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February  2021, 15(1): 73-97. doi: 10.3934/amc.2020044

A class of linear codes and their complete weight enumerators

Dandan Wang 1, , Xiwang Cao 2,1,, and  Gaojun Luo 1,

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

* Corresponding author: Xiwang Cao

Received  April 2019 Revised  June 2019 Published  February 2021 Early access  November 2019

Fund Project: This work was supported by the National Natural Science Foundation of China (Grant Nos. 11771007 and 61572027)

Let
$ {\mathbb F}_q $
 be the finite field with
$ q = p^m $
 elements, where
$ p $
 is an odd prime and
$ m $
 is a positive integer. Let
$ \operatorname{Tr}_m $
 denote the trace function from
$ {\mathbb F}_q $
 onto
$ {\mathbb F}_p $
, and the defining set
$ D\subset {\mathbb F}_q^t $
, where
$ t $
 is a positive integer. In this paper, the set
$ 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\} $
. Define the
$ p $
-ary linear code
$ {\mathcal C}_D $
 by
$ \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*} $
where
$ \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}. $
We evaluate the complete weight enumerator of the linear codes
$ {\mathcal C}_D $
, and present its weight distributions. Some examples are given to illustrate the results.
Keywords: Linear code, complete weight enumerator, cyclotomic number, Gauss sum.
Mathematics Subject Classification: Primary: 94B05; Secondary: 11T71.
Citation: Dandan Wang, Xiwang Cao, Gaojun Luo. A class of linear codes and their complete weight enumerators. Advances in Mathematics of Communications, 2021, 15 (1) : 73-97. doi: 10.3934/amc.2020044
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, III. 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17] R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

T. Storer, Cyclotomy and Difference Sets, Lectures in Advanced Mathematics, No. 2 Markham Publishing Co., Chicago, III. 1967.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Table 1.  The weight distribution of $ \mathcal{C}_D $ for $ 2\mid mt, (mt)_p = 0 $
Weight Frequency
0 1
$ p^{tm-2} $ $ p-1 $
$ (p-1)p^{tm-3} $ $ p^{tm-1}-p $
$ (p-1)(p^{tm-3}-p^{-2}G_m^t) $ $ (p-1)p^{tm-2} $
$ (p-1)p^{tm-3}+p^{-2}G_m^t $ $ (p-1)^2p^{tm-2} $
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Weight Frequency
0 1
$ p^{tm-2} $ $ p-1 $
$ (p-1)p^{tm-3} $ $ p^{tm-1}-p $
$ (p-1)(p^{tm-3}-p^{-2}G_m^t) $ $ (p-1)p^{tm-2} $
$ (p-1)p^{tm-3}+p^{-2}G_m^t $ $ (p-1)^2p^{tm-2} $
Table 2.  The weight distribution of $ \mathcal{C}_D $ for $ 2\mid mt, (mt)_p\neq0 $
Weight Frequency
0 1
$ p^{tm-2}+p^{-1}G_m^t $ $ p-1 $
$ (p-1)p^{tm-3} $ $ p^{tm-2}-p $
$ (p-1)p^{tm-3}+p^{-2}G_m^t $ $ (p-1)(p^{tm-2}+p^{-1}G_m^t) $
$ (p-1)p^{tm-3}+p^{-1}G_m^t $ $ (p-1)(p^{tm-2}-1) $
$ (p-1)p^{tm-3}+p^{-2}(p+1)G_m^t $ $ \frac{1}{2}(p-1)(p-2)(p^{tm-2}+p^{-1}G_m^t) $
$ (p-1)(p^{tm-3}+p^{-2}G_m^t) $ $ \frac{1}{2}(p-1)(p^{tm-1}-G_m^t) $
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Weight Frequency
0 1
$ p^{tm-2}+p^{-1}G_m^t $ $ p-1 $
$ (p-1)p^{tm-3} $ $ p^{tm-2}-p $
$ (p-1)p^{tm-3}+p^{-2}G_m^t $ $ (p-1)(p^{tm-2}+p^{-1}G_m^t) $
$ (p-1)p^{tm-3}+p^{-1}G_m^t $ $ (p-1)(p^{tm-2}-1) $
$ (p-1)p^{tm-3}+p^{-2}(p+1)G_m^t $ $ \frac{1}{2}(p-1)(p-2)(p^{tm-2}+p^{-1}G_m^t) $
$ (p-1)(p^{tm-3}+p^{-2}G_m^t) $ $ \frac{1}{2}(p-1)(p^{tm-1}-G_m^t) $
Table 3.  The weight distribution of $ \mathcal{C}_D $ for $ 2\nmid mt, (mt)_p = 0 $
Weight Frequency
0 1
$ p^{tm-2} $ $ p-1 $
$ (p-1)p^{tm-3} $ $ 2p^{tm-1}-p^{tm-2}-p $
$ (p-1)p^{tm-3}+p^{-2}G_m^tG $ $ \frac{1}{2}(p-1)^2p^{tm-2} $
$ (p-1)p^{tm-3}-p^{-2}G_m^tG $ $ \frac{1}{2}(p-1)^2p^{tm-2} $
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Weight Frequency
0 1
$ p^{tm-2} $ $ p-1 $
$ (p-1)p^{tm-3} $ $ 2p^{tm-1}-p^{tm-2}-p $
$ (p-1)p^{tm-3}+p^{-2}G_m^tG $ $ \frac{1}{2}(p-1)^2p^{tm-2} $
$ (p-1)p^{tm-3}-p^{-2}G_m^tG $ $ \frac{1}{2}(p-1)^2p^{tm-2} $
Table 4.  The weight distribution of $ \mathcal{C}_D $ for $ 2\nmid mt, (mt)_p\neq0 $
Weight Frequency
0 1
$ n $ $ p-1 $
$ (p-1)p^{tm-3} $ $ n+p^{-1}\eta(-(mt)_p)G_m^tG-1 $
$ n-p^{tm-3} $ $ (p-1)(2n+p^{-1}\eta(-(mt)_p)G_m^tG-1) $
$ n-p^{tm-3}+p^{-2}G_m^tG $ $\Gamma = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \frac{1}{2}(p - 1)(p - 2)n\;{\rm{if}}\;\eta ({(mt)_p}) = 1\\ \frac{1}{2}p(p - 1)n\;\;\;\;\;\;\;\;{\rm{if}}\;\eta ({(mt)_p}) = - 1 \end{array} \end{array}} \right. $
$ n-p^{tm-3}-p^{-2}G_m^tG $ $\Gamma ' = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \frac{1}{2}p(p - 1)n\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = 1\\ \frac{1}{2}(p - 1)(p - 2)n\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = - 1 \end{array} \end{array}} \right.$
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Weight Frequency
0 1
$ n $ $ p-1 $
$ (p-1)p^{tm-3} $ $ n+p^{-1}\eta(-(mt)_p)G_m^tG-1 $
$ n-p^{tm-3} $ $ (p-1)(2n+p^{-1}\eta(-(mt)_p)G_m^tG-1) $
$ n-p^{tm-3}+p^{-2}G_m^tG $ $\Gamma = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \frac{1}{2}(p - 1)(p - 2)n\;{\rm{if}}\;\eta ({(mt)_p}) = 1\\ \frac{1}{2}p(p - 1)n\;\;\;\;\;\;\;\;{\rm{if}}\;\eta ({(mt)_p}) = - 1 \end{array} \end{array}} \right. $
$ n-p^{tm-3}-p^{-2}G_m^tG $ $\Gamma ' = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \frac{1}{2}p(p - 1)n\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = 1\\ \frac{1}{2}(p - 1)(p - 2)n\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = - 1 \end{array} \end{array}} \right.$
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