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Complete weight enumerator of torsion codes

  • * Corresponding author: Jian Gao

    * Corresponding author: Jian Gao

This research is supported by the National Natural Science Foundation of China (Nos. 11701336, 11626144, 11671235, 12071264)

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  • In this paper, we introduce two classes of MacDonald codes over the finite non-chain ring $ \mathbb{F}_p+v\mathbb{F}_p+v^2\mathbb{F}_p $ and their torsion codes which are linear codes over $ \mathbb{F}_p $, where $ p $ is an odd prime and $ v^3 = v $. We give the complete weight enumerator of two classes of torsion codes. As an application, systematic authentication codes are obtained by these torsion codes.

    Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.


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  • [1] S. BaeC. Li and Q. Yue, On the complete weight enumerators of some reducible cyclic codes, Discrete Mathematics, 60 (2015), 2275-2287.  doi: 10.1016/j.disc.2015.05.016.
    [2] I. F. Blake and K. Kith, On the complete weight enumerator of Reed-Solomon codes, SIAM Journal on Discrete Mathematics, 4 (1991), 164-171.  doi: 10.1137/0404016.
    [3] Y. Cengellenmis and M. Department, MacDonald codes over the ring $\mathbb{F}_3+ v\mathbb{F}_3$, IUG Journal of Natural and Engineering Studues, 20 (2012), 109-112. 
    [4] C. Colbourn and M. Gupta, On quaternary MacDonald codes, Proceedings ITCC 2003, International Conference on Information Technology: Coding and Computing, 5 (2003), 212-215. 
    [5] A. Dertli and Y. Cengellenmis, Macdonald codes over the ring $\mathbb{F}_2+v\mathbb{F}_2$, International Journal of Algebra, 5 (2011), 985-991. 
    [6] L. Diao, J. Gao and J. Lu, On $\mathbb{Z}_{p}\mathbb{Z}_{p}[v]$-additive cyclic codes, Advances in Mathematics of Communications, 14, (2020), 555–572. doi: 10.3934/amc.2018038.
    [7] C. Ding and J. Yin, Algebraic constructions of constant composition codes, International Conference on Information Technology, 51 (2005), 1585-1589.  doi: 10.1109/TIT.2005.844087.
    [8] C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoretical Computer Science, 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.
    [9] C. DingT. HellesethT. Kløve and X. Wang, A generic construction of Cartesian authentication codes, IEEE Transactions on Information Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.
    [10] T. Helleseth and A. Kholosha, Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Transactions on Information Theory, 52 (2006), 2018-2032.  doi: 10.1109/TIT.2006.872854.
    [11] X. Hou and J. Gao, $\mathbb{Z}_{p}\mathbb{Z}_{p}[v]$-additive cyclic codes are asymptotically good, Journal of Applied Mathematics and Computing, (2020), https://doi.org/10.1007/s12190-020-01466-w.
    [12] A. Kuzmin and A. Nechaev, Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring, Discrete Applied Mathematics, 111 (2001), 117-137.  doi: 10.1016/S0166-218X(00)00348-6.
    [13] C. LiQ. Yue and F.-W. Fu, Complete weight enumerators of some cyclic codes, Designs, Codes and Crytography, 80 (2016), 295-315.  doi: 10.1007/s10623-015-0091-5.
    [14] C. LiS. BaeJ. AhnS. Yang and Z. Yao, Complete weight enumerators of some linear codes and their applications, Designs, Codes and Cryptography, 81 (2016), 153-168.  doi: 10.1007/s10623-015-0136-9.
    [15] J. Luo and T. Helleset, Constant composition codes as subcodes of cyclic codes, IEEE Transactions on Information Theory, 57 (2011), 7482-7488.  doi: 10.1109/TIT.2011.2161631.
    [16] J. E. MacDonald, Design methods for maximum minimum-distance error-correcting codes, IBM Journal of Research and Development, 4 (1960), 43-57.  doi: 10.1147/rd.41.0043.
    [17] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977.
    [18] A. M. Patel, Maximal $q$-ary linear codes with large minimum distance, IEEE Transactions on Information Theory, 21 (1975), 106-110.  doi: 10.1109/tit.1975.1055315.
    [19] R. S. Rees and D. R. Stinson, Combinatorial characterizations of authentication codes Ⅱ, Designs, Codes and Cryptography, 2 (1992), 175-187.  doi: 10.1007/BF00124896.
    [20] G. J. Simmons, Authentication theory coding theory, International Cryptology Conference, (1985), 411–4317.
    [21] X. WangJ. Gao and F.-W. Fu, Secret sharing schemes from linear codes over $\mathbb{F}_p+ v\mathbb{F}_p$, International Journal of Foundations of Computer Science, 27 (2016), 595-605.  doi: 10.1142/S0129054116500180.
    [22] X. WangJ. Gao and F.-W. Fu, Complete weight enumerators of two classes of linear codes, Cryptography and Communications, 9 (2017), 545-562.  doi: 10.1007/s12095-016-0198-1.
    [23] Y. Wang and J. Gao, MacDonald codes over the ring $\mathbb{F}_p+ v\mathbb{F}_p+v^2\mathbb{F}_p$, Computational and Applied Mathematics, 38 (2019), 169. doi: 10.1007/s40314-019-0937-y.
    [24] S. Yang and Z. Yao, Complete weight enumerators of a family of three-weight linear codes, Designs, Codes and Cryptography, 82 (2017), 663-674.  doi: 10.1007/s10623-016-0191-x.
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