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doi: 10.3934/amc.2020124
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Complete weight enumerator of torsion codes

Xiangrui Meng and  Jian Gao ,

School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255000, China

* Corresponding author: Jian Gao

Received  July 2020 Revised  September 2020 Early access December 2020

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

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.

Keywords: Complete weight enumerator, Torsion codes, authentication codes.
Mathematics Subject Classification: Primary: 94B05, 94B15; Secondary: 11T71.
Citation: Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020124
References:
[1]

S. Bae, C. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Ding, T. Helleseth, T. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

C. Li, Q. 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.  Google Scholar

[14]

C. Li, S. Bae, J. Ahn, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

G. J. Simmons, Authentication theory coding theory, International Cryptology Conference, (1985), 411–4317. Google Scholar

[21]

X. Wang, J. 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.  Google Scholar

[22]

X. Wang, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. Bae, C. 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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

C. Ding, T. Helleseth, T. 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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[13]

C. Li, Q. 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.  Google Scholar

[14]

C. Li, S. Bae, J. Ahn, S. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-correcting Codes, North-Holland, Amsterdam, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

G. J. Simmons, Authentication theory coding theory, International Cryptology Conference, (1985), 411–4317. Google Scholar

[21]

X. Wang, J. 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.  Google Scholar

[22]

X. Wang, J. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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