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

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.

Citation: Xiangrui Meng, Jian Gao. Complete weight enumerator of torsion codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2020124
References:
[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.

show all references

References:
[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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