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Additive Toeplitz codes over $ \mathbb{F}_{4} $
A class of linear codes and their complete weight enumerators
1. | Department of Math, Nanjing University of Aeronautics and Astronautics, Nanjing, Jiangsu Province 211100, China |
2. | State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China |
$ {\mathbb F}_q $ |
$ q = p^m $ |
$ p $ |
$ m $ |
$ \operatorname{Tr}_m $ |
$ {\mathbb F}_q $ |
$ {\mathbb F}_p $ |
$ D\subset {\mathbb F}_q^t $ |
$ t $ |
$ D = \{(x_1, x_2, \cdots, x_t)\in {\mathbb F}_q^t:\operatorname{Tr}_m(x_1^2+x_2^2+\cdots+x_t^2) = 0, \operatorname{Tr}_m(x_1+x_2+\cdots+x_t) = 1\} $ |
$ p $ |
$ {\mathcal C}_D $ |
$ \begin{eqnarray*} {\mathcal C}_D = \{\textbf{c}(a_1, a_2, \cdots, a_t): (a_1, a_2, \cdots, a_t)\in {\mathbb F}_q^t\}, \end{eqnarray*} $ |
$ \textbf{c}(a_1, a_2, \cdots, a_t) = (\operatorname{Tr}_m(a_1x_1+a_1x_2\cdots+a_tx_t))_{(x_1, \cdots, x_t)\in D}. $ |
$ {\mathcal C}_D $ |
References:
[1] |
J. Ahn, D. Ka and C. J. Li,
Complete weight enumerators of a class of linear codes, Des. Codes Cryptogr., 83 (2017), 83-99.
doi: 10.1007/s10623-016-0205-8. |
[2] |
I. F. Blake and K. Kith,
On the complete weight enumerator of Reed-Solomon codes, SIAM J. Discret. Math., 4 (1991), 164-171.
doi: 10.1137/0404016. |
[3] |
C. S. Ding, J. Q. Luo and H. Niederreiter,
Two-weight codes punctured from irreducible cyclic codes, Coding and Cryptology, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, NJ, 4 (2008), 119-124.
doi: 10.1142/9789812832245_0009. |
[4] |
C. S. Ding,
Optimal constant composition codes from zero-difference balanced functions, IEEE Trans. Inf. Theory, 54 (2008), 5766-5770.
doi: 10.1109/TIT.2008.2006420. |
[5] |
C. S. Ding and J. X. Yin,
A construction of optimal constant composition codes, Des. Codes Cryptogr., 40 (2006), 157-165.
doi: 10.1007/s10623-006-0004-8. |
[6] |
C. S. Ding, T. Helleseth, T. Klove and X. S. Wang,
A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.
doi: 10.1109/TIT.2007.896872. |
[7] |
C. S. Ding and X. S. Wang,
A coding theory construction of new systematic authentication codes, Theory Comput. Sci., 330 (2005), 81-99.
doi: 10.1016/j.tcs.2004.09.011. |
[8] |
K. L. Ding and C. S. Ding,
A class of two-weight and three weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.
doi: 10.1109/TIT.2015.2473861. |
[9] |
T. Helleseth and A. Kholosha,
Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[10] |
K. Kith, Complete Weight Enumeration of Reed-Solomon Codes, Master's Thesis, Department of Electrical and Computing Engineering, University of Waterloo, Waterloo, Ontario, Canada, 1989. Google Scholar |
[11] |
A. Kuzmin and A. Nechaev,
Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring, Discret. Appl. Math., 111 (2001), 117-137.
doi: 10.1016/S0166-218X(00)00348-6. |
[12] |
C. J. Li, S. H. Bae and S. D. Yang,
Some two-weight and three-weight linear codes, Advances in Mathematics of Communications, 13 (2019), 195-211.
doi: 10.3934/amc.2019013. |
[13] |
C. J. Li, Q. Yue and F. W. Fu,
Complete weight enumerators of some cyclic codes, Des. Codes Cryptogr., 80 (2016), 295-315.
doi: 10.1007/s10623-015-0091-5. |
[14] |
C. J. Li and Q. Yue,
Weight distributions of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303.
doi: 10.1109/TIT.2013.2287211. |
[15] |
C. J. Li, S. Bae, J. Ahn, S. D. Yang and Z.-A. Yao,
Complete weight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.
doi: 10.1007/s10623-015-0136-9. |
[16] |
F. Li and Q. Y. Wang,
A class of three-weight and five weight linear codes, Discrete Appl. Math., 241 (2018), 25-38.
doi: 10.1016/j.dam.2016.11.005. |
[17] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[18] |
G. J. Luo, X. W. Cao, S. D. Xu and J. F. Mi,
Binary linear codes with two or three weights from niho exponents, Cryptogr. Commun., 10 (2018), 301-318.
doi: 10.1007/s12095-017-0220-2. |
[19] |
G. J. Luo and X. W. Cao,
Complete weight enumerators of three classes of linear codes, Cryptogr. Commun., 10 (2018), 1091-1108.
doi: 10.1007/s12095-017-0270-5. |
[20] |
M. J. Shi, Y. Guan and P. Solé,
Two new families of two-weight codes, IEEE Trans. Inf. Theory, 63 (2017), 6240-6246.
doi: 10.1109/TIT.2017.2742499. |
[21] |
M. J. Shi, Y. Liu and P. Solé,
Optimal two weight codes from trace codes over a non-chain ring, Discrete Appl. Math., 219 (2017), 176-181.
doi: 10.1016/j.dam.2016.09.050. |
[22] |
M. J. Shi, R. S. Wu, Y. Liu and P. Solé,
Two and three weight codes over $ \mathbb{F}_p+u \mathbb{F}_p$, Cryptogr. Commun., 9 (2017), 637-646.
doi: 10.1007/s12095-016-0206-5. |
[23] |
T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, III. 1967. |
[24] |
S. D. Yang, X. L. Kong and C. M. Tang,
A construction of linear codes and their complete weight enumerator, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[25] |
S. D. Yang, Z.-A. Yao and C.-A. Zhao,
The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Field Appl., 44 (2017), 76-91.
doi: 10.1016/j.ffa.2016.11.004. |
[26] |
S. D. Yang and Z.-A. Yao,
Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[27] |
Z. C. Zhou, N. Li, C. L. Fan and T. Helleseth,
Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.
doi: 10.1007/s10623-015-0144-9. |
[28] |
Z. C. Zhou and C. S. Ding,
A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
show all references
References:
[1] |
J. Ahn, D. Ka and C. J. Li,
Complete weight enumerators of a class of linear codes, Des. Codes Cryptogr., 83 (2017), 83-99.
doi: 10.1007/s10623-016-0205-8. |
[2] |
I. F. Blake and K. Kith,
On the complete weight enumerator of Reed-Solomon codes, SIAM J. Discret. Math., 4 (1991), 164-171.
doi: 10.1137/0404016. |
[3] |
C. S. Ding, J. Q. Luo and H. Niederreiter,
Two-weight codes punctured from irreducible cyclic codes, Coding and Cryptology, Ser. Coding Theory Cryptol., World Sci. Publ., Hackensack, NJ, 4 (2008), 119-124.
doi: 10.1142/9789812832245_0009. |
[4] |
C. S. Ding,
Optimal constant composition codes from zero-difference balanced functions, IEEE Trans. Inf. Theory, 54 (2008), 5766-5770.
doi: 10.1109/TIT.2008.2006420. |
[5] |
C. S. Ding and J. X. Yin,
A construction of optimal constant composition codes, Des. Codes Cryptogr., 40 (2006), 157-165.
doi: 10.1007/s10623-006-0004-8. |
[6] |
C. S. Ding, T. Helleseth, T. Klove and X. S. Wang,
A generic construction of Cartesian authentication codes, IEEE Trans. Inf. Theory, 53 (2007), 2229-2235.
doi: 10.1109/TIT.2007.896872. |
[7] |
C. S. Ding and X. S. Wang,
A coding theory construction of new systematic authentication codes, Theory Comput. Sci., 330 (2005), 81-99.
doi: 10.1016/j.tcs.2004.09.011. |
[8] |
K. L. Ding and C. S. Ding,
A class of two-weight and three weight codes and their applications in secret sharing, IEEE Trans. Inf. Theory, 61 (2015), 5835-5842.
doi: 10.1109/TIT.2015.2473861. |
[9] |
T. Helleseth and A. Kholosha,
Monomial and quadratic bent functions over the finite fields of odd characteristic, IEEE Trans. Inf. Theory, 52 (2006), 2018-2032.
doi: 10.1109/TIT.2006.872854. |
[10] |
K. Kith, Complete Weight Enumeration of Reed-Solomon Codes, Master's Thesis, Department of Electrical and Computing Engineering, University of Waterloo, Waterloo, Ontario, Canada, 1989. Google Scholar |
[11] |
A. Kuzmin and A. Nechaev,
Complete weight enumerators of generalized Kerdock code and related linear codes over Galois ring, Discret. Appl. Math., 111 (2001), 117-137.
doi: 10.1016/S0166-218X(00)00348-6. |
[12] |
C. J. Li, S. H. Bae and S. D. Yang,
Some two-weight and three-weight linear codes, Advances in Mathematics of Communications, 13 (2019), 195-211.
doi: 10.3934/amc.2019013. |
[13] |
C. J. Li, Q. Yue and F. W. Fu,
Complete weight enumerators of some cyclic codes, Des. Codes Cryptogr., 80 (2016), 295-315.
doi: 10.1007/s10623-015-0091-5. |
[14] |
C. J. Li and Q. Yue,
Weight distributions of two classes of cyclic codes with respect to two distinct order elements, IEEE Trans. Inf. Theory, 60 (2014), 296-303.
doi: 10.1109/TIT.2013.2287211. |
[15] |
C. J. Li, S. Bae, J. Ahn, S. D. Yang and Z.-A. Yao,
Complete weight enumerators of some linear codes and their applications, Des. Codes Cryptogr., 81 (2016), 153-168.
doi: 10.1007/s10623-015-0136-9. |
[16] |
F. Li and Q. Y. Wang,
A class of three-weight and five weight linear codes, Discrete Appl. Math., 241 (2018), 25-38.
doi: 10.1016/j.dam.2016.11.005. |
[17] |
R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.
![]() |
[18] |
G. J. Luo, X. W. Cao, S. D. Xu and J. F. Mi,
Binary linear codes with two or three weights from niho exponents, Cryptogr. Commun., 10 (2018), 301-318.
doi: 10.1007/s12095-017-0220-2. |
[19] |
G. J. Luo and X. W. Cao,
Complete weight enumerators of three classes of linear codes, Cryptogr. Commun., 10 (2018), 1091-1108.
doi: 10.1007/s12095-017-0270-5. |
[20] |
M. J. Shi, Y. Guan and P. Solé,
Two new families of two-weight codes, IEEE Trans. Inf. Theory, 63 (2017), 6240-6246.
doi: 10.1109/TIT.2017.2742499. |
[21] |
M. J. Shi, Y. Liu and P. Solé,
Optimal two weight codes from trace codes over a non-chain ring, Discrete Appl. Math., 219 (2017), 176-181.
doi: 10.1016/j.dam.2016.09.050. |
[22] |
M. J. Shi, R. S. Wu, Y. Liu and P. Solé,
Two and three weight codes over $ \mathbb{F}_p+u \mathbb{F}_p$, Cryptogr. Commun., 9 (2017), 637-646.
doi: 10.1007/s12095-016-0206-5. |
[23] |
T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, III. 1967. |
[24] |
S. D. Yang, X. L. Kong and C. M. Tang,
A construction of linear codes and their complete weight enumerator, Finite Fields Appl., 48 (2017), 196-226.
doi: 10.1016/j.ffa.2017.08.001. |
[25] |
S. D. Yang, Z.-A. Yao and C.-A. Zhao,
The weight distributions of two classes of $p$-ary cyclic codes with few weights, Finite Field Appl., 44 (2017), 76-91.
doi: 10.1016/j.ffa.2016.11.004. |
[26] |
S. D. Yang and Z.-A. Yao,
Complete weight enumerators of a class of linear codes, Discrete Math., 340 (2017), 729-739.
doi: 10.1016/j.disc.2016.11.029. |
[27] |
Z. C. Zhou, N. Li, C. L. Fan and T. Helleseth,
Linear codes with two or three weights from quadratic Bent functions, Des. Codes Cryptogr., 81 (2016), 283-295.
doi: 10.1007/s10623-015-0144-9. |
[28] |
Z. C. Zhou and C. S. Ding,
A class of three-weight cyclic codes, Finite Fields Appl., 25 (2014), 79-93.
doi: 10.1016/j.ffa.2013.08.005. |
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