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Designs, Codes and Cryptography

, Volume 81, Issue 1, pp 1–9 | Cite as

Two classes of cyclic codes and their weight enumerator

  • Haode Yan
  • Chunlei Liu
Article
  • 375k Downloads

Abstract

Let p be an odd prime, and mk and d be positive integers such that \(2 \le k\le \frac{m+1}{2}\) and \(\hbox {gcd}(m,d)=1. \pi \) is a primitive element of the finite field \({\mathbb {F}}_{p^{m}}\). The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have 2k zeros \(\pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-1)\) is determined in the present paper. The weight enumerator of cyclic codes over \({\mathbb {F}}_{p}\) whose duals have \(2k-1\) zeros \(\pi ^{-(p^{(k-1)d}+1)/2}, \pi ^{-(p^{jd}+1)/2}\) and \(-\pi ^{-(p^{jd}+1)/2} (j=0,1,\ldots ,k-2)\) is also determined when \(2\not \mid \frac{m}{gcd(m,k-1)}\) holds.

Keywords

Cyclic code Weight enumerator Finite field 

Mathematics Subject Classification

94B15 11T71 

Notes

Acknowledgments

The authors are grateful to the referees for their careful reading of the original version of this paper, their detailed comments and suggestions, which have much improved the quality of this paper.

References

  1. 1.
    Baumert L.D., McEliece R.J.: Weights of irreducible cyclic codes. Inf. Control. 20(2), 158–175 (1972).Google Scholar
  2. 2.
    Baumert L.D., Mykkeltveit J.: Weight distribution of some irreducible cyclic codes. DSN Program Rep. 16, 128–131 (1973)Google Scholar
  3. 3.
    Calderbank A.R., Goethals J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984).Google Scholar
  4. 4.
    Carlet C., Ding C., Yuan J.: Linear codes from highly nonlinear functions and their secret sharing schemes. IEEE Trans. Inf. Theory 51(6), 2089–2102 (2005).Google Scholar
  5. 5.
    Ding C.: The weight distribution of some irreducible cyclic codes. IEEE Trans. Inf. Theory 55(3), 955–960 (2009).Google Scholar
  6. 6.
    Ding C., Yang J.: Hamming weights in irreducible cyclic codes. Discret. Math. 313(4), 434–446 (2013).Google Scholar
  7. 7.
    Ding C., Liu Y., Ma C., Zeng L.: The weight distributions of the duals of cyclic codes with two zeros. IEEE Trans. Inf. Theory 57(12), 8000–8006 (2011).Google Scholar
  8. 8.
    Feng T.: On cyclic codes of length \(2^{2^{r}}-1\) with two zeros whose dual codes have three weights. Des. Codes Cryptogr. 62, 253–258 (2012).Google Scholar
  9. 9.
    Feng K., Luo J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008).Google Scholar
  10. 10.
    Feng T., Momihara K.: Evaluation of the weight distribution of a class of cyclic codes based on index 2 Gauss sums. IEEE Trans. Inf. Theory 59(9), 5980–5984 (2013).Google Scholar
  11. 11.
    Feng T., Leung K., Xiang Q.: Binary cyclic codes with two primitive nonzeros. Sci. China Math. 56(7), 1403–1412 (2012).Google Scholar
  12. 12.
    Li C., Yue Q.: Weight distributions of two classes of cyclic codes with respect to two distinct order elements. IEEE Trans. Inf. Theory 60(1), 296–303 (2014).Google Scholar
  13. 13.
    Li C., Li N., Helleseth T., Ding C.: The weight distributions of several classes of cyclic codes from APN monomials. IEEE Trans. Inf. Theory 60(8), 4710–4721 (2014).Google Scholar
  14. 14.
    Liu Y., Yan H.: A class of five-weight cyclic codes and their weight distribution. Des. Codes Cryptogr. (2015). doi: 10.1007/s10623-015-0056-8.
  15. 15.
    Liu Y., Yan H., Liu C.: A class of six-weight cyclic codes and their weight distribution. Des. Codes Cryptogr. (2014). doi: 10.1007/s10623-014-9984-y.
  16. 16.
    Luo J., Feng K.: Cyclic codes and sequences form generalized Coulter-Matthews function. IEEE Trans. Inf. Theory 54(12), 5345–5353 (2008).Google Scholar
  17. 17.
    Luo J., Feng K.: On the weight distribution of two classes of cyclic codes. IEEE Trans. Inf. Theory 54(12), 5332–5344 (2008).Google Scholar
  18. 18.
    Ma C., Zeng L., Liu Y., Feng D., Ding C.: The weight enumerator of a class of cyclic codes. IEEE Trans. Inf. Theory 57(1), 397–402 (2011).Google Scholar
  19. 19.
    Schmidt K.: Symmetric bilinear forms over finite fields with applications to coding theory, arXiv:1410.7184 (2014).
  20. 20.
    Sharma A., Bakshi G.: The weight distribution of some irreducible cyclic codes. Finite Fields Appl. 18(1), 144–159 (2012).Google Scholar
  21. 21.
    Trachtenberg H.M.: On the crosscorrelation functions of maximal linear recurring sequences. Ph.D. Dissertation, University of Southern California, Los Angels (1970).Google Scholar
  22. 22.
    Vega G.: The weight distribution of an extended class of reducible cyclic codes. IEEE Trans. Inf. Theory 58(7), 4862–4869 (2012).Google Scholar
  23. 23.
    Wang B., Tang C., Qi Y., Yang Y., Xu M.: The weight distributions of cyclic codes and elliptic curves. IEEE Trans. Inf. Theory 58(12), 7253–7259 (2012).Google Scholar
  24. 24.
    Xiong M.: The weight distributions of a class of cyclic codes. Finite Fields Appl. 18(5), 933–945 (2012).Google Scholar
  25. 25.
    Yang J., Xiong M., Ding C., Luo J.: Weight distribution of a class of cyclic codes with arbitrary number of zeros. IEEE Trans. Inf. Theory 59(9), 5985–5993 (2013).Google Scholar
  26. 26.
    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).Google Scholar
  27. 27.
    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(1), 56–73 (2010).Google Scholar
  28. 28.
    Zheng D., Wang X., Zeng X., Hu L.: The weight distribution of a family of p-ary cyclic codes. Des. Codes Cryptogr. 75(2), 263–275 (2015).Google Scholar
  29. 29.
    Zhou Z., Ding C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014).Google Scholar
  30. 30.
    Zhou Z., Ding C., Luo J., Zhang A.: A family of five-weight cyclic codes and their weight enumerators. IEEE Trans. Inf. Theory 59(10), 6674–6682 (2013).Google Scholar
  31. 31.
    Zhu X., Yue Q., Hu L.: Weight distributions of cyclic codes of length \(l^m\). Finite Fields Appl. 31, 241–257 (2015).Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of MathematicsShanghai Jiaotong UniversityShanghaiChina

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