November  2021, 15(4): 663-676. doi: 10.3934/amc.2020088

Infinite families of 2-designs from a class of non-binary Kasami cyclic codes

1. 

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

2. 

Guangxi Key Laboratory of Cryptography and Information Security, Guilin University of Electronic Technology, Guilin, Guangxi 541004, China

3. 

School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610000, China

* Corresponding author: Xiaoni Du

Received  December 2019 Revised  February 2020 Published  November 2021 Early access  June 2020

Fund Project: The second author is supported by NSFC grant No. 61772022. The third author is supported by NSFC grant No. 11971395

Combinatorial $ t $-designs have been an important research subject for many years, as they have wide applications in coding theory, cryptography, communications and statistics. The interplay between coding theory and $ t $-designs has been attracted a lot of attention for both directions. It is well known that a linear code over any finite field can be derived from the incidence matrix of a $ t $-design, meanwhile, that the supports of all codewords with a fixed weight in a code also may hold a $ t $-design. In this paper, by determining the weight distribution of a class of linear codes derived from non-binary Kasami cyclic codes, we obtain infinite families of $ 2 $-designs from the supports of all codewords with a fixed weight in these codes, and calculate their parameters explicitly.

Citation: Rong Wang, Xiaoni Du, Cuiling Fan. Infinite families of 2-designs from a class of non-binary Kasami cyclic codes. Advances in Mathematics of Communications, 2021, 15 (4) : 663-676. doi: 10.3934/amc.2020088
References:
[1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

E. F. Assmus Jr. and H. F. Mattson Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-152.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.

[4]

M. Antweiler and L. Bömer, Complex sequences over GF$ {(p^M)} $ with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, 38 (1992), 120-130.  doi: 10.1109/18.108256.

[5]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. II. Encyclopedia of Mathematics and its Applications, Vol. 78, 2nd edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139507660.003.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[8]

C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[9]

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.

[10]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of linear codes, preprint, arXiv: 1903.07459.

[11]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros, preprint, arXiv: 1903.08153.

[12]

X. DuR. Wang and C. Fan, Infinite families of $2$-designs from a class of cyclic codes, J. Comb. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[13]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110.  doi: 10.1016/j.ffa.2004.06.002.

[14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[15]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, $2^nd$ edition, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.

[16]

T. KasamiS. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 (1967), 475-496.  doi: 10.1016/S0019-9958(67)90691-2.

[17]

J. Luo, Y. Tang, and H. Wang, Exponential sums, cycle codees and sequences: the odd characteristic Kasami case, preprint, arXiv: 0902.4508v1 [cs.IT].

[18]

R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997.

[19]

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

[20]

C. Reid and A. Rosa, Steiner systems $ {S} (2, 4, v) $-a survey, The Electronic Journal of Combinatorics, 18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39.

[21]

J. Serrin, C. J. Colbourn and R. Mathon, Steiner systems, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, (2006), 128–135. https://www.researchgate.net/publication/329786723.

[22]

V. D. Tonchev, Codes and designs, in Handbook of Coding Theory, Vol. I, II North-Holland, Amsterdam, (1998), 1229–1267. https://www.researchgate.net/publication/268549395.

[23]

V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.

[24]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

show all references

References:
[1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.
[2]

E. F. Assmus Jr. and H. F. Mattson Jr, New $5$-designs, J. Combinatorial Theory, 6 (1969), 122-152.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.

[4]

M. Antweiler and L. Bömer, Complex sequences over GF$ {(p^M)} $ with a two-level autocorrelation function and a large linear span, IEEE Trans. Inform. Theory, 38 (1992), 120-130.  doi: 10.1109/18.108256.

[5]

T. Beth, D. Jungnickel and H. Lenz, Design Theory, Vol. II. Encyclopedia of Mathematics and its Applications, Vol. 78, 2nd edition, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9781139507660.003.

[6]

C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.

[7]

C. Ding, Infinite families of $3$-designs from a type of five-weight code, Des. Codes Cryptogr., 86 (2018), 703-719.  doi: 10.1007/s10623-017-0352-6.

[8]

C. Ding and C. Li, Infinite families of $2$-designs and $3$-designs from linear codes, Discrete Math., 340 (2017), 2415-2431.  doi: 10.1016/j.disc.2017.05.013.

[9]

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.

[10]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of linear codes, preprint, arXiv: 1903.07459.

[11]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of $2$-designs from two classes of binary cyclic codes with three nonzeros, preprint, arXiv: 1903.08153.

[12]

X. DuR. Wang and C. Fan, Infinite families of $2$-designs from a class of cyclic codes, J. Comb. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[13]

R. W. Fitzgerald and J. L. Yucas, Sums of Gauss sums and weights of irreducible codes, Finite Fields Appl., 11 (2005), 89-110.  doi: 10.1016/j.ffa.2004.06.002.

[14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.
[15]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, $2^nd$ edition, Graduate Texts in Mathematics, Vol. 84, Springer-Verlag, New York, 1990.

[16]

T. KasamiS. Lin and W. W. Peterson, Some results on cyclic codes which are invariant under the affine group and their applications, Information and Control, 11 (1967), 475-496.  doi: 10.1016/S0019-9958(67)90691-2.

[17]

J. Luo, Y. Tang, and H. Wang, Exponential sums, cycle codees and sequences: the odd characteristic Kasami case, preprint, arXiv: 0902.4508v1 [cs.IT].

[18]

R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997.

[19]

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

[20]

C. Reid and A. Rosa, Steiner systems $ {S} (2, 4, v) $-a survey, The Electronic Journal of Combinatorics, 18 (2010), 1–34. https://www.researchgate.net/publication/266996333. doi: 10.37236/39.

[21]

J. Serrin, C. J. Colbourn and R. Mathon, Steiner systems, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, (2006), 128–135. https://www.researchgate.net/publication/329786723.

[22]

V. D. Tonchev, Codes and designs, in Handbook of Coding Theory, Vol. I, II North-Holland, Amsterdam, (1998), 1229–1267. https://www.researchgate.net/publication/268549395.

[23]

V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.

[24]

M. van der Vlugt, Hasse-Davenport curves, Gauss sums, and weight distributions of irreducible cyclic codes, J. Number Theory, 55 (1995), 145-159.  doi: 10.1006/jnth.1995.1133.

Table 1.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd
Weight Multiplicity
$ 0 $ $ 1 $
$ (p-1)(p^{m-1}-p^{s-1}) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{m-1}(p-1)+p^{s-1} $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ \frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)-p^{s-1} $ $ \frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d} $
$ -p^{m-2d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Weight Multiplicity
$ 0 $ $ 1 $
$ (p-1)(p^{m-1}-p^{s-1}) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{m-1}(p-1)+p^{s-1} $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ \frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)-p^{s-1} $ $ \frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)} $
$ p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d} $
$ -p^{m-2d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table 2.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s-1) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p-1)(p^s+1) $ $ p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1) $ $ \frac{1}{2}p^{3s-2d}(p^m-1) $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1) $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d} $
$ +1)(p^m-1) $
$ p^m $ $ p-1 $
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s-1) $ $ \frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ \frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1) $
$ p^{s-1}(p-1)(p^s+1) $ $ p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1) $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1) $ $ \frac{1}{2}p^{3s-2d}(p^m-1) $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1) $ $ \frac{1}{2}p^{3s-2d}(p-1)(p^m-1) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1) $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ \frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d} $
$ +1)(p^m-1) $
$ p^m $ $ p-1 $
Table 3.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s+1) $ $ \frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ \frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ \frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ \frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ \frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ \frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d} $
$ +p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d} $
$ -p^{m-4d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Weight Multiplicity
$ 0 $ $ 1 $
$ p^{s-1}(p-1)(p^s+1) $ $ \frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ \frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ \frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ \frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ \frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ \frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{m-1}(p-1) $ $ p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d} $
$ +p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d} $
$ -p^{m-4d}+1)(p^m-1) $
$ p^m $ $ p-1 $
Table 4.  The value of $ T(a,b,c,h) $ when $ d' = d $ is odd ($ j\in \mathbb{F}_p $)
Value Corresponding Condition
$ p^{m-1}+\varepsilon p^{s-1}(p-1) $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0 $
$ p^{m-1}-\varepsilon p^{s-1} $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0 $
$ 0 $ $ S(a,b,c)= p^m\; \mathrm{and}\; h\neq0 $
$ p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon $
$ p^{m-1}-p^{s+d-1}(p-1) $ $ S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0 $
$ p^{m-1}+p^{s+d-1} $ $ S(a,b,c) $ $ =-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0 $
$ p^{m-1} $ $ S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and} $
$ h+j = 0 $
$ p^m $ $ S(a,b,c)=p^m\; \mathrm{and}\; h = 0 $
Value Corresponding Condition
$ p^{m-1}+\varepsilon p^{s-1}(p-1) $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0 $
$ p^{m-1}-\varepsilon p^{s-1} $ $ S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0 $
$ 0 $ $ S(a,b,c)= p^m\; \mathrm{and}\; h\neq0 $
$ p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon $
$ p^{m-1}-p^{s+d-1}(p-1) $ $ S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0 $
$ p^{m-1}+p^{s+d-1} $ $ S(a,b,c) $ $ =-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0 $
$ p^{m-1} $ $ S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and} $
$ h+j = 0 $
$ p^m $ $ S(a,b,c)=p^m\; \mathrm{and}\; h = 0 $
Table 5.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is odd
Value Multiplicity
$ (p-1)(p^{m-1}-p^{s-1}) $ $ M_1+(p-1)M_3 $
$ p^{m-1}(p-1)+p^{s-1} $ $ (p-1)M_1+(p-1)^2M_3 $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ M_2+(p-1)M_4 $
$ p^{m-1}(p-1)-p^{s-1} $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ (p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1}) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1}) $
$ p^m $ $ p-1 $
Value Multiplicity
$ (p-1)(p^{m-1}-p^{s-1}) $ $ M_1+(p-1)M_3 $
$ p^{m-1}(p-1)+p^{s-1} $ $ (p-1)M_1+(p-1)^2M_3 $
$ (p-1)(p^{m-1}+p^{s-1}) $ $ M_2+(p-1)M_4 $
$ p^{m-1}(p-1)-p^{s-1} $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}} $ $ (p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1}) $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{m-1}(p-1)-p^{s+d-1} $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1}) $
$ p^m $ $ p-1 $
Table 6.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = d $ is even
Value Multiplicity
$ p^{s-1}(p-1)(p^s-1) $ $ M_1+(p-1)M_3 $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ (p-1)M_1+(p-1)^2M_3 $
$ p^{s-1}(p-1)(p^s+1) $ $ M_2+(p-1)M_4 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1) $ $ (p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1} $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1) $ $ M_{5,\mp1}+(p-1)M_{6,\mp1} $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9 $
$ p^m $ $ p-1 $
Value Multiplicity
$ p^{s-1}(p-1)(p^s-1) $ $ M_1+(p-1)M_3 $
$ p^{s-1}(p^{s+1}-p^s+1) $ $ (p-1)M_1+(p-1)^2M_3 $
$ p^{s-1}(p-1)(p^s+1) $ $ M_2+(p-1)M_4 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_2+(p-1)^2M_4 $
$ p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1) $ $ (p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1} $
$ p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1) $ $ M_{5,\mp1}+(p-1)M_{6,\mp1} $
$ p^{s+d-1}(p-1)(p^{s-d}+1) $ $ M_7+(p-1)M_8 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}-1) $ $ (p-1)M_7+(p-1)^2M_8 $
$ p^{m-1}(p-1) $ $ pM_9 $
$ p^m $ $ p-1 $
Table 7.  The weight distribution of $ {\overline{{\mathbb{C}}^{\bot}}}^{\bot} $ when $ d' = 2d $
Value Multiplicity
$ p^{s-1}(p-1)(p^s+1) $ $ M_1+(p-1)M_2 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_1+(p-1)^2M_2 $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ M_3+(p-1)M_4 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ (p-1)M_3+(p-1)^2M_4 $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ M_5+(p-1)M_6 $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ (p-1)M_5+(p-1)^2M_6 $
$ p^{m-1}(p-1) $ $ pM_7 $
$ p^m $ $ p-1 $
Value Multiplicity
$ p^{s-1}(p-1)(p^s+1) $ $ M_1+(p-1)M_2 $
$ p^{s-1}(p^{s+1}-p^s-1) $ $ (p-1)M_1+(p-1)^2M_2 $
$ p^{s+d-1}(p-1)(p^{s-d}-1) $ $ M_3+(p-1)M_4 $
$ p^{s+d-1}(p^{s-d+1}-p^{s-d}+1) $ $ (p-1)M_3+(p-1)^2M_4 $
$ p^{s+2d-1}(p-1)(p^{s-2d}+1) $ $ M_5+(p-1)M_6 $
$ p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1) $ $ (p-1)M_5+(p-1)^2M_6 $
$ p^{m-1}(p-1) $ $ pM_7 $
$ p^m $ $ p-1 $
Table 8.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is odd
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}} $ $ M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}} $ $ M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1) $
$ \varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table 9.  The value distribution of $ S(a,b,c) $ when $ d' = d $ is even
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon p^{s+\frac{d}{2}} $ $ M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1) $
$ \varepsilon p^{s+\frac{d}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Value Multiplicity
$ p^s $ $ M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1) $
$ -p^s $ $ M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1) $
$ \zeta^j_pp^s $ $ M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1) $
$ -\zeta^j_pp^s $ $ M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1) $
$ \varepsilon p^{s+\frac{d}{2}} $ $ M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1) $
$ \varepsilon p^{s+\frac{d}{2}}\zeta^j_p $ $ M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1) $
$ -p^{s+d} $ $ M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1) $
$ -p^{s+d}\zeta^j_p $ $ M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1) $
$ 0 $ $ M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1) $
$ p^m $ $ 1 $
Table 10.  The value distribution of $ S(a,b,c) $ when $ d' = 2d $
Value Multiplicity
$ -p^s $ $ M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)} $
$ -\zeta^j_pp^s $ $ M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d} $ $ M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d}\zeta^j_p $ $ M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ -p^{s+2d} $ $ M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ -p^{s+2d}\zeta^j_p $ $ M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ 0 $ $ M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}- $
$ 2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1) $
$ p^m $ $ 1 $
Value Multiplicity
$ -p^s $ $ M_1=\frac{p^{s+3d-1}(p^s-1)(p^s-p+1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)}{(p^d+1)(p^{2d}-1)} $
$ -\zeta^j_pp^s $ $ M_2=\frac{p^{s+3d-1}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ p^{s+d} $ $ M_3=\frac{p^{s-1}(p^{s-d}+p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ p^{s+d}\zeta^j_p $ $ M_4=\frac{p^{s-1}(p^{s-d}-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2} $
$ -p^{s+2d} $ $ M_5=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}-p+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ -p^{s+2d}\zeta^j_p $ $ M_6=\frac{p^{s-2d-1}(p^{s-d}-1)(p^{s-2d}+1)(p^m-1)}{(p^d+1)(p^{2d}-1)} $
$ 0 $ $ M_7=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}+p^{3s-5d}+p^{m-d}- $
$ 2p^{m-2d}+p^{m-3d}-p^{m-4d}+1)(p^m-1) $
$ p^m $ $ 1 $
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