# American Institute of Mathematical Sciences

doi: 10.3934/mfc.2021041
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## 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 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, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] Z. Heng, D. Li, J. 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.  Google Scholar [10] Z. Heng, Q. 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.  Google Scholar [11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridgeshire, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar [12] G. Jian, Z. 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.  Google Scholar [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.  Google Scholar [14] C. Li, Q. 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.  Google Scholar [15] R. Lidl and H. Niederreiter, Finite Fields, 2$^nd$ edition, Cambridge University Press, Cambridgeshire, 1997.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Mesnager, Y. Qi, H. Ru and C. Tang, Minimal linear codes from characteristic functions, IEEE Trans. Inform. Theory, 66 (2020), 5404-5413.  doi: 10.1109/TIT.2020.2978387.  Google Scholar [19] B. Mounits, T. 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.  Google Scholar [20] A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] D. Zheng, Q. Zhao, X. 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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [9] Z. Heng, D. Li, J. 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.  Google Scholar [10] Z. Heng, Q. 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.  Google Scholar [11] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridgeshire, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar [12] G. Jian, Z. 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.  Google Scholar [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.  Google Scholar [14] C. Li, Q. 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.  Google Scholar [15] R. Lidl and H. Niederreiter, Finite Fields, 2$^nd$ edition, Cambridge University Press, Cambridgeshire, 1997.   Google Scholar [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.  Google Scholar [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.  Google Scholar [18] S. Mesnager, Y. Qi, H. Ru and C. Tang, Minimal linear codes from characteristic functions, IEEE Trans. Inform. Theory, 66 (2020), 5404-5413.  doi: 10.1109/TIT.2020.2978387.  Google Scholar [19] B. Mounits, T. 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.  Google Scholar [20] A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] D. Zheng, Q. Zhao, X. 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.  Google Scholar
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 $q^{2m-2}\left(q-1 \right) -q^{3s-2}$ $\left(q^m-q^{m-1}+q^s-q^{s-1}\right) \left(q-1\right)$ $q^{2m-2}\left( q-1\right)$ $q^m\left(q^m-q+1 \right)-1$ $\left( q^{2m-2}+q^{3s-2}\right)\left(q-1 \right)$ $\left(q^{m-1}-q^s+q^{s-1}\right) \left(q-1\right)$
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 $\; \qquad q^{2m-2}\left( q-1\right)$ $\left( q^m-1\right)\left( q^{m-1}+1\right) \qquad \;$ $\; \qquad q^{m-1}\left( q^{m-1}+1\right) \left(q-1 \right)$ $q^{m-1}\left( q^m-1\right) \left( q-1\right) \qquad \;$
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 $\;q^{m-1}\left( q^{m-1}+(-1)^s\right) \left(q-1 \right)$ $q^{m-1}\left( q^m-(-1)^s\right) \left( q-1\right) \;$ $\;q^{2m-2}\left( q-1\right)$ $\left( q^m-(-1)^s\right)\left( q^{m-1}+(-1)^s\right) \;$
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 $\left( q^{2m-2}-q^{3s-2}\right)\left(q-1 \right)$ $\dfrac{1}{2}\left(q^m+q^s \right) \left(q-1\right)$ $q^{2m-2}\left( q-1\right)$ $q^m\left(q^m-q+1 \right)-1$ $\left( q^{2m-2}+q^{3s-2}\right) \left(q-1 \right)$ $\dfrac{1}{2}\left(q^m-q^s \right) \left(q-1\right)$
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