doi: 10.3934/amc.2022011
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Infinite families of 2-designs from a class of affine-invariant codes

1. 

College of Mathematics and Physics, Yancheng Institute of Technology, Yancheng, 224003, China

2. 

School of Mathematical Sciences, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

3. 

Key Laboratory of Mathematical Modeling and, High Performance Computing of Air Vehicles(NUAA), MIIT, Nanjing, 210016, China

* Corresponding author: Xiwang Cao

Received  September 2021 Revised  January 2022 Early access March 2022

Fund Project: The first author is supported by the National Natural Science Foundation of China (No. 12001475), Natural Science Foundation of Jiangsu Province (No. BK20201059) and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 19KJB120014). The second author is supported by the National Natural Science Foundation of China (No. 12171241)

In this paper, we first introduce a class of cyclic codes and their related linear codes which are affine-invariant. Then we obtain infinite families of 2-designs from the affine-invariant codes. And the parameters of the 2-designs are determined by considering the weight distribution of the linear codes.

Citation: Yan Liu, Xiwang Cao. Infinite families of 2-designs from a class of affine-invariant codes. Advances in Mathematics of Communications, doi: 10.3934/amc.2022011
References:
[1] E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge: Cambridge University Press, 1992.  doi: 10.1017/CBO9781316529836.
[2]

E. F. Assmus Jr. and H. F. Mattson Jr., New 5-designs, J. Comb. Theory, 6 (1969), 122-151.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. Bannai, E. Bannai, S. Suda and H. Tanaka, On relative t-designs in polynomial association schemes, Electron. J. Comb., 22 (2015), Paper 4.47, 17 pp. doi: 10.37236/4889.

[4]

E. ChoC. Ding and J. Hyun, A special characterisation of t-designs and its applications, Adv. Math. Commun., 13 (2019), 477-503.  doi: 10.3934/amc.2019030.

[5]

S. CostaT. Feng and X. Wang, New 2-designs from strong difference families, Finite Fields Appl., 50 (2018), 391-405.  doi: 10.1016/j.ffa.2017.12.011.

[6]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.

[7]

C. Ding, Codes from Difference Sets, World Scientific, Singapore, 2015.

[8]

C. Ding, Designs from Linear Codes, World Scientific, 2019.

[9]

C. Ding, An infinite family of Steiner systems S(2, 4, $2^{m}$) from cyclic codes, J. Comb. Des., 26 (2018), 127-144.  doi: 10.1002/jcd.21565.

[10]

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.

[11]

C. Ding and Z. Zhou, Parameters of 2-designs from some BCH codes, Codes, cryptography and information security, Cryptology and Information Security, 10194 (2017), 110-127.  doi: 10.1007/978-3-319-55589-8_8.

[12]

X. DuR. Wang and C. Fan, Infinite families of 2-designs from a class of cyclic codes, J. Comb. Des., 26 (2019), 1-14. 

[13]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from linear codes, Appl. Algebr. Eng. Comm., 2020. doi: 10.1007/s00200-020-00438-8.

[14]

R. FengM. Zhao and L. Zeng, Constructions of 11/2-designs from orthogonal geometry over finite fields, Discrete Math., 339 (2016), 382-390.  doi: 10.1016/j.disc.2015.09.002.

[15]

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

[16]

R. Lidl and H. Niederreiter, Finite Fieds, Addison-Wdsley Publishing Inc., 1983.

[17]

Y. Liu and H. Yan, A class of five-weight cyclic codes and their weight distribution, Des. Codes Crypogr., 79 (2016), 353-366.  doi: 10.1007/s10623-015-0056-8.

[18]

Y. Liu and X. Cao, Infinite families of 2-designs from affine-invariant codes, J. Comb. Des., 29 (2021), 683-702.  doi: 10.1002/jcd.21796.

[19]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[20]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.

[21]

C. Tang and C. Ding, An infinite family of linear codes supporting 4-designs, IEEE Trans. Inf. Theory, 67 (2021), 244-254.  doi: 10.1109/TIT.2020.3032600.

[22]

C. TangC. Ding and M. Xiong, Steiner systems $S(2, 4, \frac{3m-1}{2})$ and 2-designs from ternary linear codes of length $\frac{3m-1}{2}$, Des. Codes Crypogr., 87 (2019), 2793-2811.  doi: 10.1007/s10623-019-00651-8.

[23]

V. D. Tonchev, Codes and designs, Handbook of Coding Theory, Elsevier, Amsterdam, 2 (1998), 1229–1267.

[24]

V. D. Tonchev, Codes, In CRC Press, C. J. Colbourn, J. H. Dinitz (Eds.), Handbook of Combinatorial Designs, second edition, New York, (2007), 677–701.

[25]

R. WangX. Du and C. Fan, Infinite families of 2-designs from a class of non-binary Kasami cyclic codes, J. Combin. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[26]

R. WangX. DuC. Fan and Z. Niu, Infinite families of 2-designs from a class of linear codes related to Dembowski-Ostrom functions, Int. J. Found. Comput. S., 32 (2021), 253-267.  doi: 10.1142/S0129054121500143.

[27]

G. XuX. Cao and S. Xu, Optimal $p$-ary cyclic codes with minimum distance four from monomials, Cryptogr. Commun., 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.

[28]

X. Zhan and S. Ding, Some infinite families of 2-designs from PG(n, q), J. Comb. Des., 27 (2019), 22-26.  doi: 10.1002/jcd.21635.

[29]

X. Zhan and S. Zhou, Non-symmetric 2-designs admitting a two-dimensional projective linear group, Des. Codes Crypogr., 86 (2018), 2765-2773.  doi: 10.1007/s10623-018-0474-5.

[30]

Y. Zhang and S. Zhou, Flag-transitive non-symmetric 2-designs with (r, $\lambda$) = 1 and exceptional groups of Lie type, Electron. J. Comb., 27 (2020), Paper No. 2.9, 16 pp. doi: 10.37236/8832.

[31]

D. ZhengX. WangX. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Crypogr., 75 (2015), 263-275.  doi: 10.1007/s10623-013-9908-2.

[32]

Y. ZhuE. Bannai and E. Bannai, Tight relative 2-designs on two shells in Johnson association schemes, Discrete Math., 339 (2016), 957-973.  doi: 10.1016/j.disc.2015.10.024.

show all references

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

E. F. Assmus Jr. and H. F. Mattson Jr., New 5-designs, J. Comb. Theory, 6 (1969), 122-151.  doi: 10.1016/S0021-9800(69)80115-8.

[3]

E. Bannai, E. Bannai, S. Suda and H. Tanaka, On relative t-designs in polynomial association schemes, Electron. J. Comb., 22 (2015), Paper 4.47, 17 pp. doi: 10.37236/4889.

[4]

E. ChoC. Ding and J. Hyun, A special characterisation of t-designs and its applications, Adv. Math. Commun., 13 (2019), 477-503.  doi: 10.3934/amc.2019030.

[5]

S. CostaT. Feng and X. Wang, New 2-designs from strong difference families, Finite Fields Appl., 50 (2018), 391-405.  doi: 10.1016/j.ffa.2017.12.011.

[6]

P. Delsarte, On subfield subcodes of modified Reed-Solomon codes, IEEE Trans. Inf. Theory, 21 (1975), 575-576.  doi: 10.1109/tit.1975.1055435.

[7]

C. Ding, Codes from Difference Sets, World Scientific, Singapore, 2015.

[8]

C. Ding, Designs from Linear Codes, World Scientific, 2019.

[9]

C. Ding, An infinite family of Steiner systems S(2, 4, $2^{m}$) from cyclic codes, J. Comb. Des., 26 (2018), 127-144.  doi: 10.1002/jcd.21565.

[10]

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.

[11]

C. Ding and Z. Zhou, Parameters of 2-designs from some BCH codes, Codes, cryptography and information security, Cryptology and Information Security, 10194 (2017), 110-127.  doi: 10.1007/978-3-319-55589-8_8.

[12]

X. DuR. Wang and C. Fan, Infinite families of 2-designs from a class of cyclic codes, J. Comb. Des., 26 (2019), 1-14. 

[13]

X. Du, R. Wang, C. Tang and Q. Wang, Infinite families of 2-designs from linear codes, Appl. Algebr. Eng. Comm., 2020. doi: 10.1007/s00200-020-00438-8.

[14]

R. FengM. Zhao and L. Zeng, Constructions of 11/2-designs from orthogonal geometry over finite fields, Discrete Math., 339 (2016), 382-390.  doi: 10.1016/j.disc.2015.09.002.

[15]

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

[16]

R. Lidl and H. Niederreiter, Finite Fieds, Addison-Wdsley Publishing Inc., 1983.

[17]

Y. Liu and H. Yan, A class of five-weight cyclic codes and their weight distribution, Des. Codes Crypogr., 79 (2016), 353-366.  doi: 10.1007/s10623-015-0056-8.

[18]

Y. Liu and X. Cao, Infinite families of 2-designs from affine-invariant codes, J. Comb. Des., 29 (2021), 683-702.  doi: 10.1002/jcd.21796.

[19]

J. Luo and K. Feng, On the weight distribution of two classes of cyclic codes, IEEE Trans. Inf. Theory, 54 (2008), 5332-5344.  doi: 10.1109/TIT.2008.2006424.

[20]

K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4757-2103-4.

[21]

C. Tang and C. Ding, An infinite family of linear codes supporting 4-designs, IEEE Trans. Inf. Theory, 67 (2021), 244-254.  doi: 10.1109/TIT.2020.3032600.

[22]

C. TangC. Ding and M. Xiong, Steiner systems $S(2, 4, \frac{3m-1}{2})$ and 2-designs from ternary linear codes of length $\frac{3m-1}{2}$, Des. Codes Crypogr., 87 (2019), 2793-2811.  doi: 10.1007/s10623-019-00651-8.

[23]

V. D. Tonchev, Codes and designs, Handbook of Coding Theory, Elsevier, Amsterdam, 2 (1998), 1229–1267.

[24]

V. D. Tonchev, Codes, In CRC Press, C. J. Colbourn, J. H. Dinitz (Eds.), Handbook of Combinatorial Designs, second edition, New York, (2007), 677–701.

[25]

R. WangX. Du and C. Fan, Infinite families of 2-designs from a class of non-binary Kasami cyclic codes, J. Combin. Des., 28 (2020), 157-170.  doi: 10.1002/jcd.21682.

[26]

R. WangX. DuC. Fan and Z. Niu, Infinite families of 2-designs from a class of linear codes related to Dembowski-Ostrom functions, Int. J. Found. Comput. S., 32 (2021), 253-267.  doi: 10.1142/S0129054121500143.

[27]

G. XuX. Cao and S. Xu, Optimal $p$-ary cyclic codes with minimum distance four from monomials, Cryptogr. Commun., 8 (2016), 541-554.  doi: 10.1007/s12095-015-0159-0.

[28]

X. Zhan and S. Ding, Some infinite families of 2-designs from PG(n, q), J. Comb. Des., 27 (2019), 22-26.  doi: 10.1002/jcd.21635.

[29]

X. Zhan and S. Zhou, Non-symmetric 2-designs admitting a two-dimensional projective linear group, Des. Codes Crypogr., 86 (2018), 2765-2773.  doi: 10.1007/s10623-018-0474-5.

[30]

Y. Zhang and S. Zhou, Flag-transitive non-symmetric 2-designs with (r, $\lambda$) = 1 and exceptional groups of Lie type, Electron. J. Comb., 27 (2020), Paper No. 2.9, 16 pp. doi: 10.37236/8832.

[31]

D. ZhengX. WangX. Zeng and L. Hu, The weight distribution of a family of $p$-ary cyclic codes, Des. Codes Crypogr., 75 (2015), 263-275.  doi: 10.1007/s10623-013-9908-2.

[32]

Y. ZhuE. Bannai and E. Bannai, Tight relative 2-designs on two shells in Johnson association schemes, Discrete Math., 339 (2016), 957-973.  doi: 10.1016/j.disc.2015.10.024.

Table 1.  Value Distribution of $ T(\alpha,\beta,\gamma) $
Value Frequency
$ q_{0}^{s} $ 1
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s-1}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s-1}{2}} $ $\begin{align} & \frac{1}{2}({{p}^{m}}-1)({{p}^{2m}}-{{p}^{2m-d}}+{{p}^{2m-4d}}+{{p}^{m}}-{{p}^{m-d}}-{{p}^{m-3d}}+1) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -{{n}_{1, 2}}-{{n}_{1, 4}} \\ \end{align}$
$ q_{0}^{\frac{s+1}{2}} $ $ \frac{(p^{m+d}+p^{(m+3d)/2})(p^{2m}-p^{2m-2d}-p^{2m-3d}+p^{m-2d}+p^{m-3d}-1)}{2(p^{2d}-1)} $
$ -q_{0}^{\frac{s+1}{2}} $ $ \frac{(p^{m+d}-p^{(m+3d)/2})(p^{2m}-p^{2m-2d}-p^{2m-3d}+p^{m-2d}+p^{m-3d}-1)}{2(p^{2d}-1)} $
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+1}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+1}{2}} $ $ \frac{p^{2d}(p^{m}-1)(p^{m}-p^{m-2d}-p^{m-3d}+1)(p^{m-d}-1)}{2(p^{2d}-1)^{2}} $
$ q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m-3d}+p^{(m-3d)/2})(p^{m}-1)(p^{m-d}-1)}{2(p^{2d}-1)} $
$ -q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m-3d}-p^{(m-3d)/2})(p^{m}-1)(p^{m-d}-1)}{2(p^{2d}-1)} $
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+3}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m}-1)(p^{m-d}-1)(p^{m-3d}-1)}{2(p^{2d}-1)(p^{4d}-1)} $
Value Frequency
$ q_{0}^{s} $ 1
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s-1}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s-1}{2}} $ $\begin{align} & \frac{1}{2}({{p}^{m}}-1)({{p}^{2m}}-{{p}^{2m-d}}+{{p}^{2m-4d}}+{{p}^{m}}-{{p}^{m-d}}-{{p}^{m-3d}}+1) \\ & \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -{{n}_{1, 2}}-{{n}_{1, 4}} \\ \end{align}$
$ q_{0}^{\frac{s+1}{2}} $ $ \frac{(p^{m+d}+p^{(m+3d)/2})(p^{2m}-p^{2m-2d}-p^{2m-3d}+p^{m-2d}+p^{m-3d}-1)}{2(p^{2d}-1)} $
$ -q_{0}^{\frac{s+1}{2}} $ $ \frac{(p^{m+d}-p^{(m+3d)/2})(p^{2m}-p^{2m-2d}-p^{2m-3d}+p^{m-2d}+p^{m-3d}-1)}{2(p^{2d}-1)} $
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+1}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+1}{2}} $ $ \frac{p^{2d}(p^{m}-1)(p^{m}-p^{m-2d}-p^{m-3d}+1)(p^{m-d}-1)}{2(p^{2d}-1)^{2}} $
$ q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m-3d}+p^{(m-3d)/2})(p^{m}-1)(p^{m-d}-1)}{2(p^{2d}-1)} $
$ -q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m-3d}-p^{(m-3d)/2})(p^{m}-1)(p^{m-d}-1)}{2(p^{2d}-1)} $
$ \sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+3}{2}} $, $ -\sqrt{q_{0}^{\ast}}q_{0}^{\frac{s+3}{2}} $ $ \frac{(p^{m}-1)(p^{m-d}-1)(p^{m-3d}-1)}{2(p^{2d}-1)(p^{4d}-1)} $
Table 2.  Value Distribution of $ S(\alpha,\beta,\gamma,\delta) $
Value Frequency
$ 0 $ $ \omega $
$ p^{m} $ 1
$ \varepsilon p^{\frac{m+id}{2}} $ $ (p^{m-id-1}+\varepsilon(p-1)p^{\frac{m-id}{2}-1})n_{\varepsilon, i}, i=1,3 $
$ \varepsilon \zeta_{p}^{j}p^{\frac{m+id}{2}} $ $ (p^{m-id-1}-\varepsilon p^{\frac{m-id}{2}-1})n_{\varepsilon, i}, i=1,3 $
$ \varepsilon \sqrt{p^{\ast}}p^{\frac{m+id-1}{2}} $ $ p^{m-id-1}n_{\varepsilon, i}, i=0,2,4 $
$ \varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+id-1}{2}} $ $ (p^{m-id-1}+\varepsilon\eta(-j)p^{\frac{m-id-1}{2}})n_{\varepsilon, i}, i=0,2,4 $
Value Frequency
$ 0 $ $ \omega $
$ p^{m} $ 1
$ \varepsilon p^{\frac{m+id}{2}} $ $ (p^{m-id-1}+\varepsilon(p-1)p^{\frac{m-id}{2}-1})n_{\varepsilon, i}, i=1,3 $
$ \varepsilon \zeta_{p}^{j}p^{\frac{m+id}{2}} $ $ (p^{m-id-1}-\varepsilon p^{\frac{m-id}{2}-1})n_{\varepsilon, i}, i=1,3 $
$ \varepsilon \sqrt{p^{\ast}}p^{\frac{m+id-1}{2}} $ $ p^{m-id-1}n_{\varepsilon, i}, i=0,2,4 $
$ \varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+id-1}{2}} $ $ (p^{m-id-1}+\varepsilon\eta(-j)p^{\frac{m-id-1}{2}})n_{\varepsilon, i}, i=0,2,4 $
Table 3.  Weight distribution of $ \overline{\mathcal{C}}^{\perp} $
Weight $ (i) $ Frequency ($ A_{i} $)
$ 0 $ $ 1 $
$ p^{m} $ $ p-1 $
$ p^{m-1}(p-1) $ $ pw+p^{m}n_{0}+p^{m-2d}n_{2}+p^{m-4d}n_{4} $
$ (p-1)(p^{m-1}-p^{\frac{m+d}{2}-1}) $ $ p^{m-d}n_{1,1} $
$ (p-1)(p^{m-1}+p^{\frac{m+d}{2}-1}) $ $ p^{m-d}n_{-1,1} $
$ (p-1)p^{m-1}\pm p^{\frac{m+d}{2}-1} $ $ p^{m-d}(p-1)n_{\pm1,1} $
$ (p-1)(p^{m-1}-p^{\frac{m+3d}{2}-1}) $ $ p^{m-3d}n_{1,3} $
$ (p-1)(p^{m-1}+p^{\frac{m+3d}{2}-1}) $ $ p^{m-3d}n_{-1,3} $
$ (p-1)p^{m-1}\pm p^{\frac{m+3d}{2}-1} $ $ p^{m-3d}(p-1)n_{\pm1,3} $
$ (p-1)p^{m-1}\pm p^{\frac{m-1}{2}} $ $ \frac{p-1}{2}p^{m}n_{0} $
$ (p-1)p^{m-1}\pm p^{\frac{m+2d-1}{2}} $ $ \frac{p-1}{2}p^{m-2d}n_{2} $
$ (p-1)p^{m-1}\pm p^{\frac{m+4d-1}{2}} $ $ \frac{p-1}{2}p^{m-4d}n_{4} $
Weight $ (i) $ Frequency ($ A_{i} $)
$ 0 $ $ 1 $
$ p^{m} $ $ p-1 $
$ p^{m-1}(p-1) $ $ pw+p^{m}n_{0}+p^{m-2d}n_{2}+p^{m-4d}n_{4} $
$ (p-1)(p^{m-1}-p^{\frac{m+d}{2}-1}) $ $ p^{m-d}n_{1,1} $
$ (p-1)(p^{m-1}+p^{\frac{m+d}{2}-1}) $ $ p^{m-d}n_{-1,1} $
$ (p-1)p^{m-1}\pm p^{\frac{m+d}{2}-1} $ $ p^{m-d}(p-1)n_{\pm1,1} $
$ (p-1)(p^{m-1}-p^{\frac{m+3d}{2}-1}) $ $ p^{m-3d}n_{1,3} $
$ (p-1)(p^{m-1}+p^{\frac{m+3d}{2}-1}) $ $ p^{m-3d}n_{-1,3} $
$ (p-1)p^{m-1}\pm p^{\frac{m+3d}{2}-1} $ $ p^{m-3d}(p-1)n_{\pm1,3} $
$ (p-1)p^{m-1}\pm p^{\frac{m-1}{2}} $ $ \frac{p-1}{2}p^{m}n_{0} $
$ (p-1)p^{m-1}\pm p^{\frac{m+2d-1}{2}} $ $ \frac{p-1}{2}p^{m-2d}n_{2} $
$ (p-1)p^{m-1}\pm p^{\frac{m+4d-1}{2}} $ $ \frac{p-1}{2}p^{m-4d}n_{4} $
Table 4.  Value of $ I(\alpha,\beta,\gamma,\delta,h) $, $ j\in \mathbb{F}_{p} $
Value Corresponding Condition
$ 0 $ $ S(\alpha,\beta,\gamma,\delta)=p^{m}, h\neq0 $
$ p^{m} $ $ S(\alpha,\beta,\gamma,\delta)=p^{m}, h= 0 $
$ p^{m-1} $ $ S(\alpha,\beta,\gamma,\delta)=0 $ or $ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m-1}{2}}, h+j=0 $ or
$ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m+1}{2}}, h+j=0 $ or $ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m+3}{2}}, h+j=0 $
$ p^{m-1}+\varepsilon p^{\frac{m-1}{2}}(p-1) $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+1}{2}}, h+j=0 $
$ p^{m-1}-\varepsilon p^{\frac{m-1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+1}{2}}, h+j\neq0 $
$ p^{m-1}+\varepsilon p^{\frac{m+1}{2}}(p-1) $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+3}{2}}, h+j=0 $
$ p^{m-1}-\varepsilon p^{\frac{m+1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+3}{2}}, h+j\neq0 $
$ p^{m-1}\pm p^{\frac{m-1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m-1}{2}}, \eta(-(h+j))=\pm\varepsilon $
$ p^{m-1}\pm p^{\frac{m+1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+1}{2}}, \eta(-(h+j))=\pm\varepsilon $
$ p^{m-1}\pm p^{\frac{m+3}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+3}{2}}, \eta(-(h+j))=\pm\varepsilon $
Value Corresponding Condition
$ 0 $ $ S(\alpha,\beta,\gamma,\delta)=p^{m}, h\neq0 $
$ p^{m} $ $ S(\alpha,\beta,\gamma,\delta)=p^{m}, h= 0 $
$ p^{m-1} $ $ S(\alpha,\beta,\gamma,\delta)=0 $ or $ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m-1}{2}}, h+j=0 $ or
$ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m+1}{2}}, h+j=0 $ or $ \varepsilon\zeta_{p}^{j}\sqrt{p^{\ast}}p^{\frac{m+3}{2}}, h+j=0 $
$ p^{m-1}+\varepsilon p^{\frac{m-1}{2}}(p-1) $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+1}{2}}, h+j=0 $
$ p^{m-1}-\varepsilon p^{\frac{m-1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+1}{2}}, h+j\neq0 $
$ p^{m-1}+\varepsilon p^{\frac{m+1}{2}}(p-1) $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+3}{2}}, h+j=0 $
$ p^{m-1}-\varepsilon p^{\frac{m+1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \zeta_{p}^{j}p^{\frac{m+3}{2}}, h+j\neq0 $
$ p^{m-1}\pm p^{\frac{m-1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m-1}{2}}, \eta(-(h+j))=\pm\varepsilon $
$ p^{m-1}\pm p^{\frac{m+1}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+1}{2}}, \eta(-(h+j))=\pm\varepsilon $
$ p^{m-1}\pm p^{\frac{m+3}{2}} $ $ S(\alpha,\beta,\gamma,\delta)=\varepsilon \sqrt{p^{\ast}}\zeta_{p}^{j}p^{\frac{m+3}{2}}, \eta(-(h+j))=\pm\varepsilon $
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