# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022041
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## Constructions for several classes of few-weight linear codes and their applications

 College of Mathematical Science, Sichuan Normal University, Chengdu Sichuan, 610066, China

*Corresponding author: Qunying Liao

Received  January 2022 Revised  May 2022 Early access June 2022

Fund Project: The second author is supported by National Natural Science Foundation of China (Grant No. 12071321)

In this paper, for any odd prime $p$ and an integer $m\ge 3$, several classes of linear codes with $t$-weight $(t = 3,5,7)$ are obtained based on some defining sets, and then their complete weight enumerators are determined explicitly by employing Gauss sums and quadratic character sums. Especially for $m = 3$, a class of MDS codes with parameters $[p,3,p-2]$ are obtained. Furthermore, some of these codes can be suitable for applications in secret sharing schemes and $s$-sum sets for any odd $s>1$.

Citation: Canze Zhu, Qunying Liao. Constructions for several classes of few-weight linear codes and their applications. Advances in Mathematics of Communications, doi: 10.3934/amc.2022041
##### References:
 [1] A. Ashikhmin and A. Barg, Minimum vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584. [2] R. Blakley, Safe guarding cryptographic keys, Proc Afips National Computer Conf., 48 (1979), 313. [3] A. R. Calderbank and J.-M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. [4] B. Courteau and J. Wolfmann, On triple sum sets and two or three-weight codes, Discrete. Math., 50 (1984), 179-191.  doi: 10.1016/0012-365X(84)90047-5. [5] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [6] R. Calderbank and W. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97. [7] C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011. [8] K. Ding and C. Ding, Binary linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.  doi: 10.1109/LCOMM.2014.2361516. [9] K. Ding and C. 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. [10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077. [11] Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491. [12] Z. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002. [13] Z. Heng and Q. Yue, A construction of q-ary linear codes with two weights, Finite Fields Appl., 48 (2017), 20-42.  doi: 10.1016/j.ffa.2017.07.006. [14] 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. [15] 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. [16] M. Kiermaier, S. Kurz, P. Solé, M. Stoll and A. Wassermann, On strongly walk regular graphs, triple sum sets and their codes, (2020), arXiv: 2012.06160. [17] C. Li, S. Bae and S. Yang, Some two-weight and three-weight linear codes, Adv. Math. Commun., 13 (2019), 195-211.  doi: 10.3934/amc.2019013. [18] C. Li, Q. Yue and F. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Algebra Eng. Commun. Comput., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4. [19] R. Lidl, H. Niederreiter and F. M. Cohn, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. [20] G. Luo, X. Cao, S. Xu and J. 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. [21] J. Massey, Minimum codewords and secret sharing, Proc. 6th Joint Swedish-Russian Workshop on Information Theory, Mölle, Sweden. Aug., (1993), 276–279. [22] R. J. Mceliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584.  doi: 10.1145/358746.358762. [23] S. L. Ma, A survey of partial differences sets, Des. Codes Cryptogr., 4 (1994), 221-261.  doi: 10.1007/BF01388454. [24] A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176. [25] M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Des. Codes Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7. [26] C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678. [27] K. Torleiv, Codes for Error Detection, Series on Coding Theory and Cryptology, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812770516. [28] Q. Wang, F. Li and D. Lin, A class of linear codes with three weights, (2015), arXiv: 1512.03866. [29] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412. [30] S. Yang and Z. Yao, Complete weight enumerators of a family of three-weight linear codes, Des. Codes Cryptogr., 82 (2017), 663-674.  doi: 10.1007/s10623-016-0191-x. [31] S. Yang, X. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001. [32] C. Zhu and Q. Liao, Complete weight enumerators for several classes of two-weight and three-weight linear codes, Finite Fields Appl., 75 (2021), 101897, 31 pp. doi: 10.1016/j.ffa.2021.101897. [33] Z. Zhou, N. Li, C. 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.

show all references

##### References:
 [1] A. Ashikhmin and A. Barg, Minimum vectors in linear codes, IEEE Trans. Inf. Theory, 44 (1998), 2010-2017.  doi: 10.1109/18.705584. [2] R. Blakley, Safe guarding cryptographic keys, Proc Afips National Computer Conf., 48 (1979), 313. [3] A. R. Calderbank and J.-M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. [4] B. Courteau and J. Wolfmann, On triple sum sets and two or three-weight codes, Discrete. Math., 50 (1984), 179-191.  doi: 10.1016/0012-365X(84)90047-5. [5] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inf. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [6] R. Calderbank and W. Kantor, The geometry of two-weight codes, Bull. Lond. Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97. [7] C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theor. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011. [8] K. Ding and C. Ding, Binary linear codes with three weights, IEEE Commun. Lett., 18 (2014), 1879-1882.  doi: 10.1109/LCOMM.2014.2361516. [9] K. Ding and C. 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. [10] W. C. Huffman and V. Pless, Fundamentals of Error-Correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077. [11] Z. Heng and Q. Yue, A class of binary linear codes with at most three weights, IEEE Commun. Lett., 19 (2015), 1488-1491. [12] Z. Heng and Q. Yue, Two classes of two-weight linear codes, Finite Fields Appl., 38 (2016), 72-92.  doi: 10.1016/j.ffa.2015.12.002. [13] Z. Heng and Q. Yue, A construction of q-ary linear codes with two weights, Finite Fields Appl., 48 (2017), 20-42.  doi: 10.1016/j.ffa.2017.07.006. [14] 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. [15] 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. [16] M. Kiermaier, S. Kurz, P. Solé, M. Stoll and A. Wassermann, On strongly walk regular graphs, triple sum sets and their codes, (2020), arXiv: 2012.06160. [17] C. Li, S. Bae and S. Yang, Some two-weight and three-weight linear codes, Adv. Math. Commun., 13 (2019), 195-211.  doi: 10.3934/amc.2019013. [18] C. Li, Q. Yue and F. Fu, A construction of several classes of two-weight and three-weight linear codes, Appl. Algebra Eng. Commun. Comput., 28 (2017), 11-30.  doi: 10.1007/s00200-016-0297-4. [19] R. Lidl, H. Niederreiter and F. M. Cohn, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. [20] G. Luo, X. Cao, S. Xu and J. 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. [21] J. Massey, Minimum codewords and secret sharing, Proc. 6th Joint Swedish-Russian Workshop on Information Theory, Mölle, Sweden. Aug., (1993), 276–279. [22] R. J. Mceliece and D. V. Sarwate, On sharing secrets and Reed-Solomon codes, Comm. ACM, 24 (1981), 583-584.  doi: 10.1145/358746.358762. [23] S. L. Ma, A survey of partial differences sets, Des. Codes Cryptogr., 4 (1994), 221-261.  doi: 10.1007/BF01388454. [24] A. Shamir, How to share a secret, Comm. ACM, 22 (1979), 612-613.  doi: 10.1145/359168.359176. [25] M. Shi and P. Solé, Three-weight codes, triple sum sets, and strongly walk regular graphs, Des. Codes Cryptogr., 87 (2019), 2395-2404.  doi: 10.1007/s10623-019-00628-7. [26] C. Tang, N. Li, Y. Qi, Z. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inf. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678. [27] K. Torleiv, Codes for Error Detection, Series on Coding Theory and Cryptology, 2. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2007. doi: 10.1142/9789812770516. [28] Q. Wang, F. Li and D. Lin, A class of linear codes with three weights, (2015), arXiv: 1512.03866. [29] J. Yuan and C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. Inf. Theory, 52 (2006), 206-212.  doi: 10.1109/TIT.2005.860412. [30] S. Yang and Z. Yao, Complete weight enumerators of a family of three-weight linear codes, Des. Codes Cryptogr., 82 (2017), 663-674.  doi: 10.1007/s10623-016-0191-x. [31] S. Yang, X. Kong and C. Tang, A construction of linear codes and their complete weight enumerators, Finite Fields Appl., 48 (2017), 196-226.  doi: 10.1016/j.ffa.2017.08.001. [32] C. Zhu and Q. Liao, Complete weight enumerators for several classes of two-weight and three-weight linear codes, Finite Fields Appl., 75 (2021), 101897, 31 pp. doi: 10.1016/j.ffa.2021.101897. [33] Z. Zhou, N. Li, C. 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.
The weight distribution for $\mathcal{C}_{D_a}$ $(m{{{\; is\; odd}}},\; a\in\mathbb{F}_{p^{m}}\backslash\mathbb{F}_p)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-3}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{m-3}-p^\frac{m-3}{2}\big)\big)$ $p^{m-2}-p^{m-3}$ $2(p-1)p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-3}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{m-3}+p^\frac{m-3}{2}\big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-3}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{m-3}-p^\frac{m-3}{2}\big)\big)$ $p^{m-2}-p^{m-3}$ $2(p-1)p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-3}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{m-3}+p^\frac{m-3}{2}\big)$
The weight distribution for $\mathcal{C}_{D_a}$ $(2\mid m, \; a\in\mathbb{F}_{p^{2}}\backslash\mathbb{F}_p)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\big)$
The weight distribution for $\mathcal{C}_{D_a}$ $(2\mid m,\; a\in\mathbb{F}_{p^{m}}\backslash\mathbb{F}_{p^2})$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-3}+p^{\frac{m-6}{2}}\big((p-1)\big(1+\eta_{m}(a)I_1(a)\big)-I_{2}(a)\big)\Big)$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-2}+p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\Big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}-p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}$ $p^{m-1}-p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-2}-p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\Big)$ $p^{m-2}-p^{m-3}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-3}-p^{\frac{m-6}{2}}\big((p-1)\big(1+\eta_{m}(a)I_1(a)\big)-I_{2}(a)\big)\Big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-3}+p^{\frac{m-6}{2}}\big((p-1)\big(1+\eta_{m}(a)I_1(a)\big)-I_{2}(a)\big)\Big)$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-2}+p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\Big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}-p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}$ $p^{m-1}-p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big((p-1)p^{m-2}+p^{\frac{m-4}{2}}\big(p^2-I_{2}(a)\big)\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-2}-p^{\frac{m-4}{2}}\big((p-1)-\eta_{m}(a)I_1(a)\big)\Big)$ $p^{m-2}-p^{m-3}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\Big(p^{m-3}-p^{\frac{m-6}{2}}\big((p-1)\big(1+\eta_{m}(a)I_1(a)\big)-I_{2}(a)\big)\Big)$
The weight distribution for $\mathcal{C}_{D_a}$ $(4\mid m,\; a\in\mathbb{F}_{p^{2}}\backslash\mathbb{F}_p)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big)$
The weight distribution for $\mathcal{C}_{D_a}$ $(s\mid\frac{m}{2},\; a\in\mathbb{F}_{p^{s}}\backslash\mathbb{F}_{p^2})$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-3}-p^{\frac{m-6}{2}}\big)$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)^2p^{m-2}$ $p^{m-2}-p^{m-3}$ $p^{m-1}-p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)^2p^{m-2}$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-3}+p^{\frac{m-6}{2}}\big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-3}-p^{\frac{m-6}{2}}\big)$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)^2p^{m-2}$ $p^{m-2}-p^{m-3}$ $p^{m-1}-p^{m-2}+p^{m-3}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)^2p^{m-2}$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-4}{2}}\big)$ $p^{m-2}-p^{m-3}+p^{\frac{m-2}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-3}+p^{\frac{m-6}{2}}\big)$
The weight distribution for $\mathcal{C}_{D_a}$ $(2\mid m,\; a\in\mathbb{F}_{p^{2}}\backslash\mathbb{F}_p,\; \eta_{m}(a) = -1)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-2}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m}{2}}\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m}{2}}\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-2}{2}}\big)$
 weight $w$ frequency $A_w$ $0$ $1$ $p^{m-2}-p^{m-3}-(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}+p^{\frac{m-2}{2}}\big)$ $p^{m-2}-p^{m-3}-p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}-p^{\frac{m}{2}}\big)$ $p^{m-2}-p^{m-3}$ $p^{m-2}-1$ $p^{m-2}-p^{m-3}+p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-1}+p^{\frac{m}{2}}\big)$ $p^{m-2}-p^{m-3}+(p-1)p^{\frac{m-4}{2}}$ $\frac{1}{2}(p-1)\big(p^{m-2}-p^{\frac{m-2}{2}}\big)$
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