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A class of linear codes with a few weights

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Abstract

Linear codes have been an interesting subject of study for many years, as linear codes with few weights have applications in secrete sharing, authentication codes, association schemes, and strongly regular graphs. In this paper, a class of linear codes with a few weights over the finite field GF(p) are presented and their weight distributions are also determined, where p is an odd prime. Some of the linear codes obtained are optimal in the sense that they meet certain bounds on linear codes.

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References

  1. Anderson, R., Ding, C., Helleseth, T., Kløve, T.: How to build robust shared control systems. Des. Codes Cryptogr. 15(2), 111–124 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system I: The user language. J. Symb. Comput. 24(3), 235–265 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  3. Calderbank, A.R., Goethals, J.M.: Three-weight codes and association schemes. Philips J. Res. 39, 143–152 (1984)

    MathSciNet  MATH  Google Scholar 

  4. Calderbank, A.R., Kantor, W.M.: The geometry of two-weight codes. Bull. London Math. Soc. 18, 97–122 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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)

    Article  MathSciNet  MATH  Google Scholar 

  6. Coulter, R.S.: Further evaluations of Weil sums. Acta Arith 86(3), 217–226 (1998)

    MathSciNet  MATH  Google Scholar 

  7. Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 60(6), 3265–3275 (2015)

    Article  MathSciNet  Google Scholar 

  8. Ding, C., et al.: A class of three-weight and four-weight codes. In: Xing, C. (ed.) Proc. of the Second International Workshop on Coding Theory and Cryptography, Lecture Notes in Computer Science, vol. 5557, pp. 34–42. Springer Verlag, Berlin (2009)

  9. Ding, C.: A construction of binary linear codes from boolean functions. Discret. Math. 339(9), 2288–2303 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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.) Proc. of the First International Workshop on Coding Theory and Cryptography, pp. 119–124. World Scientific, Singapore (2008)

  11. Ding, C., Niederreiter, H.: Cyclotomic linear codes of order 3. IEEE Trans. Inf. Theory 53(6), 2274–2277 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ding, C., Wang, X.: A coding theory construction of new systematic authentication codes. Theor. Comput. Sci. 330, 81–99 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ding, C., Li, C., Li, N., Zhou, Z.: Three-weight cyclic codes and their weight distributions. Discret. Math. 339(2), 415–427 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ding, K., Ding, C.: Binary linear codes with three weights. IEEE Commun. Lett. 18(11), 1879–1882 (2014)

    Article  MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  Google Scholar 

  16. Feng, K., Luo, J.: Value distribution of exponential sums from perfect nonlinear functions and their applications. IEEE Trans. Inform. Theory 53(9), 3035–3041 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Huffman, W. C., Pless, V.: Fundamentals of Error-Correcting Codes. Cambridge University Press, Cambridge (2003)

    Book  MATH  Google Scholar 

  18. Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd ed, vol. 84. Springer-Verlag, New York (1990). Graduate Texts in Mathematics

  19. Li, C., Yue, Q., Li, F.: Weight distributions of cyclic codes with respect to pairwise coprime order elements. Finite Fields Appl. 28, 94–114 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, F., Wang, Q., Lin, D.: A class of three-weight and five-weight linear codes. arXiv:1509.06242v1 (2015)

  21. Lidl, R., Niederreiter, H.: Finite Fields. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  22. Mesnager, S.: Linear codes with few weights from weakly regular bent functions based on a generic construction. IACR Cryptology ePrint Archive, 1103 (2015)

  23. Qi, Y., Tang, C., Huang, D.: Binary linear codes with few weights. IEEE Commun. Lett. 20(2), 208–211 (2016)

    Article  Google Scholar 

  24. Tang, C., Li, N., Qi, Y., Zhou, Z., Helleseth, T.: Linear codes with two or three weights from weakly regular bent functions. IEEE Trans. Inf. Theory 62(3), 1166–1176 (2016)

    Article  MathSciNet  Google Scholar 

  25. Tang, C., Qi, Y., Huang, D.: Two-weight and three-weight linear codes from square functions. IEEE Commun. Lett. 20(1), 29–32 (2016)

    Article  Google Scholar 

  26. Wang, Q., Ding, K., Xue, R.: Binary linear codes with two weights. IEEE Commun. Lett. 19(7), 1097–1100 (2015)

    Article  Google Scholar 

  27. Xiang, C.: Linear codes from a generic construction. Cryptogr. Commun. 8(4), 525–539 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yuan, J., Ding, C.: Secret sharing schemes from three classes of linear codes. IEEE Trans. Inf. Theory 52(1), 206–212 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, Z., Ding, C.: A class of three-weight cyclic codes. Finite Fields Appl. 25, 79–93 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zhou, Z., Li, N., Fan, C., Helleseth, T.: Linear codes with two or three weights from quadratic bent functions, Des. Codes Cryptogr. doi:10.1007/s10623-015-0144-9 (2015)

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Acknowledgments

The authors are very grateful to the reviewers and the Editor, for their comments and suggestions that improved the presentation and quality of this paper. The research of K. Feng was supported by NSFC No. 11471178, 11571007 and the Tsinghua National Lab. for Information Science and Technology. The research of C. Tang was supported by NSFC No. 11401480, 11531002. C. Tang also acknowledges support from 14E013 and CXTD2014-4 of China West Normal University.

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Authors and Affiliations

  1. College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, China

    Can Xiang

  2. School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China

    Chunming Tang

  3. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China

    Keqin Feng

Authors
  1. Can Xiang
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  2. Chunming Tang
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  3. Keqin Feng
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Correspondence to Can Xiang.

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Xiang, C., Tang, C. & Feng, K. A class of linear codes with a few weights. Cryptogr. Commun. 9, 93–116 (2017). https://doi.org/10.1007/s12095-016-0200-y

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  • DOI: https://doi.org/10.1007/s12095-016-0200-y

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