Designs, Codes and Cryptography

, Volume 86, Issue 5, pp 1007–1022 | Cite as

Four classes of linear codes from cyclotomic cosets

Article

Abstract

This paper presents four classes of linear codes from coset representatives of subgroups and cyclotomic coset families of certain finite field, and determines their weight enumerators. These linear codes may have applications in consumer electronics, communications and secret sharing schemes.

Keywords

Linear code Cyclic code Weight distribution Gauss sum 

Mathematics Subject Classification

11T71 94B15 

Notes

Acknowledgements

The authors wish to thank prof. Qing Xiang for his helpful comments. The work was partially supported by National Natural Science Foundation of China (NSFC) under Grant 11101131.

References

  1. 1.
    Arno S.: The imaginary quadratic fields of class number 4. Acta Arith. 60, 321–334 (1992).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015).MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Ding C.: A construction of binary linear codes from Boolean functions. Discret. Math. 339(9), 2288–2303 (2016).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ding K., Ding C.: Binary linear codes with three weights. IEEE Commun. Lett. 18, 1879–1882 (2014).CrossRefGoogle Scholar
  5. 5.
    Ding C., Niederreiter H.: Cyclotomic linear codes of order 3. IEEE Trans. Inf. Theory 53(6), 2274–2277 (2007).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ding C., Yang J.: Hamming weights in Irreducible cyclic codes. Discret. Math. 313, 434–446 (2013).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ding C., Luo J., Niederreiter H.: Two weight codes punctured from irreducible cyclic codes. In: Li Y., Ling S., Niederreiter H., Wang H., Xing C., Zhang S (Eds.) Proceedings of the First International Workshop on Coding Theory and Cryptography, pp. 119–124. World Scientific, Singapore (2008)Google Scholar
  8. 8.
    Feng T., Xiang Q.: Strongly regular graphs from unions of cyclotomic classes. J. Comb. Theory (B) 102, 982–995 (2012).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Heng Z., Yue Q.: A class of binary linear codes with at most three weights. IEEE Commun. Lett. 19, 1488–1491 (2015).CrossRefGoogle Scholar
  10. 10.
    Heng Z., Yue Q.: Evaluation of the Hamming weights of a classes of linear codes based on Gauss sums. Des. Codes Cryptogr. (2016). doi: 10.1007/s10623-016-0222-7.
  11. 11.
    Heng Z., Yue Q.: Optimal linear codes, constant-weight codes and constant-composition codes over \({\mathbb{F}}_{q}\). arXiv:1605.04063v1.
  12. 12.
    Kløve T.: Codes for Error Detection. World Scientific, Hackensack (2007).CrossRefMATHGoogle Scholar
  13. 13.
    Langevin P.: Calcus de certaines sommes de Gauss. J. Number Theory 63, 59–64 (1997).MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Li C., Bae S., Ahn J., et al.: Complete weight enumerartors of some linear codes and their applications. Des. Codes Cryptogr. 81(1), 153–168 (2016).MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Li F., Wang Q., Lin D.: A class of three-weight and five-wiehgt linear codes. arXiv:1509.06242.
  16. 16.
    Lidl R., Niederreiter H.: Finite Fields. Cambridge University Press, Cambridge (1997).MATHGoogle Scholar
  17. 17.
    Mbodj O.D.: Quadratic Gauss Sums. Finite Fields Appl. 4, 347–361 (1998).MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Meijer P., van der Vlugt M.: The evaluation of Gauss sums for characters of 2-power order. J. Number Theory 100, 381–395 (2003).MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Neukirch J.: Algebraic Number Theory. Springer, Berlin (1999).CrossRefMATHGoogle Scholar
  20. 20.
    Wang Q., Ding K., Xue R.: Binary linear codes with two weight. IEEE Commun. Lett. 19, 1097–1100 (2015).CrossRefGoogle Scholar
  21. 21.
    Xiang C.: Linear codes from a generic construction. Cryptogr. Commun. 8, 525–539 (2016).MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Yang J., Xia L.: Complete solving of explicit evaluation of Gauss sums in the index 2 case. Sci. China Ser. A 53, 2525–2542 (2010).MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Yuan J., Ding C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006).MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Zhou Z., Li N., Fan C., Helleseth T.: Linear codes with two or three weight from quafratic bent functions. Des. Codes Cryptogr. 81(2), 283–295 (2016).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and StatisticsHubei UniversityWuhanChina
  2. 2.Department of MathematicsNingbo UniversityNingboChina

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