Skip to main content
SpringerLink
Log in

On cyclic codes over the ring \(\mathbb Z _p[u]/\langle u^k\rangle \)

  • Published:
Designs, Codes and Cryptography Aims and scope Submit manuscript

Abstract

In this paper, we study cyclic codes over the ring \(\mathbb Z _p[u]/\langle u^k\rangle .\) We find a set of generators for these codes. We also study the rank and the Hamming distance of these codes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abualrub T., Oehmke R.H.: On the generators of \(Z_4\) cyclic codes of length \(2^e.\) IEEE Trans. Inf. Theory 49(9), 2126–2133 (2003).

  2. Abualrub T., Siap I.: Cyclic codes over the rings \(Z_2+uZ_2\) and \(Z_2+uZ_2+u^2Z_2.\) Des. Codes Cryptogr. 42(3), 273–287 (2007).

  3. Abualrub T., Ghrayeb A., Oehmke R.H.: A mass formula and rank of \(Z_4\) cyclic codes of length \(2^e.\) IEEE Trans. Inf. Theory 50(12), 3306–3312 (2004).

  4. Al-Ashker M., Hamoudeh M.: Cyclic codes over \(Z_2+uZ_2+u^2Z_2+\cdots +u^{k-1}Z_2.\) Turk. J. Math. 35(4), 737–749 (2011).

  5. Blackford T.: Cyclic codes over \(Z_4\) of oddly even length. Discret. Appl. Math. 128(1), 27–46 (2003). International Workshop on Coding and Cryptography (WCC 2001) (Paris).

  6. Bonnecaze A., Udaya P.: Cyclic codes and self-dual codes over \(F_2+uF_2.\) IEEE Trans. Inf. Theory 45(4), 1250–1255 (1999).

  7. Calderbank A.R., Sloane N.J.A.: Modular and \(p\)-adic cyclic codes. Des. Codes Cryptogr. 6(1), 21–35 (1995).

  8. Calderbank R.A., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error correction via codes over \({\rm GF}(4).\) IEEE Trans. Inf. Theory 44(4), 1369–1387 (1998).

    Google Scholar 

  9. Conway J.H., Sloane N.J.A.: Self-dual codes over the integers modulo 4. J. Comb. Theory A 62(1), 30–45 (1993).

    Google Scholar 

  10. Dinh H.Q., Lopez-Permouth S.: Cyclic and negacyclic codes over finite chain rings. IEEE Trans. Inf. Theory 50(8), 1728–1744 (2004).

    Google Scholar 

  11. Dougherty S.T., Shiromoto K.: Maximum distance codes over rings of order 4. IEEE Trans. Inf. Theory 47(1), 400–404 (2001).

    Google Scholar 

  12. McDonald B.R.: Finite rings with identity. In: Pure and Applied Mathematics, vol. 28. Marcel Dekker, New York (1974).

  13. Pless V.S., Qian Z.: Cyclic codes and quadratic residue codes over \(Z_4.\) IEEE Trans. Inf. Theory 42(5), 1594–1600 (1996).

    Google Scholar 

  14. van Lint J.H.: Repeated-root cyclic codes. IEEE Trans. Inf. Theory 37(2), 343–345 (1991).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Indian School of Mines, Dhanbad, 826 004, Jharkhand, India

    Abhay Kumar Singh & Pramod Kumar Kewat

Authors
  1. Abhay Kumar Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pramod Kumar Kewat
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pramod Kumar Kewat.

Additional information

Communicated by J.-L. Kim.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Singh, A.K., Kewat, P.K. On cyclic codes over the ring \(\mathbb Z _p[u]/\langle u^k\rangle \) . Des. Codes Cryptogr. 74, 1–13 (2015). https://doi.org/10.1007/s10623-013-9843-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-013-9843-2

Keywords

Mathematics Subject Classification

Search

Navigation