# American Institute of Mathematical Sciences

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.  Google Scholar [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.  Google Scholar [3] E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [12] X. Du, R. 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.  Google Scholar [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.  Google Scholar [14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar [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.  Google Scholar [16] T. Kasami, S. 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.  Google Scholar [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]. Google Scholar [18] R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [23] V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [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.  Google Scholar

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##### References:
 [1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, 1992.  doi: 10.1017/CBO9781316529836.  Google Scholar [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.  Google Scholar [3] E. F. Assmus Jr. and H. F. Mattson Jr, Coding and combinatorics, SIAM Rev., 16 (1974), 349-388.  doi: 10.1137/1016056.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] C. Ding, Designs from Linear Codes, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2019. doi: 10.1142/11101.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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. Google Scholar [12] X. Du, R. 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.  Google Scholar [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.  Google Scholar [14] W. C. Huffman and V. Pless, Fundamentals of Error-correcting Codes, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511807077.  Google Scholar [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.  Google Scholar [16] T. Kasami, S. 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.  Google Scholar [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]. Google Scholar [18] R. Lidl and H. Niederreiter, Finite Fields, 2nd edition, Encyclopedia of Mathematics and its Applications, Vol. 20, Cambridge University Press, Cambridge, 1997. Google Scholar [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. Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [23] V. D. Tonchev, Codes, in Handbook of Combinatorial Designs, $2^nd$ edition, Chapman and Hall/CRC, Boca Raton, FL, 2007.  Google Scholar [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.  Google Scholar
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 $0$ $1$ $(p-1)(p^{m-1}-p^{s-1})$ $\frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1)$ $p^{m-1}(p-1)+p^{s-1}$ $\frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1)$ $(p-1)(p^{m-1}+p^{s-1})$ $\frac{p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)}$ $p^{m-1}(p-1)-p^{s-1}$ $\frac{p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)}{2(p^d-1)}$ $p^{m-1}(p-1)\pm (-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $\frac{1}{2}p^{3s-2d}(p-1)(p^m-1)$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1)$ $p^{m-1}(p-1)-p^{s+d-1}$ $\frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)}$ $p^{m-1}(p-1)$ $p(p^{3s-d}-p^{3s-2d}+p^{3s-2d-1}+p^{3s-3d}$ $-p^{m-2d}+1)(p^m-1)$ $p^m$ $p-1$
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 $0$ $1$ $p^{s-1}(p-1)(p^s-1)$ $\frac{1}{2}p^{m+d}(p^s+1)(p^m-1)/(p^d+1)$ $p^{s-1}(p^{s+1}-p^s+1)$ $\frac{1}{2}p^{m+d}(p-1)(p^s+1)(p^m-1)/(p^d+1)$ $p^{s-1}(p-1)(p^s+1)$ $p^{m+d}(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1)$ $p^{s-1}(p^{s+1}-p^s-1)$ $p^{m+d}(p-1)(p^m-2p^{m-d}+1)(p^s-1)/2(p^d-1)$ $p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}} \pm 1)$ $\frac{1}{2}p^{3s-2d}(p^m-1)$ $p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm 1)$ $\frac{1}{2}p^{3s-2d}(p-1)(p^m-1)$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $p^{m-2d}(p^{s-d}-1)(p^m-1)/(p^{2d}-1)$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}-1)$ $\frac{p^{m-2d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^{2d}-1)}$ $p^{m-1}(p-1)$ $p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}$ $+1)(p^m-1)$ $p^m$ $p-1$
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 $0$ $1$ $p^{s-1}(p-1)(p^s+1)$ $\frac{p^{m+3d}(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)}$ $p^{s-1}(p^{s+1}-p^s-1)$ $\frac{p^{m+3d}(p-1)(p^m-p^{m-2d}-p^{m-3d}+p^s-p^{s-d}+1)(p^s-1)}{(p^d+1)(p^{2d}-1)}$ $p^{s+d-1}(p-1)(p^{s-d}-1)$ $\frac{p^{m-d}(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2}$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}+1)$ $\frac{p^{m-d}(p-1)(p^s+p^{s-d}+p^{s-2d}+1)(p^m-1)}{(p^d+1)^2}$ $p^{s+2d-1}(p-1)(p^{s-2d}+1)$ $\frac{p^{m-4d}(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)}$ $p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1)$ $\frac{p^{m-4d}(p-1)(p^{s-d}-1)(p^m-1)}{(p^d+1)(p^{2d}-1)}$ $p^{m-1}(p-1)$ $p(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{3s-4d}$ $+p^{3s-5d}+p^{m-d}-2p^{m-2d}+p^{m-3d}$ $-p^{m-4d}+1)(p^m-1)$ $p^m$ $p-1$
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 $p^{m-1}+\varepsilon p^{s-1}(p-1)$ $S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j=0$ $p^{m-1}-\varepsilon p^{s-1}$ $S(a,b,c)=\varepsilon p^s\zeta_p^j\; \mathrm{and}\; h+j\neq0$ $0$ $S(a,b,c)= p^m\; \mathrm{and}\; h\neq0$ $p^{m-1}\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $S(a,b,c)=\varepsilon \sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta_p^j\; \mathrm{and}\; \eta'(h+j)=\pm\varepsilon$ $p^{m-1}-p^{s+d-1}(p-1)$ $S(a,b,c)=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j=0$ $p^{m-1}+p^{s+d-1}$ $S(a,b,c)$ $=-p^{s+d}\zeta_p^j \; \mathrm{and}\; h+j\neq0$ $p^{m-1}$ $S(a,b,c)=0\; \mathrm{or}\; S(a,b,c)=\varepsilon\sqrt{p^*}\zeta_p^jp^{s+\frac{d-1}{2}}\; \mathrm{and}$ $h+j = 0$ $p^m$ $S(a,b,c)=p^m\; \mathrm{and}\; h = 0$
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 $(p-1)(p^{m-1}-p^{s-1})$ $M_1+(p-1)M_3$ $p^{m-1}(p-1)+p^{s-1}$ $(p-1)M_1+(p-1)^2M_3$ $(p-1)(p^{m-1}+p^{s-1})$ $M_2+(p-1)M_4$ $p^{m-1}(p-1)-p^{s-1}$ $(p-1)M_2+(p-1)^2M_4$ $p^{m-1}(p-1)\pm(-1)^{\frac{p-1}{2}}p^{s+\frac{d-1}{2}}$ $(p-1)M_5+\frac{(p-1)^2}{2}(M_{6,1}+M_{6,-1})$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $M_7+(p-1)M_8$ $p^{m-1}(p-1)-p^{s+d-1}$ $(p-1)M_7+(p-1)^2M_8$ $p^{m-1}(p-1)$ $pM_9+2M_5+(p-1)(M_{6,1}+M_{6,-1})$ $p^m$ $p-1$
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 $p^{s-1}(p-1)(p^s-1)$ $M_1+(p-1)M_3$ $p^{s-1}(p^{s+1}-p^s+1)$ $(p-1)M_1+(p-1)^2M_3$ $p^{s-1}(p-1)(p^s+1)$ $M_2+(p-1)M_4$ $p^{s-1}(p^{s+1}-p^s-1)$ $(p-1)M_2+(p-1)^2M_4$ $p^{s+\frac{d}{2}-1}(p^{s-\frac{d}{2}+1}-p^{s-\frac{d}{2}}\pm1)$ $(p-1)M_{5,\pm1}+(p-1)^2M_{6,\pm1}$ $p^{s+\frac{d}{2}-1}(p-1)(p^{s-\frac{d}{2}}\pm1)$ $M_{5,\mp1}+(p-1)M_{6,\mp1}$ $p^{s+d-1}(p-1)(p^{s-d}+1)$ $M_7+(p-1)M_8$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}-1)$ $(p-1)M_7+(p-1)^2M_8$ $p^{m-1}(p-1)$ $pM_9$ $p^m$ $p-1$
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 $p^{s-1}(p-1)(p^s+1)$ $M_1+(p-1)M_2$ $p^{s-1}(p^{s+1}-p^s-1)$ $(p-1)M_1+(p-1)^2M_2$ $p^{s+d-1}(p-1)(p^{s-d}-1)$ $M_3+(p-1)M_4$ $p^{s+d-1}(p^{s-d+1}-p^{s-d}+1)$ $(p-1)M_3+(p-1)^2M_4$ $p^{s+2d-1}(p-1)(p^{s-2d}+1)$ $M_5+(p-1)M_6$ $p^{s+2d-1}(p^{s-2d+1}-p^{s-2d}-1)$ $(p-1)M_5+(p-1)^2M_6$ $p^{m-1}(p-1)$ $pM_7$ $p^m$ $p-1$
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 $p^s$ $M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1)$ $-p^s$ $M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1)$ $\zeta^j_pp^s$ $M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1)$ $-\zeta^j_pp^s$ $M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1)$ $\varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}$ $M_5=\frac{1}{2}p^{3s-2d-1}(p^m-1)$ $\varepsilon\sqrt{p^*}p^{s+\frac{d-1}{2}}\zeta^j_p$ $M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d+1}{2}}(p^{s-\frac{d+1}{2}}+\varepsilon\eta'(-j))(p^m-1)$ $-p^{s+d}$ $M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1)$ $-p^{s+d}\zeta^j_p$ $M_8= p^{s-d-1}(p^{m-2d}-1)(p^m-1)/(p^{2d}-1)$ $0$ $M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1)$ $p^m$ $1$
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 $p^s$ $M_1= \frac{1}{2}p^{s+d-1}(p^s+1)(p^s+p-1)(p^m-1)/(p^d+1)$ $-p^s$ $M_2=\frac{1}{2}p^{s+d-1}(p^s-1)(p^s-p+1)(p^m-2p^{m-d}+1)/(p^d-1)$ $\zeta^j_pp^s$ $M_3=\frac{1}{2}p^{s+d-1}(p^m-1)^2/(p^d+1)$ $-\zeta^j_pp^s$ $M_4=\frac{1}{2}p^{s+d-1}(p^m-2p^{m-d}+1)(p^m-1)/(p^d-1)$ $\varepsilon p^{s+\frac{d}{2}}$ $M_{5,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}+\varepsilon(p-1))(p^m-1)$ $\varepsilon p^{s+\frac{d}{2}}\zeta^j_p$ $M_{6,\varepsilon}=\frac{1}{2}p^{m-\frac{3d}{2}-1}(p^{s-\frac{d}{2}}-\varepsilon)(p^m-1)$ $-p^{s+d}$ $M_7=p^{s-d-1}(p^{s-d}-1)(p^{s-d}-p+1)(p^m-1)/(p^{2d}-1)$ $-p^{s+d}\zeta^j_p$ $M_8= p^{s-d-1}(p^{s-d}-1)(p^{s-d}+1)(p^m-1)/(p^{2d}-1)$ $0$ $M_9=(p^{3s-d}-p^{3s-2d}+p^{3s-3d}-p^{m-2d}+1)(p^m-1)$ $p^m$ $1$
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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