February  2021, 15(1): 9-22. doi: 10.3934/amc.2020039

A construction of $ p $-ary linear codes with two or three weights

1. 

School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China

2. 

School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China

3. 

Electronics Technology Institute, Beijing, 100195, China

* Corresponding author: Chunming Tang

Received  January 2019 Revised  July 2019 Published  February 2021 Early access  November 2019

Fund Project: This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 11531002, 11701129). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005)

Applied in consumer electronics, communication, data storage system, secret sharing, authentication codes, association schemes, and strongly regular graphs, linear codes attract much interest. We consider the construction of linear codes with two or three weights. Let
$ m_1,\ldots, m_t $
be
$ t $
positive integers and
$ T = \mathbb{F}_{q_1}\times \cdots \times \mathbb{F}_{q_t} $
, where
$ q_i = p^{m_i} $
for
$ 1\leq i\leq t $
and
$ p $
is an odd prime. A linear code
$ \begin{equation*} \mathcal{C}_D = \{ \mathbf{c}(\mathbf{a}): \mathbf{a} = (a_1,\ldots,a_t)\in T\}, \end{equation*} $
can be constructed by a defining set
$ D $
, where
$ D $
is a subset of
$ T $
and
$ \mathbf{c}(\mathbf{a}) = (\sum_{i = 1}^{t}\mathrm{Tr}_1^{m_i}(a_ix_i))_{ \mathbf{x} = (x_1,\ldots,x_t)\in D} $
. We construct linear codes with two or three weights from the following three defining sets:
$ D_0 = \{\mathbf{x}\in T\backslash \{\mathbf{0}\}: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2) = 0\} $
,
$ D_{SQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in SQ\} $
,
$ D_{NSQ} = \{\mathbf{x}\in T: \sum_{i = 1}^t\mathrm{Tr}_1^{m_i}(x_i^2)\in NSQ\} $
,
where
$ SQ $
is the set of all the squares in
$ \mathbb{F}_p^* $
and
$ NSQ $
is the set of all the nonsquares in
$ \mathbb{F}_p^* $
. We also determine the weight distributions of these codes. The punctured codes of codes from the defining set
$ D_0 $
contain optimal codes meeting certain bounds. This paper generalizes results of [22].
Citation: Hongming Ru, Chunming Tang, Yanfeng Qi, Yuxiao Deng. A construction of $ p $-ary linear codes with two or three weights. Advances in Mathematics of Communications, 2021, 15 (1) : 9-22. doi: 10.3934/amc.2020039
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]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.

[3]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[4]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.

[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[6]

G. D. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 8308, Springer, Heidelberg, 2013, 85–98. doi: 10.1007/978-3-642-45239-0_6.

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C. Ding, Codes from Difference Sets, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9283.

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C. Ding, A construction of binary linear codes from Boolean functions, Discrete Math., 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029.

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

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C. DingT. HellesethT. Kl$\phi$ve and X. Wang, A general construction of Cartesian authentication codes, IEEE Trans. Inform. Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.

[11]

C. DingC. LiN. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.

[12]

C. DingY. LiuC. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.

[13]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, in Coding and Cryptology, Ser. Coding Theory Cryptol., 4, World Sci. Publ., Hackensack, NJ, 2008,119–124. doi: 10.1142/9789812832245_0009.

[14]

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.

[15]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoret. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[16]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.

[17]

C. Ding and J. Yin, Algebraic constructions of constant composition codes, IEEE Trans. Inform. Theory, 51 (2005), 1585-1589.  doi: 10.1109/TIT.2005.844087.

[18]

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

[19]

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.

[20]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[21]

C. LiQ. Yue and F. W. 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.

[22]

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.

[23]

S. LiT. Feng and G. Ge, On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inform. Theory, 60 (2014), 3903-3912.  doi: 10.1109/TIT.2014.2318297.

[24]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20, Addison-Wesley Publishing Company, Reading, MA, 1983. doi: 10.1017/CBO9780511525926.

[25]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.

[26]

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

[27]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptogr. Commun., 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[28]

C. TangN. LiY. QiZ. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inform. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[29]

G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory, 58 (2012), 4862-4869.  doi: 10.1109/TIT.2012.2193376.

[30]

M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.

[31]

J. YangM. XiongC. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.

[32]

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

[33]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.

[34]

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.

[35]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD Codes and self-orthogonal codes from a generic construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.

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]

L. D. Baumert and R. J. McEliece, Weights of irreducible cyclic codes, Information and Control, 20 (1972), 158-175.  doi: 10.1016/S0019-9958(72)90354-3.

[3]

A. R. Calderbank and J. M. Goethals, Three-weight codes and association schemes, Philips J. Res., 39 (1984), 143-152. 

[4]

A. R. Calderbank and W. M. Kantor, The geometry of two-weight codes, Bull. London Math. Soc., 18 (1986), 97-122.  doi: 10.1112/blms/18.2.97.

[5]

C. CarletC. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722.

[6]

G. D. Cohen, S. Mesnager and A. Patey, On minimal and quasi-minimal linear codes, in Cryptography and Coding, Lecture Notes in Comput. Sci., 8308, Springer, Heidelberg, 2013, 85–98. doi: 10.1007/978-3-642-45239-0_6.

[7]

C. Ding, Codes from Difference Sets, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2015. doi: 10.1142/9283.

[8]

C. Ding, A construction of binary linear codes from Boolean functions, Discrete Math., 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029.

[9]

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

[10]

C. DingT. HellesethT. Kl$\phi$ve and X. Wang, A general construction of Cartesian authentication codes, IEEE Trans. Inform. Theory, 53 (2007), 2229-2235.  doi: 10.1109/TIT.2007.896872.

[11]

C. DingC. LiN. Li and Z. Zhou, Three-weight cyclic codes and their weight distributions, Discrete Math., 339 (2016), 415-427.  doi: 10.1016/j.disc.2015.09.001.

[12]

C. DingY. LiuC. Ma and L. Zeng, The weight distributions of the duals of cyclic codes with two zeros, IEEE Trans. Inform. Theory, 57 (2011), 8000-8006.  doi: 10.1109/TIT.2011.2165314.

[13]

C. Ding, J. Luo and H. Niederreiter, Two-weight codes punctured from irreducible cyclic codes, in Coding and Cryptology, Ser. Coding Theory Cryptol., 4, World Sci. Publ., Hackensack, NJ, 2008,119–124. doi: 10.1142/9789812832245_0009.

[14]

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.

[15]

C. Ding and X. Wang, A coding theory construction of new systematic authentication codes, Theoret. Comput. Sci., 330 (2005), 81-99.  doi: 10.1016/j.tcs.2004.09.011.

[16]

C. Ding and J. Yang, Hamming weights in irreducible cyclic codes, Discrete Math., 313 (2013), 434-446.  doi: 10.1016/j.disc.2012.11.009.

[17]

C. Ding and J. Yin, Algebraic constructions of constant composition codes, IEEE Trans. Inform. Theory, 51 (2005), 1585-1589.  doi: 10.1109/TIT.2005.844087.

[18]

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

[19]

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.

[20]

Z. HengC. Ding and Z. Zhou, Minimal linear codes over finite fields, Finite Fields Appl., 54 (2018), 176-196.  doi: 10.1016/j.ffa.2018.08.010.

[21]

C. LiQ. Yue and F. W. 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.

[22]

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.

[23]

S. LiT. Feng and G. Ge, On the weight distribution of cyclic codes with Niho exponents, IEEE Trans. Inform. Theory, 60 (2014), 3903-3912.  doi: 10.1109/TIT.2014.2318297.

[24]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and its Applications, 20, Addison-Wesley Publishing Company, Reading, MA, 1983. doi: 10.1017/CBO9780511525926.

[25]

C. MaL. ZengY. LiuD. Feng and C. Ding, The weight enumerator of a class of cyclic codes, IEEE Trans. Inform. Theory, 57 (2011), 397-402.  doi: 10.1109/TIT.2010.2090272.

[26]

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

[27]

S. Mesnager, Linear codes with few weights from weakly regular bent functions based on a generic construction, Cryptogr. Commun., 9 (2017), 71-84.  doi: 10.1007/s12095-016-0186-5.

[28]

C. TangN. LiY. QiZ. Zhou and T. Helleseth, Linear codes with two or three weights from weakly regular bent functions, IEEE Trans. Inform. Theory, 62 (2016), 1166-1176.  doi: 10.1109/TIT.2016.2518678.

[29]

G. Vega, The weight distribution of an extended class of reducible cyclic codes, IEEE Trans. Inform. Theory, 58 (2012), 4862-4869.  doi: 10.1109/TIT.2012.2193376.

[30]

M. Xiong, The weight distributions of a class of cyclic codes, Finite Fields Appl., 18 (2012), 933-945.  doi: 10.1016/j.ffa.2012.06.001.

[31]

J. YangM. XiongC. Ding and J. Luo, Weight distribution of a class of cyclic codes with arbitrary number of zeros, IEEE Trans. Inform. Theory, 59 (2013), 5985-5993.  doi: 10.1109/TIT.2013.2266731.

[32]

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

[33]

X. ZengL. HuW. JiangQ. Yue and X. Cao, The weight distribution of a class of $p$-ary cyclic codes, Finite Fields Appl., 16 (2010), 56-73.  doi: 10.1016/j.ffa.2009.12.001.

[34]

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.

[35]

Z. ZhouC. TangX. Li and C. Ding, Binary LCD Codes and self-orthogonal codes from a generic construction, IEEE Trans. Inform. Theory, 65 (2019), 16-27.  doi: 10.1109/TIT.2018.2823704.

Table 1.  The weight distribution of $ \mathcal{C}_{D_0} $ for $ \sum_{i = 1}^tm_i $ odd
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)(p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
Table 2.  The weight distribution of $ \mathcal{C}_{D_0} $ for $ \sum_{i = 1}^tm_i $ even
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $ $ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)(p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $ $ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$
Table 3.  The weight distribution of $ \mathcal{C}_{\overline{D}_0} $ for $ \sum_{i = 1}^tm_i $ odd
Weight Frequency
0 1
$ p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
Weight Frequency
0 1
$ p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ p^{\sum_{i=1}^tm_i-2}- p^{\frac{\sum_{i=1}^tm_i-3}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ p^{\sum_{i=1}^tm_i-2}+ p^{\frac{\sum_{i=1}^tm_i-3}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - p^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
Table 4.  The weight distribution of $ \mathcal{C}_{\overline{D}_0} $ for $ \sum_{i = 1}^tm_i $ even
Weight Frequency
0 1
$ p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}} $ $ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$
Weight Frequency
0 1
$ p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^tm_i-1} +M (p-1)p^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ p^{\sum_{i=1}^tm_i-2} +Mp^{\frac{\sum_{i=1}^tm_i-2}{2}} $ $ (p-1)(p^{\sum_{i=1}^tm_i-1} -Mp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* M=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}$
Table 5.  The weight distribution of $ \mathcal{C}_{D_{SQ}} $ for $ \sum_{i = 1}^tm_i $ odd
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A (p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A (p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} +A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$
Table 6.  The weight distribution of $ \mathcal{C}_{D_{SQ}} $ for $ \sum_{i = 1}^tm_i $ even
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ \frac{p+1}{2}p^{\sum_{i=1}^tm_i-1} +\frac{p-1}{2}Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -2Bp^{\frac{\sum_{i=1}^tm_i-2}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} -Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* B=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}.$
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ \frac{p+1}{2}p^{\sum_{i=1}^tm_i-1} +\frac{p-1}{2}Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -2Bp^{\frac{\sum_{i=1}^tm_i-2}{2}} $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} -Bp^{\frac{\sum_{i=1}^tm_i-2}{2}}) $
$^* B=(-1)^{\frac{(p-1) \sum_{i=1}^{t} m_{i}}{4}+t}.$
Table 7.  The weight distribution of $ \mathcal{C}_{D_{NSQ}} $ for $ \sum_{i = 1}^tm_i $ odd
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$
Weight Frequency
0 1
$ (p-1)p^{\sum_{i=1}^tm_i-2} $ $ p^{\sum_{i=1}^t m_i-1}-1 $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} - p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1}+ Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$ (p-1)p^{\sum_{i=1}^tm_i-2} -A(p^{\frac{\sum_{i=1}^tm_i-1}{2}} + p^{\frac{\sum_{i=1}^tm_i-3}{2}}) $ $ \frac{p-1}{2}(p^{\sum_{i=1}^tm_i-1} - Ap^{\frac{\sum_{i=1}^tm_i-1}{2}}) $
$^* A=(-1) \frac{(p-1)\left(\sum_{i=1}^{t} m_{i}+3\right)}{4}+t+1.$
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