American Institute of Mathematical Sciences

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February  2021, 15(1): 99-112. doi: 10.3934/amc.2020045

Complete weight enumerators of a class of linear codes over finite fields

Shudi Yang 1, , Xiangli Kong 1,, and  Xueying Shi 2,

1. 

School of Mathematical Sciences, Qufu Normal University, Shandong, 273165, China
2. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 211100, China

*Corresponding author: Xiangli Kong

Received  April 2019 Published  February 2021 Early access  November 2019

Fund Project: The work is partially supported by the National Natural Science Foundation of China (11701317, 11801303, 11571380) and the Natural Science Foundation of Shandong Province of China (ZR2016AM04). This work is also partially supported by Guangzhou Science and Technology Program (201607010144) and the Natural Science Foundation of Jiangsu Higher Education Institutions of China (17KJB110018)

We investigate a class of linear codes by choosing a proper defining set and determine their complete weight enumerators and weight enumerators. These codes have at most three weights and some of them are almost optimal so that they are suitable for applications in secret sharing schemes. This is a supplement of the results raised by Wang et al. (2017) and Kong et al. (2019).

Keywords: Linear code, complete weight enumerator, weight enumerator, exponential sum.
Mathematics Subject Classification: Primary: 94B05, 11T71; Secondary: 11T23.
Citation: Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045
References:
[1]

J. Ahn, D. Ka and C. J. Li, Complete weight enumerators of a class of linear codes, Designs, Codes and Cryptography, 83 (2017), 83-99.  doi: 10.1007/s10623-016-0205-8.

[2]

S. Bae, C. J. Li and Q. Yue, On the complete weight enumerators of some reducible cyclic codes, Discrete Mathematics, 338 (2015), 2275-2287.  doi: 10.1016/j.disc.2015.05.016.

[3]

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.

[4]

R. S. Coulter, Explicit evaluations of some Weil sums, Acta Arithmetica, 83 (1998), 241-251.  doi: 10.4064/aa-83-3-241-251.

[5]

R. S. Coulter, Further evaluations of Weil sums, Acta Arithmetica, 86 (1998), 217-226.  doi: 10.4064/aa-86-3-217-226.

[6]

C. S. Ding, Linear codes from some 2-designs, IEEE Transactions on Information Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[7]

C. S. Ding, T. Helleseth, T. Kløve and X. S. Wang, A generic construction of Cartesian authentication codes, IEEE Transactions on Information Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.

[8]

C. S. Ding and X. S. 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. S. Ding and J. X. Yin, A construction of optimal constant composition codes, Designs, Codes and Cryptography, 40 (2006), 157-165.  doi: 10.1007/s10623-006-0004-8.

[10]

K. L. Ding and C. S. Ding, Binary linear codes with three weights, IEEE Communications Letters, 18 (2014), 1879-1882.  doi: 10.1109/LCOMM.2014.2361516.

[11]

K. Ding and C. S. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Transactions on Information Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[12]

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.

[13]

Z. L. Heng and Q. Yue, Complete weight distributions of two classes of cyclic codes, Cryptography and Communications, 9 (2017), 323-343.  doi: 10.1007/s12095-015-0177-y.

[14]

K. Kith, Complete Weight Enumeration of Reed-Solomon Codes, Master's thesis, University of Waterloo in Waterloo, 1989.

[15]

X. L. Kong and S. D. Yang, Complete weight enumerators of a class of linear codes with two or three weights, Discrete Mathematics, 342 (2019), 3166-3176.  doi: 10.1016/j.disc.2019.06.025.

[16]

C. J. Li, S. Bae, J. Ahn, S. D. Yang and Z. A. 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.

[17]

C. J. Li, S. 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.

[18]

C. J. Li, Q. Yue and F. W. Fu, Complete weight enumerators of some cyclic codes, Designs, Codes and Cryptography, 80 (2016), 295-315.  doi: 10.1007/s10623-015-0091-5.

[19]

C. J. Li, Q. Yue and Z. L. Heng, Weight distributions of a class of cyclic codes from $ \mathbb{F}_l $-conjugates, Advances in Mathematics of Communications, 9 (2015), 341-352.  doi: 10.3934/amc.2015.9.341.

[20]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983.

[21]

H. B. Liu, Q. Y. Liao and X. F. Wang, Complete weight enumerator for a class of linear codes from defining sets and their applications, Journal of Systems Science and Complexity, 32 (2019), 947-969.  doi: 10.1007/s11424-018-7414-3.

[22]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[23]

A. Sharma and G. K. Bakshi, The weight distribution of some irreducible cyclic codes, Finite Fields and Their Applications, 18 (2012), 144-159.  doi: 10.1016/j.ffa.2011.07.002.

[24]

M. J. Shi and Y. P. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.

[25]

M. J. Shi, L. Q. Qian, L. Sok, N. Aydin and P. Solé, On constacyclic codes over $ \mathbb{Z}_4[u]/\langle u^2-1 \rangle$ and their Gray images, Finite Fields and Their Applications, 45 (2017), 86-95.  doi: 10.1016/j.ffa.2016.11.016.

[26]

M. J. Shi, R. S. Wu, L. Q. Qian, L. Sok and P. Solé, New classes of $p$-ary few weight codes, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1.

[27]

L. Sok, M. J. Shi and P. Solé, Construction of optimal LCD codes over large finite fields, Finite Fields and Their Applications, 50 (2018), 138-153.  doi: 10.1016/j.ffa.2017.11.007.

[28]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, Journal of Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

[29]

G. Vega, The weight distribution for any irreducible cyclic code of length $p^m$, Applicable Algebra in Engineering, Communication and Computing, 29 (2018), 363-370.  doi: 10.1007/s00200-017-0347-6.

[30]

Q. Y. Wang, F. Li, K. L. Ding and D. D. Lin, Complete weight enumerators of two classes of linear codes, Discrete Mathematics, 340 (2017), 467-480.  doi: 10.1016/j.disc.2016.09.003.

[31]

Y. S. Wu, Q. Yue and S. Q. Fan, Further factorization of $x^n -1$ over a finite field, Finite Fields and Their Applications, 54 (2018), 197-215.  doi: 10.1016/j.ffa.2018.07.007.

[32]

Y. S. Wu, Q. Yue, X. M. Zhu and S. D. Yang, Weight enumerators of reducible cyclic codes and their dual codes, Discrete Mathematics, 342 (2019), 671-682.  doi: 10.1016/j.disc.2018.10.035.

[33]

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 Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[34]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight enumerator of the duals of a class of cyclic codes with three zeros, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 347-367.  doi: 10.1007/s00200-015-0255-6.

[35]

S. D. Yang, X. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields and Their Applications, 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[36]

S. D. Yang and Z. A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[37]

J. Yuan and C. S. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

show all references

References:
[1]

J. Ahn, D. Ka and C. J. Li, Complete weight enumerators of a class of linear codes, Designs, Codes and Cryptography, 83 (2017), 83-99.  doi: 10.1007/s10623-016-0205-8.

[2]

S. Bae, C. J. Li and Q. Yue, On the complete weight enumerators of some reducible cyclic codes, Discrete Mathematics, 338 (2015), 2275-2287.  doi: 10.1016/j.disc.2015.05.016.

[3]

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.

[4]

R. S. Coulter, Explicit evaluations of some Weil sums, Acta Arithmetica, 83 (1998), 241-251.  doi: 10.4064/aa-83-3-241-251.

[5]

R. S. Coulter, Further evaluations of Weil sums, Acta Arithmetica, 86 (1998), 217-226.  doi: 10.4064/aa-86-3-217-226.

[6]

C. S. Ding, Linear codes from some 2-designs, IEEE Transactions on Information Theory, 61 (2015), 3265-3275.  doi: 10.1109/TIT.2015.2420118.

[7]

C. S. Ding, T. Helleseth, T. Kløve and X. S. Wang, A generic construction of Cartesian authentication codes, IEEE Transactions on Information Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.

[8]

C. S. Ding and X. S. 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. S. Ding and J. X. Yin, A construction of optimal constant composition codes, Designs, Codes and Cryptography, 40 (2006), 157-165.  doi: 10.1007/s10623-006-0004-8.

[10]

K. L. Ding and C. S. Ding, Binary linear codes with three weights, IEEE Communications Letters, 18 (2014), 1879-1882.  doi: 10.1109/LCOMM.2014.2361516.

[11]

K. Ding and C. S. Ding, A class of two-weight and three-weight codes and their applications in secret sharing, IEEE Transactions on Information Theory, 61 (2015), 5835-5842.  doi: 10.1109/TIT.2015.2473861.

[12]

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.

[13]

Z. L. Heng and Q. Yue, Complete weight distributions of two classes of cyclic codes, Cryptography and Communications, 9 (2017), 323-343.  doi: 10.1007/s12095-015-0177-y.

[14]

K. Kith, Complete Weight Enumeration of Reed-Solomon Codes, Master's thesis, University of Waterloo in Waterloo, 1989.

[15]

X. L. Kong and S. D. Yang, Complete weight enumerators of a class of linear codes with two or three weights, Discrete Mathematics, 342 (2019), 3166-3176.  doi: 10.1016/j.disc.2019.06.025.

[16]

C. J. Li, S. Bae, J. Ahn, S. D. Yang and Z. A. 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.

[17]

C. J. Li, S. 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.

[18]

C. J. Li, Q. Yue and F. W. Fu, Complete weight enumerators of some cyclic codes, Designs, Codes and Cryptography, 80 (2016), 295-315.  doi: 10.1007/s10623-015-0091-5.

[19]

C. J. Li, Q. Yue and Z. L. Heng, Weight distributions of a class of cyclic codes from $ \mathbb{F}_l $-conjugates, Advances in Mathematics of Communications, 9 (2015), 341-352.  doi: 10.3934/amc.2015.9.341.

[20]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983.

[21]

H. B. Liu, Q. Y. Liao and X. F. Wang, Complete weight enumerator for a class of linear codes from defining sets and their applications, Journal of Systems Science and Complexity, 32 (2019), 947-969.  doi: 10.1007/s11424-018-7414-3.

[22]

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes. I, North-Holland Mathematical Library, Vol. 16. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[23]

A. Sharma and G. K. Bakshi, The weight distribution of some irreducible cyclic codes, Finite Fields and Their Applications, 18 (2012), 144-159.  doi: 10.1016/j.ffa.2011.07.002.

[24]

M. J. Shi and Y. P. Zhang, Quasi-twisted codes with constacyclic constituent codes, Finite Fields and Their Applications, 39 (2016), 159-178.  doi: 10.1016/j.ffa.2016.01.010.

[25]

M. J. Shi, L. Q. Qian, L. Sok, N. Aydin and P. Solé, On constacyclic codes over $ \mathbb{Z}_4[u]/\langle u^2-1 \rangle$ and their Gray images, Finite Fields and Their Applications, 45 (2017), 86-95.  doi: 10.1016/j.ffa.2016.11.016.

[26]

M. J. Shi, R. S. Wu, L. Q. Qian, L. Sok and P. Solé, New classes of $p$-ary few weight codes, Bulletin of the Malaysian Mathematical Sciences Society, 42 (2019), 1393-1412.  doi: 10.1007/s40840-017-0553-1.

[27]

L. Sok, M. J. Shi and P. Solé, Construction of optimal LCD codes over large finite fields, Finite Fields and Their Applications, 50 (2018), 138-153.  doi: 10.1016/j.ffa.2017.11.007.

[28]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, Journal of Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

[29]

G. Vega, The weight distribution for any irreducible cyclic code of length $p^m$, Applicable Algebra in Engineering, Communication and Computing, 29 (2018), 363-370.  doi: 10.1007/s00200-017-0347-6.

[30]

Q. Y. Wang, F. Li, K. L. Ding and D. D. Lin, Complete weight enumerators of two classes of linear codes, Discrete Mathematics, 340 (2017), 467-480.  doi: 10.1016/j.disc.2016.09.003.

[31]

Y. S. Wu, Q. Yue and S. Q. Fan, Further factorization of $x^n -1$ over a finite field, Finite Fields and Their Applications, 54 (2018), 197-215.  doi: 10.1016/j.ffa.2018.07.007.

[32]

Y. S. Wu, Q. Yue, X. M. Zhu and S. D. Yang, Weight enumerators of reducible cyclic codes and their dual codes, Discrete Mathematics, 342 (2019), 671-682.  doi: 10.1016/j.disc.2018.10.035.

[33]

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 Fields and Their Applications, 44 (2017), 76-91.  doi: 10.1016/j.ffa.2016.11.004.

[34]

S. D. Yang, Z. A. Yao and C. A. Zhao, The weight enumerator of the duals of a class of cyclic codes with three zeros, Applicable Algebra in Engineering, Communication and Computing, 26 (2015), 347-367.  doi: 10.1007/s00200-015-0255-6.

[35]

S. D. Yang, X. L. Kong and C. M. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields and Their Applications, 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001.

[36]

S. D. Yang and Z. A. Yao, Complete weight enumerators of a class of linear codes, Discrete Mathematics, 340 (2017), 729-739.  doi: 10.1016/j.disc.2016.11.029.

[37]

J. Yuan and C. S. Ding, Secret sharing schemes from three classes of linear codes, IEEE Transactions on Information Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412.

Table 1.  Weight distribution of $ \mathcal{C} _{D_0} $ for odd $ e $
Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} -1 $
$ (p-1) (p^{e-2}- p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q+ \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
$ (p-1) (p^{e-2}+ p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q- \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
Table Options

Download as excel

Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} -1 $
$ (p-1) (p^{e-2}- p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q+ \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
$ (p-1) (p^{e-2}+ p^{-2} \varepsilon_0 \eta_e(a) \sqrt{p^*} ) $ $ \dfrac{p-1}{2p} {\big( {{ q- \varepsilon_0 \eta_e(a) \sqrt{p^*}}} \big) } $
Table 2.  Weight distribution of $ \mathcal{C} _{D_0} $ for even $ e $
Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} + (p-1)p^{-1} \varepsilon_0 \eta_e(a)-1 $
$ (p-1) (p^{e-2}+ p^{-1} \varepsilon_0 \eta_e(a) ) $ $ (p-1) {\big( {{p^{e-1}- p^{-1} \varepsilon_0 \eta_e(a)}} \big) } $
Table Options

Download as excel

Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} + (p-1)p^{-1} \varepsilon_0 \eta_e(a)-1 $
$ (p-1) (p^{e-2}+ p^{-1} \varepsilon_0 \eta_e(a) ) $ $ (p-1) {\big( {{p^{e-1}- p^{-1} \varepsilon_0 \eta_e(a)}} \big) } $
Table 3.  Weight distribution of $ \mathcal{C} _{D_c} $ for odd $ e $ and $ c \neq 0 $
Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} -1 $
$ n_c-p^{e-2} + p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2} n_c $
$ n_c-p^{e-2} - p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2}{\big( {{p^{e-1} - p^{-1}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) }} \big) } $
Table Options

Download as excel

Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ p^{e-1} -1 $
$ n_c-p^{e-2} + p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2} n_c $
$ n_c-p^{e-2} - p^{-2}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) $ $ \frac{p-1}{2}{\big( {{p^{e-1} - p^{-1}\varepsilon_0 \eta_e(a) \sqrt{p^*}\eta(-c) }} \big) } $
Table 4.  Weight distribution of $ \mathcal{C} _{D_c} $ for even $ e $ and $ c\neq 0 $
Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ \frac{p+1}{2}p^{e-1}+\frac{p-1}{2} p^{-1}\varepsilon_0 \eta_e(a)-1 $
$ (p-1)p^{e-2} -2 p^{-1}\varepsilon_0 \eta_e(a) $ $ \frac{p-1}{2}n_c $
Table Options

Download as excel

Weight $ w $ Frequency $ A_w $
$ 0 $ $ 1 $
$ (p-1) p^{e-2} $ $ \frac{p+1}{2}p^{e-1}+\frac{p-1}{2} p^{-1}\varepsilon_0 \eta_e(a)-1 $
$ (p-1)p^{e-2} -2 p^{-1}\varepsilon_0 \eta_e(a) $ $ \frac{p-1}{2}n_c $
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