Skew quasi-cyclic codes over Galois rings | Designs, Codes and Cryptography Skip to main content
Springer Nature Link
Log in

Skew quasi-cyclic codes over Galois rings

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

Abstract

Recently there has been a lot of interest on algebraic codes in the setting of skew polynomial rings. In this paper we have studied skew quasi-cyclic (QC) codes over Galois rings. We have given a necessary and sufficient condition for skew cyclic codes over Galois rings to be free, and determined a distance bound for free skew cyclic codes. A sufficient condition for 1-generator skew QC codes to be free is determined. Some distance bounds for free 1-generator skew QC codes are discussed. A canonical decomposition of skew QC codes is presented.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
¥17,985 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (Japan)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abualrub T., Ghrayeb A., Aydin N., Siap I.: On the construction of skew quasi-cyclic codes. IEEE Trans. Inform. Theory 56, 2081–2090 (2010)

    Article  MathSciNet  Google Scholar 

  2. Aydin N., Ray-Chaudhuri D.K.: Quasi-cyclic codes over \({\mathbb{Z}_4}\) and some new binary codes. IEEE Trans. Inform. Theory 7, 2065–2069 (2002)

    Article  MathSciNet  Google Scholar 

  3. Bhaintwal M., Wasan S.K.: On quasi-cyclic codes over \({\mathbb{Z}_q}\). Appl. Algebra Engrg. Comm. Comput. 20, 459–480 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Boucher D., Geiselmann W., Ulmer F.: Skew-cyclic codes. Appl. Algebra Engrg. Comm. Comput. 18, 379–389 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Boucher D., Solé P., Ulmer F.: Skew constacyclic codes over Galois rings. Adv. Math. Commun. 2, 273–292 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  6. Boucher D., Ulmer F.: Coding with skew polynomial rings. J. Symbolic Comput. 44, 1644–1656 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Boucher D., Ulmer F.: Codes as modules over skew polynomial rings. In: Parker M. G. (ed.) Cryptography and coding 2009. Lecture Notes in Computer Science, vol. 5921, pp. 38–55. Springer-Verlag, Heidelberg (2009).

  8. Chaussade L., Loidreau P., Ulmer F.: Skew codes of prescribed distance or rank. Des. Codes Cryptogr. 50, 267–284 (2009)

    Article  MathSciNet  Google Scholar 

  9. Cohn P.M.: Skew fields: theory of general division rings. Encyclopedia in Mathematics, vol. 57. Cambridge University Press, Cambridge (1995).

  10. Conan J., Séguin G.: Structural properties and enumeration of quasi-cyclic codes. Appl. Algebra Engrg. Commun. Comput. 4, 25–39 (1993)

    Article  MATH  Google Scholar 

  11. Jacobson N.: Theory of Rings. Am. Math. Soc., New York (1943)

    MATH  Google Scholar 

  12. Jacobson N.: Finite Dimensional Division Algebras over Fields. Springer, New York (1996)

    Book  MATH  Google Scholar 

  13. Lally K.: Quasi-cyclic codes of index l over \({\mathbb{F}_q}\) viewed as \({\mathbb{F}_q[x]}\)-submodules of \({\mathbb{F}_{q^l}[x]/\langle x^m - 1 \rangle}\). In: Fossorier M., Hoeholdt T., Poli A. (eds.) Proceedings of AAECC-15, Lecture Note in Computer Science, vol. 2643, pp. 244–253, Springer-Verlag, Heidelberg (2003).

  14. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes I: finite fields. IEEE Trans. Inform. Theory 47, 2751–2760 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes II: chain rings. Des. Codes Cryptogr. 30, 113–130 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Ling S., Solé P.: On the algebraic structures of quasi-cyclic codes III: generator theory. IEEE Trans. Inform. Theory 51, 2692–2700 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  17. McDonald B.R.: Finite Rings with Identity. Marcel Dekker, New York (1974)

    MATH  Google Scholar 

  18. Siap I., Abualrub T., Aydin N., Seneviratne P.: Skew cyclic codes of arbitrary length. Int. J. Inform. Coding Theory (2010) (to appear).

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India

    Maheshanand Bhaintwal

Authors
  1. Maheshanand Bhaintwal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Maheshanand Bhaintwal.

Additional information

Communicated by J. D. Key.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Bhaintwal, M. Skew quasi-cyclic codes over Galois rings. Des. Codes Cryptogr. 62, 85–101 (2012). https://doi.org/10.1007/s10623-011-9494-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10623-011-9494-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation