Skip to main content
Log in

Decoding of Cyclic Codes Over the Ring \(\frac{{{F_2}\left[ u \right]}}{{\langle {u^t}\rangle }}\)

  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

In this paper we resolve an open problem about decoding cyclic codes over the ring F2+uF2 with u2 = 0. This problem was first proposed by AbuAlrub et al. in (Des Codes Crypt 42: 273-287, 2007). Also we extend this decoding procedure for cyclic codes of arbitrary length over the ringe \(\frac{{{F_2}\left[ u \right]}}{{\langle {u^t}\rangle }} = {F_2} + u{F_2} + {u^2}{F_2} + \cdots {u^{t - 1}}{F_2}\), where ut = 0.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. T. Abualrub and I. Saip, Cyclic codes over the rings Z 2+uZ 2 and Z 2+uZ 2+u 2 Z 2, Des. Codes Crypt., 42(3) (2013), 273–287.

    Article  Google Scholar 

  2. A. Bonnecaze and P. Udaya, Cyclic codes and self-dual codes over F 2 + uF 2, IEEE Trans. Inf. Theory, 45 (1999), 1250–1255.

    Article  MATH  Google Scholar 

  3. E. Byrne, M. Greferath, J. Pernas, and J. Zumbrgel, Algebraic decoding of negacyclic codes over Z 4, Des. Codes. Crypt., 66(1-3) (2007), 3–16.

    Article  MATH  Google Scholar 

  4. A. R. Jr. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane, and P. Sole, The Z4-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory, 40(2) (1994), 301–319.

    Article  MATH  Google Scholar 

  5. J. A. Huckaba, Commutative ring with zero divisors, Pure and Applied Mathematics, Marcel Dekker, New York (1988).

    Google Scholar 

  6. G. H. Norton and A. Salagean, On the Hamming distance of linear codes over a finite chain ring, IEEE Trans. Inform. Theory, 46(3) (2000), 1060–1067.

    Article  MathSciNet  MATH  Google Scholar 

  7. H. Quang Dinh and S. R. Lpez-Permouth, Cyclic and negacyclic codes over finite chain rings, IEEE Trans. Inform. Theory, 50(8) (2004), 1728–1744.

    Article  MathSciNet  MATH  Google Scholar 

  8. S. Ling and C. Xing, Coding theory a first course, Cambridge University Press (2004).

    Book  Google Scholar 

  9. P. Udaya and A. Bonnecaze, Decoding of cyclic codes over F 2 + uF 2, IEEE Trans. Inform. Theory, 45 (1999), 2148–2157.

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Zhu, Y. Wang, and M. Shi, Some results on cyclic codes over F 2 +vF 2, IEEE Trans. Inform. Theory, 56(4) (2010), 1680–1684.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Karim Samei.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Samei, K., Alimoradi, M.R. Decoding of Cyclic Codes Over the Ring \(\frac{{{F_2}\left[ u \right]}}{{\langle {u^t}\rangle }}\). Indian J Pure Appl Math 50, 113–120 (2019). https://doi.org/10.1007/s13226-019-0310-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-019-0310-2

Key words

Navigation