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Complete weight enumerators of a class of two-weight linear codes

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Abstract

Recently, linear codes constructed from defining sets have been investigated extensively and they have many applications. For an odd prime \(p\), we determine the complete weight enumerator and weight enumerator of a class of \(p\)-ary linear codes by choosing a proper defining set. The results show that they have at most two weights and are suitable for applications in secret sharing schemes.

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References

  1. Ahn, J., Ka, D., Li, C.: Complete weight enumerators of a class of linear codes. Des. Codes Crypt. 83, 83–99 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bae, S., Li, C., Yue, Q.: On the complete weight enumerators of some reducible cyclic codes. Discret. Math. 338(12), 2275–2287 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Blake, I.F., Kith, K.: On the complete weight enumerator of Reed-Solomon codes. SIAM J. Discret. Math. 4(2), 164–171 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Coulter, R.S.: Explicit evaluations of some Weil sums. Acta Arithmetica 83(3), 241–251 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  5. Coulter, R.S.: The number of rational points of a class of artincschreier curves. Finite Fields Appl. 8, 397–413 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ding, C.: Linear codes from some 2-designs. IEEE Trans. Inf. Theory 61(6), 3265–3275 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ding, C., Helleseth, T., Kløve, T., Wang, X.: A generic construction of Cartesian authentication codes. IEEE Trans. Inf. Theory 53(6), 2229–2235 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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 

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

    Article  Google Scholar 

  10. 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  MATH  Google Scholar 

  11. Dinh, H.Q., Li, C., Yue, Q.: Recent progress on weight distributions of cyclic codes over finite fields. J. Algebra Comb. Discret. Struct. Appl. 2(1), 39–63 (2015)

    MathSciNet  MATH  Google Scholar 

  12. Feng, K., Luo, J.: Weight distribution of some reducible cyclic codes. Finite Fields Appl. 14(2), 390–409 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Heng, Z., Yue, Q.: Complete weight distributions of two classes of cyclic codes. Cryptogr. Commun. 9(3), 323–343 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kith, K.: Complete weight enumeration of Reed-Solomon codes. Master’s thesis, Department of Electrical and Computing Engineering, University of Waterloo (1989)

  16. Kuzmin, A., Nechaev, A.: Complete weight enumerators of generalized Kerdock code and linear recursive codes over Galois ring. In: Workshop on Coding and Cryptography, pp. 332–336 (1999)

  17. Li, C., Bae, S., Ahn, J., Yang, S., Yao, Z.: Complete weight enumerators of some linear codes and their applications. Des. Codes Crypt. 81, 153–168 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Li, C., Yue, Q., Fu, F.W.: Complete weight enumerators of some cyclic codes. Des. Codes Crypt. 80, 295–315 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, p 20. Addison-Wesley, Reading (1983)

    Google Scholar 

  20. MacWilliams, F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes, vol. 16. North-Holland Publishing, Amsterdam (1977)

    Google Scholar 

  21. Storer, T.: Cyclotomy and Difference Sets. Markham Publishing Company, Markham (1967)

    MATH  Google Scholar 

  22. Vega, G.: The weight distribution of an extended class of reducible cyclic codes. IEEE Trans. Inf. Theory 58(7), 4862–4869 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  23. Wang, Q., Li, F., Ding, K., Lin, D.: Complete weight enumerators of two classes of linear codes. Discret. Math. 340, 467–480 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  24. Yang, S., Kong, X., Tang, C.: A construction of linear codes and their complete weight enumerators. Finite Fields Appl. 48, 196–226 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, S., Yao, Z., Zhao, C.: The weight enumerator of the duals of a class of cyclic codes with three zeros. Applic. Algebra Eng. Commun. Comput. 26(4), 347–367 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Yang, S., Yao, Z., Zhao, C.: The weight distributions of two classes of \( p \)-ary cyclic codes with few weights. Finite Fields Appl. 44, 76–91 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  27. Yang, S., Yao, Z.: Complete weight enumerators of a class of linear codes. Discret. Math. 340, 729–739 (2017)

    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. Zheng, D., Wang, X., Yu, L., Liu, H.: The weight enumerators of several classes of \(p\)-ary cyclic codes. Discret. Math. 338(7), 1264–1276 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

The work is partially supported by the National Natural Science Foundation of China (11701317, 61772015, 61472457, 11571380), China Postdoctoral Science Foundation Funded Project (2017M611801) and Jiangsu Planned Projects for Postdoctoral Research Funds (1701104C). This work is also partially supported by Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04).

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Authors and Affiliations

  1. School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, People’s Republic of China

    Shudi Yang & Xiangli Kong

  2. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, People’s Republic of China

    Qin Yue & Yansheng Wu

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  1. Shudi Yang
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  2. Qin Yue
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  3. Yansheng Wu
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  4. Xiangli Kong
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Correspondence to Shudi Yang.

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Yang, S., Yue, Q., Wu, Y. et al. Complete weight enumerators of a class of two-weight linear codes. Cryptogr. Commun. 11, 609–620 (2019). https://doi.org/10.1007/s12095-018-0315-4

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  • DOI: https://doi.org/10.1007/s12095-018-0315-4

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