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Quadratic Residue Codes over \(\mathbb {F}_p+v\mathbb {F}_p+v^{2}\mathbb {F}_p\)

  • Yan Liu
  • Minjia ShiEmail author
  • Patrick Solé
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9061)

Abstract

This article studies quadratic residue codes of prime length \(q\) over the ring \(R=\mathbb {F}_{p}+v\mathbb {F}_{p}+v^{2}\mathbb {F}_{p},\) where \(p,q\) are distinct odd primes. After studying the structure of cyclic codes of length \(n\) over \(R,\) quadratic residue codes over \(R\) are defined by their generating idempotents and their extension codes are discussed. Examples of codes and idempotents for small values of \(p\) and \(q\) are given. As a by-product almost MDS codes over \(\mathbb {F}_7\) and \(\mathbb {F}_{13}\) are constructed.

Keywords

Cyclic codes Quadratic residue codes Generating idempotents Dual codes 

References

  1. 1.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error-Correcting Codes. North Holland Publishing Co, Amsterdam (1977)zbMATHGoogle Scholar
  2. 2.
    Pless, V., Qian, Z.: Cyclic codes and quadratic residue codes over \(\mathbb{Z}_4\). IEEE Trans. Inform. Theory 42(5), 1594–1600 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Tan, X.: A family of quadratic residue codes over \(\mathbb{Z}_{2^{m}}\). In: Fourth International Conference on Emerging Intelligent Data Web Technologies, pp. 236–240 (2013)Google Scholar
  4. 4.
    Kaya, A., Yildiz, B., Siap, I.: Quadratic Residue Codes over \(\mathbb{F}_p+v\mathbb{F}_p\) and their Gray Images. arXiv preprint arXiv: 1305, 4508 (2013)Google Scholar
  5. 5.
    Zhang, T., Zhu, S.X.: Quadratic residue codes over \(\mathbb{F}_{p}+v\mathbb{F}_{p}\). J. Univ. Sci. Technol. China 42(3), 208–213 (2012). In ChineseGoogle Scholar
  6. 6.
    Shi, M.J., Solé, P., Wu, B.: Cyclic codes and the weight enumerator of linear codes over \(\mathbb{F}_{2}+v\mathbb{F}_{2}+v^{2}\mathbb{F}_{2}\). Appl. Comput. Math 12(2), 247–255 (2013)MathSciNetGoogle Scholar
  7. 7.
    Kaya, A., Yildiz, B., Siap, I.: New extremal binary self-dual codes of length 68 from quadratic residue codes over \(\mathbb{F}_{2}+u\mathbb{F}_{2}+u^{2}\mathbb{F}_{2}\). Finite Fields Appl. 29, 160–177 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cao, D.C., Zhu, S.X.: Constacyclic codes over the ring \(\mathbb{F}_{q}+v\mathbb{F}_{q}+v^{2}\mathbb{F}_{q}\). J. Hefei Univ. Technol. 36(12), 1534–1536 (2013). In ChinesezbMATHMathSciNetGoogle Scholar
  9. 9.
    de Boer, M.: Almost MDS codes. Des. Codes Crypt. 9(2), 143–155 (1996)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesAnhui UniversityHefeiChina
  2. 2.CNRS/LTCI Telecom Paris TechParisFrance

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