# American Institute of Mathematical Sciences

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

## 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

* 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.
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.

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