American Institute of Mathematical Sciences

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doi: 10.3934/amc.2020121

Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $

Junchao Zhou 1,2, , Yunge Xu 1, , Lisha Wang 1,, and  Nian Li 1,

1. 

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan, 430062, China
2. 

Faculty of Mathematics and Statistics, Hubei Engineering University, Xiaogan, 432000, China

* Corresponding author: Lisha Wang

Received  February 2020 Revised  September 2020 Published  December 2020

Fund Project: This work was supported by the National Natural Science Foundation of China (Nos. 61702166, 61761166010) and Science Research Project of Hubei Provincial Department of Education (No. B2020150)

In this paper, based on the theory of $ \mathbb{Z}_{4} $-valued quadratic forms we propose several classes of generalized Boolean bent functions over $ \mathbb{Z}_{4} $, and new families of codebooks are constructed from these functions. The codebooks constructed in this paper are nearly optimal with respect to the Welch bound, and their parameters are new. Furthermore, some Boolean bent functions are also derived.

Keywords: Codebook, generalized bent function, nearly optimal, Boolean bent function, $ \mathbb{Z}_4 $-valued quadratic form.
Mathematics Subject Classification: Primary: 11T71; Secondary: 11T23.
Citation: Junchao Zhou, Yunge Xu, Lisha Wang, Nian Li. Nearly optimal codebooks from generalized Boolean bent functions over $ \mathbb{Z}_{4} $. Advances in Mathematics of Communications, doi: 10.3934/amc.2020121
References:
[1]

E. H. Brown, Generalizations of the Kervaire invariant, Annals. Math., 95 (1972), 368-383.  doi: 10.2307/1970804.  Google Scholar

[2]

X. Cao, W. Chou and X. Zhang, More constructions of near optimal codebooks associated with binary sequences, Adv. Math. Commun., 11 (2017), 187-202.  doi: 10.3934/amc.2017012.  Google Scholar

[3]

C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[4]

P. Charpin, E. Pasalic and C. Tavernier, On bent and semi-bent quadratic Boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.  doi: 10.1109/TIT.2005.858929.  Google Scholar

[5]

C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inf. Theory, 52 (2006), 4229-4235.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[6]

C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inf. Theory, 53 (2007), 4245-4250.  doi: 10.1109/TIT.2007.907343.  Google Scholar

[7]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[8]

C. Ding and J. Yin, Signal sets from functions with optimimum nonlinearity, IEEE Trans. Communications, 55 (2007), 936-940.  doi: 10.1109/TCOMM.2007.894113.  Google Scholar

[9]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Sole, The ${\mathbb Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[10]

Z. Heng, Nearly optimal codebooks based on generalized Jacobi sums, Dis. Appli. Math., 250 (2018), 227-240.  doi: 10.1016/j.dam.2018.05.017.  Google Scholar

[11]

Z. Heng, C. Ding and Q. Yue, New constructions of asymptotically optimal codebooks with multiplicative characters, IEEE Trans. Inf. Theory, 63 (2017), 6179-6187.  doi: 10.1109/TIT.2017.2693204.  Google Scholar

[12]

Z. Heng and Q. Yue, Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 41-53.  doi: 10.1007/s12095-016-0194-5.  Google Scholar

[13]

S. Hong, H. Park, T. Helleseth and Y. Kim, Near-optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping, IEEE Trans. Inf. Theory, 60 (2014), 3698-3705.  doi: 10.1109/TIT.2014.2314298.  Google Scholar

[14]

H. Hu and J. Wu, New constructions of codebooks nearly meeting the Welch bound with equality, IEEE Trans. Inf. Theory, 60 (2014), 1348-1355.  doi: 10.1109/TIT.2013.2292745.  Google Scholar

[15]

V. I. Levenshtein, Bounds for packing of metric spaces and some of their applications, Probl. Cybern., 40 (1983), 43-110.   Google Scholar

[16]

C. Li, Q. Yue and Y. Huang, Two families of nearly optimal codebooks, Des. Codes Cryptogr., 75 (2015), 43-57.  doi: 10.1007/s10623-013-9891-7.  Google Scholar

[17]

N. Li, X. Tang and T. Helleseth, New constructions of quadratic bent functions in polynomial form, IEEE Trans. Inf. Theory, 60 (2014), 5760-5767.  doi: 10.1109/TIT.2014.2339861.  Google Scholar

[18]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, England, 1997.  Google Scholar

[19]

W. Lu, X. Wu, X. Cao and M. Chen, Six constructions of asymptotically optimal codebooks via the character sums, Des. Codes Cryptogr., 88 (2020), 1139-1158. doi: 10.1007/s10623-020-00735-w.  Google Scholar

[20]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum, IEEE Trans. Inf. Theory, 64 (2018), 6498-6505.  doi: 10.1109/TIT.2017.2777492.  Google Scholar

[21]

G. Luo and X. Cao, New constructions of codebooks asymptotically achieving the Welch bound, 2018 IEEE International Symposium on Information Theory, (2018), 2346–2350. Google Scholar

[22]

S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[23]

Y. Qi, S. Mesnager, C. Tang, Codebooks from generalized bent $\mathbb{Z}_{4}$-valued quadratic forms, Discret. Math., 343 (2020), 111736. doi: 10.1016/j.disc.2019.111736.  Google Scholar

[24]

L. Qian, X. Cao, W. Lu and X. Wu, New constructions of asymptotically optimal codebooks via character sums over a local ring, preprint, arXiv: 1906.05523. Google Scholar

[25]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[26]

D. V. Sarwate, Meeting the Welch bound with equality, Sequences and Their Applications, Springer London, 1999, 79–102.  Google Scholar

[27]

K. Schmidt, ${\mathbb Z}_4$-valued quadratic forms and quaternary sequence families, IEEE Trans. Inf. Theory, 55 (2009), 5803-5810.  doi: 10.1109/TIT.2009.2032818.  Google Scholar

[28]

P. Sol$\acute{e}$ and N. Tokareva, Connections between quaternary and binary bent functions, 2009. Available from: https://eprint.iacr.org/2009/544.pdf. Google Scholar

[29]

P. Tan, Z. Zhou and D. Zhang, A construction of codebooks nearly achieving the Levenshtein bound, IEEE Signal Process. Lett., 23 (2016), 1306-1309.  doi: 10.1109/LSP.2016.2595106.  Google Scholar

[30]

L. Welch, Lower bounds on the maximum cross correlation of signal, IEEE Trans. Inf. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.  Google Scholar

[31]

X. Wu, W. Lu and X. Cao, Two constructions of asymptotically optimal codebooks via the trace functions, Cryptogr. Commun., 12 (2020), 1195-1211.  doi: 10.1007/s12095-020-00448-w.  Google Scholar

[32]

P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inf. Theory, 51 (2005), 1900-1907.  doi: 10.1109/TIT.2005.846411.  Google Scholar

[33]

C. Xiang, C. Ding and S. Mesnager, Optimal codebooks from binary codes meeting the Levenshtein bound, IEEE Trans. Inf. Theory, 61 (2015), 6526-6535.  doi: 10.1109/TIT.2015.2487451.  Google Scholar

[34]

W. Yin, C. Xiang and F. Fu, A further construction of asymptotically optimal codebooks with multiplicative characters, Appl. Algebra Eng. Commun. Comput., 30 (2019), 453-469.  doi: 10.1007/s00200-019-00387-x.  Google Scholar

[35]

N. Y. Yu, A construction of codebooks associated with binary sequences, IEEE Trans. Inf. Theory, 58 (2012), 5522-5533.  doi: 10.1109/TIT.2012.2196021.  Google Scholar

[36]

A. Zhang and K. Feng, Two classes of codebooks nearly meeting the Welch bound, IEEE Trans. Inf. Theory, 58 (2012), 2507-2511.  doi: 10.1109/TIT.2011.2176531.  Google Scholar

[37]

A. Zhang and K. Feng, Construction of cyclotomic codebooks nearly meeting the Welch bound, Des. Codes Cryptogr., 63 (2012), 209-224.  doi: 10.1007/s10623-011-9549-2.  Google Scholar

[38]

Z. Zhou, C. Ding and N. Li, New families of codebooks achieving the Levenshtein bound, IEEE Trans. Inf. Theory, 60 (2014), 7382-7387.  doi: 10.1109/TIT.2014.2353052.  Google Scholar

[39]

Z. Zhou and X. Tang, New nearly optimal codebooks from relative difference sets, Adv. Math. Commun., 5 (2011), 521-527.  doi: 10.3934/amc.2011.5.521.  Google Scholar

show all references

References:
[1]

E. H. Brown, Generalizations of the Kervaire invariant, Annals. Math., 95 (1972), 368-383.  doi: 10.2307/1970804.  Google Scholar

[2]

X. Cao, W. Chou and X. Zhang, More constructions of near optimal codebooks associated with binary sequences, Adv. Math. Commun., 11 (2017), 187-202.  doi: 10.3934/amc.2017012.  Google Scholar

[3]

C. Carlet and S. Mesnager, Four decades of research on bent functions, Des. Codes Cryptogr., 78 (2016), 5-50.  doi: 10.1007/s10623-015-0145-8.  Google Scholar

[4]

P. Charpin, E. Pasalic and C. Tavernier, On bent and semi-bent quadratic Boolean functions, IEEE Trans. Inf. Theory, 51 (2005), 4286-4298.  doi: 10.1109/TIT.2005.858929.  Google Scholar

[5]

C. Ding, Complex codebooks from combinatorial designs, IEEE Trans. Inf. Theory, 52 (2006), 4229-4235.  doi: 10.1109/TIT.2006.880058.  Google Scholar

[6]

C. Ding and T. Feng, A generic construction of complex codebooks meeting the Welch bound, IEEE Trans. Inf. Theory, 53 (2007), 4245-4250.  doi: 10.1109/TIT.2007.907343.  Google Scholar

[7]

C. Ding and T. Feng, Codebooks from almost difference sets, Des. Codes Cryptogr., 46 (2008), 113-126.  doi: 10.1007/s10623-007-9140-z.  Google Scholar

[8]

C. Ding and J. Yin, Signal sets from functions with optimimum nonlinearity, IEEE Trans. Communications, 55 (2007), 936-940.  doi: 10.1109/TCOMM.2007.894113.  Google Scholar

[9]

A. R. Hammons, P. V. Kumar, A. R. Calderbank, N. J. A. Sloane and P. Sole, The ${\mathbb Z}_4$-linearity of Kerdock, Preparata, Goethals, and related codes, IEEE Trans. Inf. Theory, 40 (1994), 301-319.  doi: 10.1109/18.312154.  Google Scholar

[10]

Z. Heng, Nearly optimal codebooks based on generalized Jacobi sums, Dis. Appli. Math., 250 (2018), 227-240.  doi: 10.1016/j.dam.2018.05.017.  Google Scholar

[11]

Z. Heng, C. Ding and Q. Yue, New constructions of asymptotically optimal codebooks with multiplicative characters, IEEE Trans. Inf. Theory, 63 (2017), 6179-6187.  doi: 10.1109/TIT.2017.2693204.  Google Scholar

[12]

Z. Heng and Q. Yue, Optimal codebooks achieving the Levenshtein bound from generalized bent functions over $\mathbb{Z}_4$, Cryptogr. Commun., 9 (2017), 41-53.  doi: 10.1007/s12095-016-0194-5.  Google Scholar

[13]

S. Hong, H. Park, T. Helleseth and Y. Kim, Near-optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping, IEEE Trans. Inf. Theory, 60 (2014), 3698-3705.  doi: 10.1109/TIT.2014.2314298.  Google Scholar

[14]

H. Hu and J. Wu, New constructions of codebooks nearly meeting the Welch bound with equality, IEEE Trans. Inf. Theory, 60 (2014), 1348-1355.  doi: 10.1109/TIT.2013.2292745.  Google Scholar

[15]

V. I. Levenshtein, Bounds for packing of metric spaces and some of their applications, Probl. Cybern., 40 (1983), 43-110.   Google Scholar

[16]

C. Li, Q. Yue and Y. Huang, Two families of nearly optimal codebooks, Des. Codes Cryptogr., 75 (2015), 43-57.  doi: 10.1007/s10623-013-9891-7.  Google Scholar

[17]

N. Li, X. Tang and T. Helleseth, New constructions of quadratic bent functions in polynomial form, IEEE Trans. Inf. Theory, 60 (2014), 5760-5767.  doi: 10.1109/TIT.2014.2339861.  Google Scholar

[18]

R. Lidl and H. Niederreiter, Finite Fields, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, England, 1997.  Google Scholar

[19]

W. Lu, X. Wu, X. Cao and M. Chen, Six constructions of asymptotically optimal codebooks via the character sums, Des. Codes Cryptogr., 88 (2020), 1139-1158. doi: 10.1007/s10623-020-00735-w.  Google Scholar

[20]

G. Luo and X. Cao, Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum, IEEE Trans. Inf. Theory, 64 (2018), 6498-6505.  doi: 10.1109/TIT.2017.2777492.  Google Scholar

[21]

G. Luo and X. Cao, New constructions of codebooks asymptotically achieving the Welch bound, 2018 IEEE International Symposium on Information Theory, (2018), 2346–2350. Google Scholar

[22]

S. Mesnager, Bent Functions: Fundamentals and Results, Springer-Verlag, 2016. doi: 10.1007/978-3-319-32595-8.  Google Scholar

[23]

Y. Qi, S. Mesnager, C. Tang, Codebooks from generalized bent $\mathbb{Z}_{4}$-valued quadratic forms, Discret. Math., 343 (2020), 111736. doi: 10.1016/j.disc.2019.111736.  Google Scholar

[24]

L. Qian, X. Cao, W. Lu and X. Wu, New constructions of asymptotically optimal codebooks via character sums over a local ring, preprint, arXiv: 1906.05523. Google Scholar

[25]

O. S. Rothaus, On "bent" functions, J. Combin. Theory Ser. A, 20 (1976), 300-305.  doi: 10.1016/0097-3165(76)90024-8.  Google Scholar

[26]

D. V. Sarwate, Meeting the Welch bound with equality, Sequences and Their Applications, Springer London, 1999, 79–102.  Google Scholar

[27]

K. Schmidt, ${\mathbb Z}_4$-valued quadratic forms and quaternary sequence families, IEEE Trans. Inf. Theory, 55 (2009), 5803-5810.  doi: 10.1109/TIT.2009.2032818.  Google Scholar

[28]

P. Sol$\acute{e}$ and N. Tokareva, Connections between quaternary and binary bent functions, 2009. Available from: https://eprint.iacr.org/2009/544.pdf. Google Scholar

[29]

P. Tan, Z. Zhou and D. Zhang, A construction of codebooks nearly achieving the Levenshtein bound, IEEE Signal Process. Lett., 23 (2016), 1306-1309.  doi: 10.1109/LSP.2016.2595106.  Google Scholar

[30]

L. Welch, Lower bounds on the maximum cross correlation of signal, IEEE Trans. Inf. Theory, 20 (1974), 397-399.  doi: 10.1109/TIT.1974.1055219.  Google Scholar

[31]

X. Wu, W. Lu and X. Cao, Two constructions of asymptotically optimal codebooks via the trace functions, Cryptogr. Commun., 12 (2020), 1195-1211.  doi: 10.1007/s12095-020-00448-w.  Google Scholar

[32]

P. Xia, S. Zhou and G. B. Giannakis, Achieving the Welch bound with difference sets, IEEE Trans. Inf. Theory, 51 (2005), 1900-1907.  doi: 10.1109/TIT.2005.846411.  Google Scholar

[33]

C. Xiang, C. Ding and S. Mesnager, Optimal codebooks from binary codes meeting the Levenshtein bound, IEEE Trans. Inf. Theory, 61 (2015), 6526-6535.  doi: 10.1109/TIT.2015.2487451.  Google Scholar

[34]

W. Yin, C. Xiang and F. Fu, A further construction of asymptotically optimal codebooks with multiplicative characters, Appl. Algebra Eng. Commun. Comput., 30 (2019), 453-469.  doi: 10.1007/s00200-019-00387-x.  Google Scholar

[35]

N. Y. Yu, A construction of codebooks associated with binary sequences, IEEE Trans. Inf. Theory, 58 (2012), 5522-5533.  doi: 10.1109/TIT.2012.2196021.  Google Scholar

[36]

A. Zhang and K. Feng, Two classes of codebooks nearly meeting the Welch bound, IEEE Trans. Inf. Theory, 58 (2012), 2507-2511.  doi: 10.1109/TIT.2011.2176531.  Google Scholar

[37]

A. Zhang and K. Feng, Construction of cyclotomic codebooks nearly meeting the Welch bound, Des. Codes Cryptogr., 63 (2012), 209-224.  doi: 10.1007/s10623-011-9549-2.  Google Scholar

[38]

Z. Zhou, C. Ding and N. Li, New families of codebooks achieving the Levenshtein bound, IEEE Trans. Inf. Theory, 60 (2014), 7382-7387.  doi: 10.1109/TIT.2014.2353052.  Google Scholar

[39]

Z. Zhou and X. Tang, New nearly optimal codebooks from relative difference sets, Adv. Math. Commun., 5 (2011), 521-527.  doi: 10.3934/amc.2011.5.521.  Google Scholar

Table 1.  The parameters of codebooks nearly achieving the Welch bound
Parameters $ (N, K) $ Constraints $ I_{max} $ Ref.
$ \big(p^n, K=\frac{p-1}{2p}(p^n+p^{n/2})+1\big) $ $ p $ is an odd prime $ \frac{(p+1)p^{n/2}}{2pK} $ [13]
$ \big(q^2, \frac{(q-1)^2}{2}\big) $ $ q $ is a power of an odd prime $ \frac{q+1}{(q-1)^2} $ [36]
$ \big(q(q+4), \frac{(q+3)(q+1)}{2}\big) $ $ q $ is a prime power $ \frac{1}{q+1} $ [16]
$ (q, \frac{q-1}{2}) $ $ q $ is a prime power $ \frac{\sqrt{q}+1}{q-1} $ [16]
$ (p^n-1, \frac{p^n-1}{2}) $ $ p $ is an odd prime $ \frac{\sqrt{p^n}+1}{p^n-1} $ [35]
$ (q^l+q^{l-1}-1, q^{l-1}) $ $ l>2 $ $ \frac{1}{\sqrt{q^{l-1}}} $ [39]
$ \big((q-1)^k+q^{k-1}, q^{k-1}\big) $ $ k>2, q\geq4 $ $ \frac{\sqrt{q^{k+1}}}{(q-1)^{k}+(-1)^{k+1}} $ [11]
$ \big((q-1)^k+K, K\big) $ $ k>2, K=\frac{(q-1)^{k}+(-1)^{k+1}}{q} $ $ \frac{\sqrt{q^{k-1}}}{K} $ [11]
$ \big((q^s-1)^n+K, K\big) $ $ s>1, n>1 $ and $ \frac{\sqrt{q^{sn+1}}}{(q^s-1)^{n}+(-1)^{n+1}} $ [20]
$ K=\frac{(q^s-1)^{n}+(-1)^{n+1}}{q} $
$ \big((q^s-1)^n+q^{sn-1}, q^{sn-1}\big) $ $ s>1, n>1 $ $ \frac{\sqrt{q^{sn+1}}}{(q^s-1)^{n}+(-1)^{n+1}} $ [20]
$ \big(q-1, \frac{q(r-1)}{2r}\big) $ $ r=p^t, q=r^s, p \nmid s $ $ \frac{\sqrt{r}}{\sqrt{q}(\sqrt{r}-1) K} $ [31]
$ \big(q^2, \frac{q(q+1)(r-1)}{2r}\big) $ $ r=p^t, q=r^s $ $ \frac{(r+1)q}{2rK} $ [31]
$ (q^3, q^2) $, $ (q^3+q^2, q^2) $ $ q $ is a prime power $ \frac{1}{q} $ [19]
$ \big((q-1)q^2, (q-1)q\big) $, $ q $ is a prime power $ \frac{1}{q-1} $ [19]
$ \big((q^2-1)q, (q-1)q\big) $
$ \big(q^3-q^2-q+1, (q-1)^2\big) $, $ q $ is a prime power $ \frac{q}{(q-1)^2} $ [19]
$ \big(q^3-2q+1, (q-1)^2\big) $,
$ \big((q-1)^2q, (q-1)^2\big) $,
$ \big((q-1)q^2, (q-1)^2\big) $
$ \big((q-1)^2q, (q-1)(q-2)\big) $, $ q $ is a prime power $ \frac{q}{(q-1)(q-2)} $ [19]
$ \big(q^3-q^2-2q+2, (q-1)(q-2)\big) $
$ \big((q-1)^3, (q-2)^2\big) $, $ q $ is a prime power $ \frac{q}{(q-2)^2} $ [19]
$ \big(q^3-2q^2-q+3, (q-2)^2\big) $
$ (2^{\frac{n}{r}+n}+2^n, 2^n) $ $ 1<r<n, r\mid n $ $ \frac{1}{\sqrt{2^n}} $ This paper
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Parameters $ (N, K) $ Constraints $ I_{max} $ Ref.
$ \big(p^n, K=\frac{p-1}{2p}(p^n+p^{n/2})+1\big) $ $ p $ is an odd prime $ \frac{(p+1)p^{n/2}}{2pK} $ [13]
$ \big(q^2, \frac{(q-1)^2}{2}\big) $ $ q $ is a power of an odd prime $ \frac{q+1}{(q-1)^2} $ [36]
$ \big(q(q+4), \frac{(q+3)(q+1)}{2}\big) $ $ q $ is a prime power $ \frac{1}{q+1} $ [16]
$ (q, \frac{q-1}{2}) $ $ q $ is a prime power $ \frac{\sqrt{q}+1}{q-1} $ [16]
$ (p^n-1, \frac{p^n-1}{2}) $ $ p $ is an odd prime $ \frac{\sqrt{p^n}+1}{p^n-1} $ [35]
$ (q^l+q^{l-1}-1, q^{l-1}) $ $ l>2 $ $ \frac{1}{\sqrt{q^{l-1}}} $ [39]
$ \big((q-1)^k+q^{k-1}, q^{k-1}\big) $ $ k>2, q\geq4 $ $ \frac{\sqrt{q^{k+1}}}{(q-1)^{k}+(-1)^{k+1}} $ [11]
$ \big((q-1)^k+K, K\big) $ $ k>2, K=\frac{(q-1)^{k}+(-1)^{k+1}}{q} $ $ \frac{\sqrt{q^{k-1}}}{K} $ [11]
$ \big((q^s-1)^n+K, K\big) $ $ s>1, n>1 $ and $ \frac{\sqrt{q^{sn+1}}}{(q^s-1)^{n}+(-1)^{n+1}} $ [20]
$ K=\frac{(q^s-1)^{n}+(-1)^{n+1}}{q} $
$ \big((q^s-1)^n+q^{sn-1}, q^{sn-1}\big) $ $ s>1, n>1 $ $ \frac{\sqrt{q^{sn+1}}}{(q^s-1)^{n}+(-1)^{n+1}} $ [20]
$ \big(q-1, \frac{q(r-1)}{2r}\big) $ $ r=p^t, q=r^s, p \nmid s $ $ \frac{\sqrt{r}}{\sqrt{q}(\sqrt{r}-1) K} $ [31]
$ \big(q^2, \frac{q(q+1)(r-1)}{2r}\big) $ $ r=p^t, q=r^s $ $ \frac{(r+1)q}{2rK} $ [31]
$ (q^3, q^2) $, $ (q^3+q^2, q^2) $ $ q $ is a prime power $ \frac{1}{q} $ [19]
$ \big((q-1)q^2, (q-1)q\big) $, $ q $ is a prime power $ \frac{1}{q-1} $ [19]
$ \big((q^2-1)q, (q-1)q\big) $
$ \big(q^3-q^2-q+1, (q-1)^2\big) $, $ q $ is a prime power $ \frac{q}{(q-1)^2} $ [19]
$ \big(q^3-2q+1, (q-1)^2\big) $,
$ \big((q-1)^2q, (q-1)^2\big) $,
$ \big((q-1)q^2, (q-1)^2\big) $
$ \big((q-1)^2q, (q-1)(q-2)\big) $, $ q $ is a prime power $ \frac{q}{(q-1)(q-2)} $ [19]
$ \big(q^3-q^2-2q+2, (q-1)(q-2)\big) $
$ \big((q-1)^3, (q-2)^2\big) $, $ q $ is a prime power $ \frac{q}{(q-2)^2} $ [19]
$ \big(q^3-2q^2-q+3, (q-2)^2\big) $
$ (2^{\frac{n}{r}+n}+2^n, 2^n) $ $ 1<r<n, r\mid n $ $ \frac{1}{\sqrt{2^n}} $ This paper
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