May  2022, 5(2): 129-144. doi: 10.3934/mfc.2021041

Two-weight and three-weight linear codes constructed from Weil sums

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

* Corresponding author: Shudi Yang

Received  October 2021 Revised  December 2021 Published  May 2022 Early access  January 2022

Fund Project: The work is partially supported by National Natural Science Foundation of China under Grant 12071247 and by Research and Innovation Fund for Graduate Dissertations of Qufu Normal University in 2021 under Grant LWCXS202133

Linear codes with few weights are widely used in strongly regular graphs, secret sharing schemes, association schemes and authentication codes. In this paper, we construct several two-weight and three-weight linear codes over finite fields by choosing suitable different defining sets. We also give some examples and some of the codes are optimal or almost optimal. Their applications to secret sharing schemes are also investigated.

Citation: Tonghui Zhang, Hong Lu, Shudi Yang. Two-weight and three-weight linear codes constructed from Weil sums. Mathematical Foundations of Computing, 2022, 5 (2) : 129-144. doi: 10.3934/mfc.2021041
References:
[1]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.

[2]

G. R. Blakley, Safeguarding cryptographic keys, 1979 International Workshop on Managing Requirements Knowledge (MARK), 48 (1979), 313-317.  doi: 10.1109/MARK.1979.8817296.

[3]

Y. Cheng and X. Cao, Linear codes with few weights from weakly regular plateaued functions, Discrete Math., 344 (2021), 112597.  doi: 10.1016/j.disc.2021.112597.

[4]

R. B. Chilwant, T. S. Sarvagod, K. R. Kumbhar, P. N. Gunjgur and A. V. Vidhate, SISA: A secret-sharing scheme application for cloud environment, in 2019 International Conference on Communication and Electronics Systems, (2019), 638–643. doi: 10.1109/ICCES45898.2019.9002527.

[5]

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

[6]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order $3$, IEEE Trans. Inform. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[7]

C. Ding and J. Yuan, Covering and secret sharing with linear codes, Discrete Mathematics and Theoretical Computer Science, 2731 (2003), 11-25.  doi: 10.1007/3-540-45066-1_2.

[8]

T. A. Gulliver, Two new optimal ternary two-weight codes and strongly regular graphs, Discrete Math., 149 (1996), 83-92.  doi: 10.1016/0012-365X(94)00264-J.

[9]

Z. HengD. LiJ. Du and F. Chen, A family of projective two-weight linear codes, Des. Codes Cryptogr., 89 (2021), 1993-2007.  doi: 10.1007/s10623-021-00896-2.

[10]

Z. HengQ. Yue and C. Li, Three classes of linear codes with two or three weights, Discrete Math., 339 (2016), 2832-2847.  doi: 10.1016/j.disc.2016.05.033.

[11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridgeshire, 2003.  doi: 10.1017/CBO9780511807077.
[12]

G. JianZ. Lin and R. Feng, Two-weight and three-weight linear codes based on Weil sums, Finite Fields Appl., 57 (2019), 92-107.  doi: 10.1016/j.ffa.2019.02.001.

[13]

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

[14]

C. LiQ. Yue and F. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4.

[15] R. Lidl and H. Niederreiter, Finite Fields, 2$^nd$ edition, Cambridge University Press, Cambridgeshire, 1997. 
[16]

Y. Liu and Q. Zhao, E-voting scheme using secret sharing and K-anonymity, World Wide Web, 22 (2019), 1657-1667.  doi: 10.1007/s11280-018-0575-0.

[17]

H. Lu and S. Yang, Two classes of linear codes from Weil sums, IEEE Access, 8 (2020), 180471-180480.  doi: 10.1109/ACCESS.2020.3028141.

[18]

S. MesnagerY. QiH. Ru and C. Tang, Minimal linear codes from characteristic functions, IEEE Trans. Inform. Theory, 66 (2020), 5404-5413.  doi: 10.1109/TIT.2020.2978387.

[19]

B. MounitsT. Etzion and S. Litsyn, New upper bounds on codes via association schemes and linear programming, Adv. Math. Commun., 1 (2007), 173-195.  doi: 10.3934/amc.2007.1.173.

[20]

A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.

[21]

S. Yang, Complete weight enumerators of linear codes based on Weil sums, IEEE Communications Letters, 25 (2021), 346-350.  doi: 10.1109/LCOMM.2020.3027754.

[22]

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

[23]

D. ZhengQ. ZhaoX. Wang and Y. Zhang, A class of two or three weights linear codes and their complete weight enumerators, Discrete Math., 344 (2021), 112355.  doi: 10.1016/j.disc.2021.112355.

show all references

References:
[1]

A. Ashikhmin and A. Barg, Minimal vectors in linear codes, IEEE Trans. Inform. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584.

[2]

G. R. Blakley, Safeguarding cryptographic keys, 1979 International Workshop on Managing Requirements Knowledge (MARK), 48 (1979), 313-317.  doi: 10.1109/MARK.1979.8817296.

[3]

Y. Cheng and X. Cao, Linear codes with few weights from weakly regular plateaued functions, Discrete Math., 344 (2021), 112597.  doi: 10.1016/j.disc.2021.112597.

[4]

R. B. Chilwant, T. S. Sarvagod, K. R. Kumbhar, P. N. Gunjgur and A. V. Vidhate, SISA: A secret-sharing scheme application for cloud environment, in 2019 International Conference on Communication and Electronics Systems, (2019), 638–643. doi: 10.1109/ICCES45898.2019.9002527.

[5]

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

[6]

C. Ding and H. Niederreiter, Cyclotomic linear codes of order $3$, IEEE Trans. Inform. Theory, 53 (2007), 2274-2277.  doi: 10.1109/TIT.2007.896886.

[7]

C. Ding and J. Yuan, Covering and secret sharing with linear codes, Discrete Mathematics and Theoretical Computer Science, 2731 (2003), 11-25.  doi: 10.1007/3-540-45066-1_2.

[8]

T. A. Gulliver, Two new optimal ternary two-weight codes and strongly regular graphs, Discrete Math., 149 (1996), 83-92.  doi: 10.1016/0012-365X(94)00264-J.

[9]

Z. HengD. LiJ. Du and F. Chen, A family of projective two-weight linear codes, Des. Codes Cryptogr., 89 (2021), 1993-2007.  doi: 10.1007/s10623-021-00896-2.

[10]

Z. HengQ. Yue and C. Li, Three classes of linear codes with two or three weights, Discrete Math., 339 (2016), 2832-2847.  doi: 10.1016/j.disc.2016.05.033.

[11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridgeshire, 2003.  doi: 10.1017/CBO9780511807077.
[12]

G. JianZ. Lin and R. Feng, Two-weight and three-weight linear codes based on Weil sums, Finite Fields Appl., 57 (2019), 92-107.  doi: 10.1016/j.ffa.2019.02.001.

[13]

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

[14]

C. LiQ. Yue and F. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Algebra Engrg. Comm. Comput., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4.

[15] R. Lidl and H. Niederreiter, Finite Fields, 2$^nd$ edition, Cambridge University Press, Cambridgeshire, 1997. 
[16]

Y. Liu and Q. Zhao, E-voting scheme using secret sharing and K-anonymity, World Wide Web, 22 (2019), 1657-1667.  doi: 10.1007/s11280-018-0575-0.

[17]

H. Lu and S. Yang, Two classes of linear codes from Weil sums, IEEE Access, 8 (2020), 180471-180480.  doi: 10.1109/ACCESS.2020.3028141.

[18]

S. MesnagerY. QiH. Ru and C. Tang, Minimal linear codes from characteristic functions, IEEE Trans. Inform. Theory, 66 (2020), 5404-5413.  doi: 10.1109/TIT.2020.2978387.

[19]

B. MounitsT. Etzion and S. Litsyn, New upper bounds on codes via association schemes and linear programming, Adv. Math. Commun., 1 (2007), 173-195.  doi: 10.3934/amc.2007.1.173.

[20]

A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.

[21]

S. Yang, Complete weight enumerators of linear codes based on Weil sums, IEEE Communications Letters, 25 (2021), 346-350.  doi: 10.1109/LCOMM.2020.3027754.

[22]

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

[23]

D. ZhengQ. ZhaoX. Wang and Y. Zhang, A class of two or three weights linear codes and their complete weight enumerators, Discrete Math., 344 (2021), 112355.  doi: 10.1016/j.disc.2021.112355.

Table 1.  The weight distribution of $ C_{D_1} $
weight multiplicity
0 1
$ q^{2m-2}\left(q-1 \right) -q^{3s-2} $ $ \left(q^m-q^{m-1}+q^s-q^{s-1}\right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right)\left(q-1 \right) $ $ \left(q^{m-1}-q^s+q^{s-1}\right) \left(q-1\right) $
weight multiplicity
0 1
$ q^{2m-2}\left(q-1 \right) -q^{3s-2} $ $ \left(q^m-q^{m-1}+q^s-q^{s-1}\right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right)\left(q-1 \right) $ $ \left(q^{m-1}-q^s+q^{s-1}\right) \left(q-1\right) $
Table 2.  The weight distribution of $ C_{D_2} $ when $ q\equiv 1\pmod{4} $
weight multiplicity
0 1
$ \; \qquad q^{2m-2}\left( q-1\right) $ $ \left( q^m-1\right)\left( q^{m-1}+1\right) \qquad \; $
$ \; \qquad q^{m-1}\left( q^{m-1}+1\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-1\right) \left( q-1\right) \qquad \; $
weight multiplicity
0 1
$ \; \qquad q^{2m-2}\left( q-1\right) $ $ \left( q^m-1\right)\left( q^{m-1}+1\right) \qquad \; $
$ \; \qquad q^{m-1}\left( q^{m-1}+1\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-1\right) \left( q-1\right) \qquad \; $
Table 3.  The weight distribution of $ C_{D_2} $ when $ q\equiv 3\pmod{4} $
weight multiplicity
0 1
$ \;q^{m-1}\left( q^{m-1}+(-1)^s\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-(-1)^s\right) \left( q-1\right) \; $
$ \;q^{2m-2}\left( q-1\right) $ $ \left( q^m-(-1)^s\right)\left( q^{m-1}+(-1)^s\right) \; $
weight multiplicity
0 1
$ \;q^{m-1}\left( q^{m-1}+(-1)^s\right) \left(q-1 \right) $ $ q^{m-1}\left( q^m-(-1)^s\right) \left( q-1\right) \; $
$ \;q^{2m-2}\left( q-1\right) $ $ \left( q^m-(-1)^s\right)\left( q^{m-1}+(-1)^s\right) \; $
Table 4.  The weight distribution of $ C_{D_2} $ when $ q $ is even and $ q>2 $
weight multiplicity
0 1
$ \left( q^{2m-2}-q^{3s-2}\right)\left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m+q^s \right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right) \left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m-q^s \right) \left(q-1\right) $
weight multiplicity
0 1
$ \left( q^{2m-2}-q^{3s-2}\right)\left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m+q^s \right) \left(q-1\right) $
$ q^{2m-2}\left( q-1\right) $ $ q^m\left(q^m-q+1 \right)-1 $
$ \left( q^{2m-2}+q^{3s-2}\right) \left(q-1 \right) $ $ \dfrac{1}{2}\left(q^m-q^s \right) \left(q-1\right) $
[1]

Stefka Bouyuklieva, Zlatko Varbanov. Some connections between self-dual codes, combinatorial designs and secret sharing schemes. Advances in Mathematics of Communications, 2011, 5 (2) : 191-198. doi: 10.3934/amc.2011.5.191

[2]

Fengwei Li, Qin Yue, Fengmei Liu. The weight distributions of constacyclic codes. Advances in Mathematics of Communications, 2017, 11 (3) : 471-480. doi: 10.3934/amc.2017039

[3]

Bagher Bagherpour, Shahrooz Janbaz, Ali Zaghian. Optimal information ratio of secret sharing schemes on Dutch windmill graphs. Advances in Mathematics of Communications, 2019, 13 (1) : 89-99. doi: 10.3934/amc.2019005

[4]

Fengwei Li, Qin Yue, Xiaoming Sun. The values of two classes of Gaussian periods in index 2 case and weight distributions of linear codes. Advances in Mathematics of Communications, 2021, 15 (1) : 131-153. doi: 10.3934/amc.2020049

[5]

Dae San Kim. Infinite families of recursive formulas generating power moments of ternary Kloosterman sums with square arguments arising from symplectic groups. Advances in Mathematics of Communications, 2009, 3 (2) : 167-178. doi: 10.3934/amc.2009.3.167

[6]

Axel Heim, Vladimir Sidorenko, Uli Sorger. Computation of distributions and their moments in the trellis. Advances in Mathematics of Communications, 2008, 2 (4) : 373-391. doi: 10.3934/amc.2008.2.373

[7]

Petr Lisoněk, Layla Trummer. Algorithms for the minimum weight of linear codes. Advances in Mathematics of Communications, 2016, 10 (1) : 195-207. doi: 10.3934/amc.2016.10.195

[8]

Chengju Li, Sunghan Bae, Shudi Yang. Some two-weight and three-weight linear codes. Advances in Mathematics of Communications, 2019, 13 (1) : 195-211. doi: 10.3934/amc.2019013

[9]

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

[10]

Pankaj Kumar, Monika Sangwan, Suresh Kumar Arora. The weight distributions of some irreducible cyclic codes of length $p^n$ and $2p^n$. Advances in Mathematics of Communications, 2015, 9 (3) : 277-289. doi: 10.3934/amc.2015.9.277

[11]

Chengju Li, Qin Yue, Ziling Heng. Weight distributions of a class of cyclic codes from $\Bbb F_l$-conjugates. Advances in Mathematics of Communications, 2015, 9 (3) : 341-352. doi: 10.3934/amc.2015.9.341

[12]

Thomas Hillen, Kevin J. Painter, Amanda C. Swan, Albert D. Murtha. Moments of von mises and fisher distributions and applications. Mathematical Biosciences & Engineering, 2017, 14 (3) : 673-694. doi: 10.3934/mbe.2017038

[13]

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

[14]

Toshiharu Sawashima, Tatsuya Maruta. Nonexistence of some ternary linear codes with minimum weight -2 modulo 9. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021052

[15]

Leetika Kathuria, Madhu Raka. Existence of cyclic self-orthogonal codes: A note on a result of Vera Pless. Advances in Mathematics of Communications, 2012, 6 (4) : 499-503. doi: 10.3934/amc.2012.6.499

[16]

Beniamin Mounits, Tuvi Etzion, Simon Litsyn. New upper bounds on codes via association schemes and linear programming. Advances in Mathematics of Communications, 2007, 1 (2) : 173-195. doi: 10.3934/amc.2007.1.173

[17]

Ye Wang, Ran Tao. Constructions of linear codes with small hulls from association schemes. Advances in Mathematics of Communications, 2022, 16 (2) : 349-364. doi: 10.3934/amc.2020114

[18]

Ryutaroh Matsumoto. Strongly secure quantum ramp secret sharing constructed from algebraic curves over finite fields. Advances in Mathematics of Communications, 2019, 13 (1) : 1-10. doi: 10.3934/amc.2019001

[19]

Tim Alderson, Alessandro Neri. Maximum weight spectrum codes. Advances in Mathematics of Communications, 2019, 13 (1) : 101-119. doi: 10.3934/amc.2019006

[20]

Rumi Melih Pelen. Three weight ternary linear codes from non-weakly regular bent functions. Advances in Mathematics of Communications, 2022  doi: 10.3934/amc.2022020

 Impact Factor: 

Metrics

  • PDF downloads (183)
  • HTML views (154)
  • Cited by (0)

Other articles
by authors

[Back to Top]