Designs, Codes and Cryptography

, Volume 87, Issue 1, pp 15–29 | Cite as

Several classes of linear codes with a few weights from defining sets over \(\mathbb {F}_p+u\mathbb {F}_p\)

  • Haibo Liu
  • Qunying LiaoEmail author


Recently, linear codes with a few weights have been extensively studied due to their applications in secret sharing schemes, authentication codes, constant composition codes. Results have shown that some optimal codes can be acquired if the defining sets are well chosen over finite fields. In this paper, we investigate the Lee-weight distribution of linear codes over the ring \(\mathbb {F}_p +u\mathbb {F}_p\) (p is an odd prime) based on defining sets by employing exponential sums. We then determine the explicit complete weight enumerator for the images of these linear codes under the Gray map. A class of constant weight codes that meets the Griesmer bound for constructing optimal constant composition codes achieving the LVFC bound is also presented.


Exponential sum Linear code Weight distribution Complete weight enumerator Constant composition code 

Mathematics Subject Classification

94B05 94B15 94B60 



The authors wish to thank the reviewers for valuable comments and suggestions which greatly helped us to improve this paper. This work is supported by the Natural Science Foundation of China with No. 11401408, Project of science and Technology Department of Sichuan Province with No. 2016JY0134.


  1. 1.
    Ahn J., Ka D., Li C.: Complete weight enumerators of a class of linear codes. Des. Codes Cryptogr. 83(1), 1–17 (2017).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Anderson R., Ding C., Helleseth T., Kløve T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Andrews G.E.: The Theory of Partitions. Cambridge University Press, Cambridge (1998).zbMATHGoogle Scholar
  4. 4.
    Assmus E.F., Mattson H.F.: Coding and combinatorics. Siam Rev. 16(3), 349–388 (1974).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bae S., Li C., Yue Q.: On the complete weight enumerators of some reducible cyclic codes. Discret. Math. 338(12), 2275–2287 (2015).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Byrne E., Greferath M., O’Sullivan M.E.: The linear programming bound for codes over finite Frobenius rings. Des. Codes Cryptogr. 42(3), 289–301 (2007).MathSciNetzbMATHGoogle Scholar
  7. 7.
    Calderbank A.R., Goethals J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984).MathSciNetzbMATHGoogle Scholar
  8. 8.
    Carlet C., Ding C., Yuan J.: Linear codes from perfect nonlinear mappings and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).MathSciNetzbMATHGoogle Scholar
  9. 9.
    Ding C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Ding K., Ding C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014).Google Scholar
  11. 11.
    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).MathSciNetzbMATHGoogle Scholar
  12. 12.
    Ding C., Wang X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330(1), 81–99 (2005).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Ding C., Helleseth T., Kløve T., X Wang: A general construction of authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007).zbMATHGoogle Scholar
  14. 14.
    Hammons A.R., Kumar P.V., Calderbank A.R., Sloane N.J.A., Solé P.: The $\mathbb{Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes. IEEE Trans. Inf. Theory 40, 301–319 (1994).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Helleseth T., Kholosha A.: Monomial and quadratic bent functions over the finite fields of odd characteristic. IEEE Trans. Inf. Theory 52(5), 2018–2032 (2006).MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kerdock A.M.: A class of low-rate nonlinear binary codes. Inf. Control 20(2), 182–187 (1972).MathSciNetzbMATHGoogle Scholar
  17. 17.
    Li C., Yue Q., Li F.: Hamming weights of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 60(7), 3895–3902 (2014).MathSciNetzbMATHGoogle Scholar
  18. 18.
    Li C., Yue Q., Fu F.W.: Complete weight enumerators of some cyclic codes. Des. Codes Cryptogr. 80(2), 295–315 (2016).MathSciNetzbMATHGoogle Scholar
  19. 19.
    Li C., Bae S., Ahn J., Yang S., Yao Z.: Complete weight enumerators of some linear codes and their applications. Des. Codes Cryptogr. 81(1), 153–168 (2016).MathSciNetzbMATHGoogle Scholar
  20. 20.
    Li F., Wang Q., Lin D.: A class of three-weight and five-weight linear codes. Discret. Appl. Math. (2017).Google Scholar
  21. 21.
    Li F., Wang Q., Lin D.: Complete weight enumerators of a class of three-weight linear codes. J. Appl. Math. Comput. 55(1–2), 733–747 (2017).MathSciNetzbMATHGoogle Scholar
  22. 22.
    Lidl R., Niederreiter H., Cohn F.M.: Finite Fields. Cambridge University Press, Cambridge (1997).Google Scholar
  23. 23.
    Liu H., Maouche Y.: Several classes of trace codes with either optimal two weights or a few weights over $\mathbb{F}_{q}+u\mathbb{F}_{q}$. arXiv:1703.04968 [cs.IT].Google Scholar
  24. 24.
    Luo Y., Fu F., Vinck A.J.H., Chen W.: On constant-composition codes over $\mathbb{Z}_p$. IEEE Trans. Inf. Theory 49(11), 3010–3016 (2009).zbMATHGoogle Scholar
  25. 25.
    Nechaev A.A.: Kerdock’s code in cyclic form. Diskr. Mat. 4, 123–139 (1989).zbMATHGoogle Scholar
  26. 26.
    Preparata F.P.: A class of optimum nonlinear double-error-correcting codes. Inf. Control 13(4), 378–400 (1968).MathSciNetzbMATHGoogle Scholar
  27. 27.
    Shi M., Liu Y., Solé P.: Optimal two-weight codes from trace codes over $\mathbb{F}_2+ u\mathbb{F}_2$. IEEE Commun. Lett. 20(12), 2346–2349 (2016).Google Scholar
  28. 28.
    Shi M., Luo Y., Solé P.: Construction of one-Lee weight and two-Lee weight codes over $\mathbb{F}_p+ v\mathbb{F}_p$. J. Sys. Sci. Compl. 30(2), 484–493 (2017).zbMATHGoogle Scholar
  29. 29.
    Sollé P.: Codes Over Rings. World Scientific Press, Singapore (2009).Google Scholar
  30. 30.
    Wang Q., Ding K., Xue R.: Binary linear codes with two weights. IEEE Commun. Lett. 19(7), 1097–1100 (2015).Google Scholar
  31. 31.
    Yang S., Yao Z.A.: Complete weight enumerators of a family of three-weight linear codes. Des. Codes Cryptogr. 82(3), 663–674 (2017).MathSciNetzbMATHGoogle Scholar
  32. 32.
    Yang S., Yao Z.A., Zhao C.A.: A class of three-weight linear codes and their complete weight enumerators. Cryptogr. Commun. 9(1), 133–149 (2017).MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Software ScienceSichuan Normal UniversityChengduChina

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