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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.

[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.

[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.

[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.

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} $
Table Options

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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) $
Table Options

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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} $
Table Options

Download as excel

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 $ $\Gamma ' = \left\{ {\begin{array}{*{20}{l}} \begin{array}{l} \frac{1}{2}p(p - 1)n\;\;\;\;\;\;\;\;\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = 1\\ \frac{1}{2}(p - 1)(p - 2)n\;\;\;{\rm{if}}\;\;\eta ({(mt)_p}) = - 1 \end{array} \end{array}} \right.$
Table Options

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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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