacyclic codes over the ring \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) are cyclic, quasi-cyclic, permutation equivalent to a QC code over \({\mathbb {Z}}_{4}\). Further, the generators of these \(\lambda \)-constacyclic codes over \({\mathbb {Z}}_{4}[u]/\langle u^{k}\rangle \) are obtained.

Keywords

Constacyclic code Quasi-cyclic code Gray map Skew constacyclic code 

Mathematics Subject Classification

94B05 94B15 94B35 94B60 

Notes

Acknowledgements

The authors are thankful to the University Grants Commission (UGC), Govt. of India for financial support and Indian Institute of Technology Patna for providing the research facilities. The authors would like to thank the anonymous referees and the editor for their valuable suggestions to improve the presentation of the manuscript.

References

  1. 1.
    Abualrub, T., Siap, I.: Reversible cyclic codes over \({\mathbb{Z}}_{4}\). Australas. J. Comb. 38, 195–205 (2007)zbMATHGoogle Scholar
  2. 2.
    Abualrub, T., Siap, I.: Constacyclic and cyclic codes over \({\mathbb{F}}_{2} +u{\mathbb{F}}_{2}.\). J. Frankl. Inst. 346(5), 520–529 (2009)CrossRefzbMATHGoogle Scholar
  3. 3.
    Abualrub, T., Siap, I.: Cyclic codes over the rings \({\mathbb{Z}}_{2} + u{\mathbb{Z}}_{2}\) and \({\mathbb{Z}}_{2} + u{\mathbb{Z}}_{2} + u^{2}{\mathbb{Z}}_{2}\). Des. Code Cryptogr. 42(3), 273–287 (2007)CrossRefGoogle Scholar
  4. 4.
    Ashraf, M., Mohammed, G.: \((1+2u)\)-constacyclic codes over \({\mathbb{Z}}_{4} +u{\mathbb{Z}}_{4}\). arXiv:1504.03445v1. (2015)
  5. 5.
    Aydin, N., Cengellenmis, Y., Dertli, A.: On some constacyclic codes over \({\mathbb{Z}}_{4}[u]/\langle u^2 -1\rangle \), their \({\mathbb{Z}}_{4}\) images, and new codes. Des. Codes Cryptogr. 86(6), 1249–1255 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boucher, D., Geiselmann, W., Ulmer, F.: Skew cyclic codes. Appl. Algebra Eng. Comm. 18(4), 379–389 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boucher, D., Sole, P., Ulmer, F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2(3), 273–292 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gao, J.: Skew cyclic codes over \({\mathbb{F}}_p + v{\mathbb{F}}_p\). J. Appl. Math. Inf. 31(3), 337–342 (2013)Google Scholar
  9. 9.
    Gursoy, F., Siap, I., Yildiz, B.: Construction of skew cyclic codes over \({\mathbb{F}}_{q}+v{\mathbb{F}}_{q}\). Adv. Math. Commun. 8(3), 313–322 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jitman, S., Ling, S., Udomkavanich, P.: Skew constacyclic codes over finite chain ring. Adv. Math. Commun. 6(1), 39–63 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Karadeniz, S., Yildiz, B.: \((1+v)\)-constacyclic codes over \({\mathbb{F}}_{2} +u{\mathbb{F}}_{2}+v{\mathbb{F}}_{2}+uv{\mathbb{F}}_{2}\). J. Franklin Inst. 348(9), 2625–2632 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ozen, M., Uzekmek, F.Z., Aydin, N., Ozzaim, N.T.: Cyclic and some constacyclic codes over the ring \({\mathbb{Z}}_{4}[u]/\langle u^2 -1\rangle \). Finite Fields Appl. 45, 27–39 (2016)CrossRefzbMATHGoogle Scholar
  13. 13.
    Singh, A.K., Kewat, P.K.: On cyclic codes over the ring \({\mathbb{Z}}_{p}[u]/\langle u^{k} \rangle \). Des. Code Cryptogr. 74(1), 1–13 (2015)CrossRefzbMATHGoogle Scholar
  14. 14.
    Siap, I., Abualrub, T., Aydin, N., Seneviratne, P.: Skew cyclic codes of arbitrary length. Int. J. Inf. Coding Theory 2(1), 10–20 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shi, M., Sok, L., Aydin, N., Sole, P.: On constacyclic codes over \({\mathbb{Z}}_{4}[u]/\langle u^2 -1\rangle \). Finite Fields Appl. 45, 86–95 (2015)CrossRefzbMATHGoogle Scholar
  16. 16.
    Yildiz, B., Aydin, N.: On cyclic codes over \({\mathbb{Z}}_{4} +u {\mathbb{Z}}_{4}\) and thier \({\mathbb{Z}}_{4}\)-images. Int. J. Inf. Coding Theory 2(4), 226–237 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Yu, H., Wang, Y., Shi, M.: \((1+u)\)- Constacyclic Codes Over \({\mathbb{Z}}_{4} +u{\mathbb{Z}}_{4}\). Springer, New York (2016).  https://doi.org/10.1186/s40064-016-2717-0 Google Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2018

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology PatnaPatnaIndia

Personalised recommendations