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. CarletC. 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. 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.

[15]

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.

[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. LiS. 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. LiQ. 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. LidlH. Niederreiter and F. M. Cohn, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. 
[20]

G. LuoX. CaoS. 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. TangN. LiY. QiZ. 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. YangX. 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. ZhouN. LiC. 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. CarletC. 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. 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.

[15]

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.

[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. LiS. 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. LiQ. 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. LidlH. Niederreiter and F. M. Cohn, Finite Fields, Encyclopedia of Mathematics and its Applications, 20. Cambridge University Press, Cambridge, 1997. 
[20]

G. LuoX. CaoS. 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. TangN. LiY. QiZ. 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. YangX. 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. ZhouN. LiC. 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.

Table 1.  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) $
Table 2.  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) $
Table 3.  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) $
Table 4.  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) $
Table 5.  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) $
Table 6.  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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