Several classes of linear codes and their weight distributions

  • Xiaoqiang Wang
  • Dabin ZhengEmail author
  • Hongwei Liu
Original Paper


In this paper, several classes of two-weight or three-weight linear codes over \( {{\mathbb {F}}}_p\) from quadratic or non-quadratic functions are constructed and their weight distributions are determined. From the constructed codes, we obtain some optimal linear codes with respect to the Singleton bound and the Griesmer bound. These two- or three-weight linear codes may have applications in secret sharing, authentication codes, association schemes and strongly regular graphs.


Linear code Weight distribution DO polynomial Exponential sum Quadratic form 

Mathematics Subject Classification

05A05 11T06 11T55 



The authors would like to thank Prof. Q. Xiang for providing Reference [13], and the reviewers and the editor for their helpful comments and valuable suggestions, which have greatly improved the presentation of this paper. This work of X. Wang and H. Liu was supported by the self-determined research funds of CCNU from the collegess basic research and operation of MOE (Grant No. CCNU18TS028).


  1. 1.
    Bouyukliev, I., Fack, V., Winne, J., Willems, W.: Projective two-weight codes with small parameters and their corresponding graphs. Des. Codes Cryptogr. 41, 59–78 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calderbank, A.R., Kantor, W.M.: The geometry of two-weight codes. Bull Lond. Math. Soc. 18, 97–122 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ding, C., Niederreiter, H.: Cyclotomic linear codes of order 3. IEEE Trans. Inf. Theory 53(6), 2274–2277 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    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, World Scientific, Singapore, pp. 119–124 (2008)Google Scholar
  5. 5.
    Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ding, C.: A construction of binary linear codes from Boolean functions. Discrete Math. 339(9), 2288–2303 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Ding, K., Ding, C.: Binary linear codes with three weights. IEEE Commun. Lett. 18, 1879–1882 (2014)CrossRefGoogle Scholar
  8. 8.
    Ding, K., Ding, C.: A class of two-weight and three-weight codes and their applications in secret sharing. IEEE Trans. Inf. Theory 61(11), 5835–5842 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Faldum, A., Willems, W.: A characterization of MMD codes. IEEE Trans. Inf. Theory 44(4), 1555–1558 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Klapper, A.: Cross-correlations of quadratic form sequences in odd characteristic. Des. Codes Cryptogr. 3, 289–305 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dembowski, P., Ostrom, T.G.: Planes of order \(n\) with collineation groups of order \(n^2\). Math. Zeitschrift 193(3), 239–258 (1968)CrossRefzbMATHGoogle Scholar
  12. 12.
    Draper, S., Hou, X.: Explicit evalution of certain exponential sums of quadratic functions over \({\mathbb{F}}_{p^m}\), \(p\) odd. arXiv:0708.3619
  13. 13.
    Games, R.A.: The geometry of quadrics and correlations of sequences. IEEE Trans. Inf. Theory 32(2), 423–426 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Heng, Z., Yue, Q.: A class of binary linear codes with at most three weights. IEEE Commun. Lett. 19, 1488–1491 (2015)CrossRefGoogle Scholar
  15. 15.
    Heng, Z., Yue, Q., Li, C.: Three classes of linear codes with two or three weights. Discrete Math. 339, 2832–2847 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Heng, Z., Yue, Q.: Evaluation of the Hamming weights of a classes of linear codes based on Gauss sums. Des. Codes Cryptogr. 83(2), 307–326 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Kløve, T.: Codes for Error Detection. World Scientific, Hackensack (2007)CrossRefzbMATHGoogle Scholar
  18. 18.
    Lidl, R., Niederreiter, H.: Finite Fields, Encyclopedia of Mathematics, vol. 20. Cambridge University Press, Cambridge (1983)zbMATHGoogle Scholar
  19. 19.
    Li, F., Wang, Q., Lin, D.: A class of three-weight and five-weight linear codes. Discrete Appl Math 241(31), 25–38 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    MacWilliams, F.J., Sloane, N.J.A.: The Theory of Error Correcting Codes. Elsevier, Amsterdam (1977)zbMATHGoogle Scholar
  21. 21.
    Wang, Q., Ding, K., Xue, R.: Binary linear codes with two weight. IEEE Commun. Lett. 19, 1097–1100 (2015)CrossRefGoogle Scholar
  22. 22.
    Xia, Y., Li, C.: Three-weight ternary linear codes from a family of power functions. Finite Fields Appl. 46, 17–37 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Xiang, C.: Linear codes from a generic construction. Cryptogr. Commun. 8, 525–539 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Yuan, J., Carlet, C., Ding, C.: The weight distribution of a class of linear codes from perfect nonlinear functions. IEEE Trans. Inf. Theory 52(2), 712–717 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Zeng, X., Hu, L., Jiang, W., Yue, Q., Cao, X.: The weight distribution of a class of p-ary cyclic codes. Finite Fields Appl. 16, 56–73 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Zhang, D., Fan, C., Peng, D., Tang, X.: Complete weight enumerators of some linear codes from quadratic forms. Cryptogr. Commun. 9, 151–163 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zheng, D., Bao, J.: Four classes of linear codes from cyclotomic cosets. Des. Codes Cryptogr. 86, 1007–1022 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Zhou, Z., Ding, C.: Seven classes of three-weight cyclic codes. IEEE Trans. Commun. 61(10), 4120–4126 (2013)CrossRefGoogle Scholar
  30. 30.
    Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsCentral China Normal UniversityWuhanChina
  2. 2.Hubei Province Key Laboratory of Applied Mathematics, Faculty of Mathematics and StatisticsHubei UniversityWuhanChina

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