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  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, 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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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