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Complete weight enumerator of torsion codes
School of Mathematics and Statistics, Shandong University of Technology, Zibo, Shandong 255000, China |
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.
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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. 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. |
| [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. 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. |
| [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. |
| [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. 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. |
| [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. |
| [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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