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doi: 10.3934/amc.2020110
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Two constructions of low-hit-zone frequency-hopping sequence sets

Wenjuan Yin 1, , Can Xiang 2,, and  Fang-Wei Fu 1,

1. 

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2. 

College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, China

* Corresponding author: Can Xiang

Received  March 2020 Revised  July 2020 Early access September 2020

In this paper, we present two constructions of low-hit-zone frequen-cy-hopping sequence (LHZ FHS) sets. The constructions in this paper generalize the previous constructions based on $ m $-sequences and $ d $-form functions with difference-balanced property, and generate several classes of optimal LHZ FHS sets and LHZ FHS sets with optimal periodic partial Hamming correlation (PPHC).

Keywords: Frequency-hopping sequence (FHS), Hamming correlation, low-hit-zone (LHZ), $ d $-form function, difference-balanced.
Mathematics Subject Classification: Primary: 94A05; Secondary: 94B60.
Citation: Wenjuan Yin, Can Xiang, Fang-Wei Fu. Two constructions of low-hit-zone frequency-hopping sequence sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2020110
References:
[1]

J. H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by Cartesian product, IEEE Trans. Inf. Theory, 59 (2012), 726-732.  doi: 10.1109/TIT.2012.2213065.  Google Scholar

[2]

C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610.  doi: 10.1109/TIT.2007.899545.  Google Scholar

[3]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.  Google Scholar

[4]

C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304.  doi: 10.1109/TIT.2009.2021366.  Google Scholar

[5]

C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.  Google Scholar

[6]

G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: Auto- and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.  Google Scholar

[7]

T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 48 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[8]

H. Hu, S. Shao, G. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Trans. Inf. Theory, 60 (2014), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[9]

H. Han, S. Zhang, L. Zhou and X. Liu, Decimated $m$-sequences families with optimal partial Hamming correlation, Cryptogr. Commun., 12 (2020), 405-413.  doi: 10.1007/s12095-019-00400-7.  Google Scholar

[10]

H. Han, D. Peng, U. Parampalli, Z. Ma and H. Liang, Construction of low-hit-zone frequency hopping sequences with optimal partial Hamming correlation by interleaving techniques, Des. Codes Crypt., 84 (2017), 401-414.  doi: 10.1007/s10623-016-0274-8.  Google Scholar

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H. Han, D. Peng and U. Parampalli, New sets of optimal low-hit-zone frequency-hopping sequences based on $m$-sequences, Cryptogr. Commun., 9 (2017), 511-522.  doi: 10.1007/s12095-016-0192-7.  Google Scholar

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A. Lin, From Cyclic Hadamard Difference Sets to Perfectly Balanced Sequences, Ph.D dissertation, Dept. Comput. Sci., Univ. Southern California, Los Angeles, CA, USA, 1998. Google Scholar

[13]

X. Liu, D. Peng and H. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Cryptogr., 73 (2014), 167-176.  doi: 10.1007/s10623-013-9817-4.  Google Scholar

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X. Liu and L. Zhou, New bound on partial Hamming correlation of low-hit-zone frequency hopping sequences and optimal constructions, IEEE Commun. Lett., 22 (2018), 878-881.  doi: 10.1109/LCOMM.2018.2810868.  Google Scholar

[15]

W. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Cryptogr., 60 (2011), 145-153.  doi: 10.1007/s10623-010-9422-8.  Google Scholar

[16]

X. Niu, D. Peng, F. Liu and X. Liu, Lower bounds on the maximum partial correlations of frequency hopping sequence set with low hit zone, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E93-A (2010), 2227-2231.  doi: 10.1587/transfun.E93.A.2227.  Google Scholar

[17]

X. Niu, D. Peng and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E95-A (2012), 1835-1842.  doi: 10.1587/transfun.E95.A.1835.  Google Scholar

[18]

X. Niu, D. Peng and Z. Zhou, Frequency/time hopping sequence sets with optimal partial Hamming correlation properties, Sci. China Inf. Sci., 55 (2012), 2207-2215.  doi: 10.1007/s11432-012-4620-9.  Google Scholar

[19]

X. Niu, H. Lu and X. Liu, New extension interleaved constructions of optimal frequency hopping sequence sets with low hit zone, IEEE Access, 7 (2019), 73870-73879.  doi: 10.1109/ACCESS.2019.2919353.  Google Scholar

[20]

Y. Ouyang, X. Xie, H. Hu and M. Mao, Construction of three classes of stictly optimal frequency-hopping sequence sets, preprint, arXiv: 1905.04940.  Google Scholar

[21]

D. Peng and P. Fan, Lower bounds on the Hamming auto-and cross correlations of frequency-hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362.  Google Scholar

[22]

D. Peng, P. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone, Sci. China: Series F Inf. Sci., 49 (2006), 208-218.  doi: 10.1007/s11432-006-0208-6.  Google Scholar

[23]

M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, New York, NY, 2001. Google Scholar

[24]

C. Wang, D. Peng, H. Han and L. Zhou, New sets of low-hit-zone frequency-hopping sequence with optimal maximum periodic partial Hamming correlation, Sci. China Inf. Sci., 58 (2015), 1-15.  doi: 10.1007/s11432-015-5326-6.  Google Scholar

[25]

C. Wang, D. Peng and L. Zhou, New constructions of optimal frequency-hopping sequence sets with low-hit-zone, Int. J. Found. Comput. Sci., 27 (2016), 53-66.  doi: 10.1142/S0129054116500040.  Google Scholar

[26]

C. Wang, D. Peng, X. Niu and H. Han, Optimal construction of frequency-hopping sequence sets with low-hit-zone under periodic partial Hamming correlation, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E100-A (2017), 304-307.  doi: 10.1587/transfun.E100.A.304.  Google Scholar

[27]

L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Constructions of optimal low-hit-zone frequency hopping sequence sets, Des. Codes Cryptogr., 85 (2017), 219-232.  doi: 10.1007/s10623-016-0299-z.  Google Scholar

[28]

L. Zhou, D. Peng, H. Liang, C. Wang and H. Han, Generalized methods to construct low-hit-zone frequency-hopping sequence sets and optimal constructions, Cryptogr. Commun., 9 (2017), 707-728.  doi: 10.1007/s12095-017-0211-3.  Google Scholar

[29]

Z. Zhou, X. Tang, D. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 3831-3840.  doi: 10.1109/TIT.2011.2137290.  Google Scholar

[30]

Z. Zhou, X. Tang, X. Niu and U. Parampalli, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458.  doi: 10.1109/TIT.2011.2167126.  Google Scholar

show all references

References:
[1]

J. H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by Cartesian product, IEEE Trans. Inf. Theory, 59 (2012), 726-732.  doi: 10.1109/TIT.2012.2213065.  Google Scholar

[2]

C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 53 (2007), 2606-2610.  doi: 10.1109/TIT.2007.899545.  Google Scholar

[3]

C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 54 (2008), 3741-3745.  doi: 10.1109/TIT.2008.926410.  Google Scholar

[4]

C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inf. Theory, 55 (2009), 3297-3304.  doi: 10.1109/TIT.2009.2021366.  Google Scholar

[5]

C. Ding, Y. Yang and X. Tang, Optimal sets of frequency hopping sequences from linear cyclic codes, IEEE Trans. Inf. Theory, 56 (2010), 3605-3612.  doi: 10.1109/TIT.2010.2048504.  Google Scholar

[6]

G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: Auto- and cross-correlation properties, IEEE Trans. Inf. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.  Google Scholar

[7]

T. Helleseth and G. Gong, New nonbinary sequences with ideal two-level autocorrelation, IEEE Trans. Inf. Theory, 48 (2002), 2868-2872.  doi: 10.1109/TIT.2002.804052.  Google Scholar

[8]

H. Hu, S. Shao, G. Gong and T. Helleseth, The proof of Lin's conjecture via the decimation-Hadamard transform, IEEE Trans. Inf. Theory, 60 (2014), 5054-5064.  doi: 10.1109/TIT.2014.2327625.  Google Scholar

[9]

H. Han, S. Zhang, L. Zhou and X. Liu, Decimated $m$-sequences families with optimal partial Hamming correlation, Cryptogr. Commun., 12 (2020), 405-413.  doi: 10.1007/s12095-019-00400-7.  Google Scholar

[10]

H. Han, D. Peng, U. Parampalli, Z. Ma and H. Liang, Construction of low-hit-zone frequency hopping sequences with optimal partial Hamming correlation by interleaving techniques, Des. Codes Crypt., 84 (2017), 401-414.  doi: 10.1007/s10623-016-0274-8.  Google Scholar

[11]

H. Han, D. Peng and U. Parampalli, New sets of optimal low-hit-zone frequency-hopping sequences based on $m$-sequences, Cryptogr. Commun., 9 (2017), 511-522.  doi: 10.1007/s12095-016-0192-7.  Google Scholar

[12]

A. Lin, From Cyclic Hadamard Difference Sets to Perfectly Balanced Sequences, Ph.D dissertation, Dept. Comput. Sci., Univ. Southern California, Los Angeles, CA, USA, 1998. Google Scholar

[13]

X. Liu, D. Peng and H. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Cryptogr., 73 (2014), 167-176.  doi: 10.1007/s10623-013-9817-4.  Google Scholar

[14]

X. Liu and L. Zhou, New bound on partial Hamming correlation of low-hit-zone frequency hopping sequences and optimal constructions, IEEE Commun. Lett., 22 (2018), 878-881.  doi: 10.1109/LCOMM.2018.2810868.  Google Scholar

[15]

W. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Cryptogr., 60 (2011), 145-153.  doi: 10.1007/s10623-010-9422-8.  Google Scholar

[16]

X. Niu, D. Peng, F. Liu and X. Liu, Lower bounds on the maximum partial correlations of frequency hopping sequence set with low hit zone, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E93-A (2010), 2227-2231.  doi: 10.1587/transfun.E93.A.2227.  Google Scholar

[17]

X. Niu, D. Peng and Z. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E95-A (2012), 1835-1842.  doi: 10.1587/transfun.E95.A.1835.  Google Scholar

[18]

X. Niu, D. Peng and Z. Zhou, Frequency/time hopping sequence sets with optimal partial Hamming correlation properties, Sci. China Inf. Sci., 55 (2012), 2207-2215.  doi: 10.1007/s11432-012-4620-9.  Google Scholar

[19]

X. Niu, H. Lu and X. Liu, New extension interleaved constructions of optimal frequency hopping sequence sets with low hit zone, IEEE Access, 7 (2019), 73870-73879.  doi: 10.1109/ACCESS.2019.2919353.  Google Scholar

[20]

Y. Ouyang, X. Xie, H. Hu and M. Mao, Construction of three classes of stictly optimal frequency-hopping sequence sets, preprint, arXiv: 1905.04940.  Google Scholar

[21]

D. Peng and P. Fan, Lower bounds on the Hamming auto-and cross correlations of frequency-hopping sequences, IEEE Trans. Inf. Theory, 50 (2004), 2149-2154.  doi: 10.1109/TIT.2004.833362.  Google Scholar

[22]

D. Peng, P. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone, Sci. China: Series F Inf. Sci., 49 (2006), 208-218.  doi: 10.1007/s11432-006-0208-6.  Google Scholar

[23]

M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, New York, NY, 2001. Google Scholar

[24]

C. Wang, D. Peng, H. Han and L. Zhou, New sets of low-hit-zone frequency-hopping sequence with optimal maximum periodic partial Hamming correlation, Sci. China Inf. Sci., 58 (2015), 1-15.  doi: 10.1007/s11432-015-5326-6.  Google Scholar

[25]

C. Wang, D. Peng and L. Zhou, New constructions of optimal frequency-hopping sequence sets with low-hit-zone, Int. J. Found. Comput. Sci., 27 (2016), 53-66.  doi: 10.1142/S0129054116500040.  Google Scholar

[26]

C. Wang, D. Peng, X. Niu and H. Han, Optimal construction of frequency-hopping sequence sets with low-hit-zone under periodic partial Hamming correlation, IEICE Trans. Fundam. Electron. Commun. Comput. Sci., E100-A (2017), 304-307.  doi: 10.1587/transfun.E100.A.304.  Google Scholar

[27]

L. Zhou, D. Peng, H. Liang, C. Wang and Z. Ma, Constructions of optimal low-hit-zone frequency hopping sequence sets, Des. Codes Cryptogr., 85 (2017), 219-232.  doi: 10.1007/s10623-016-0299-z.  Google Scholar

[28]

L. Zhou, D. Peng, H. Liang, C. Wang and H. Han, Generalized methods to construct low-hit-zone frequency-hopping sequence sets and optimal constructions, Cryptogr. Commun., 9 (2017), 707-728.  doi: 10.1007/s12095-017-0211-3.  Google Scholar

[29]

Z. Zhou, X. Tang, D. Peng and U. Parampalli, New constructions for optimal sets of frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 3831-3840.  doi: 10.1109/TIT.2011.2137290.  Google Scholar

[30]

Z. Zhou, X. Tang, X. Niu and U. Parampalli, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458.  doi: 10.1109/TIT.2011.2167126.  Google Scholar

Table 1.  The parameters of some existing LHZ FHS sets with optimal PPHC properties and our new ones
parameters$ (L, M, r, Z_H, W;H_{pmz}) $ Constraints Reference
$ \Big(k_1L_1, M_1N_1, q, Z_1-1, W, \left\lceil\frac{W}{L}\right\rceil\Big) $ $ gcd(Z_1+1, L)=1, k_1(Z_1+1)\equiv 1\; (mod\; L_1) $,
$ k_1\equiv1\; (mod\; Z_1), M_1Z_1=L_1 $		 [13]
$ \Big(L_1L_2, p_2, p_1p_2, \min\{L_1, L_2\}-1, W; \left\lceil\frac{W}{T_1L_2}\right\rceil\Big) $ $ gcd(L_1, L_2)=1, p_1(\frac{L_1}{T_1}-1+\eta)(\min\{L_1, L_2\}p_2-1)= $
$ L_1L_2(\min\{L_1, L_2\}-p_1), 0 < \eta\leq 1 $
$ sl=q^n-1, 2\leq r_0 < \frac{(q^n-2)q}{q-1}, r_0\equiv t\; mod\; l $,		 [24]
$ \Big(r_0(q^n-1), s, q, l-1, \omega\frac{q^{n}-1}{q-1}; \omega\frac{q^{n-1}-1}{q-1}\Big) $ $ (l+1)r_0t^{-1}\equiv1\; (mod\; (q^n-1)), gcd(t, q^n-1)=1 $
$ gcd((l+1)t^{-1}\; mod\; (q^n-1), \frac{q^n-1}{q-1})=1, 1\leq \omega\leq(q-1)r_0 $		 [10]
$ \left((q_1-1)(q_2^n-1), q_1, q_1q_2^{n-1}, q_2^n-2, W;\left\lceil\frac{W(q_2-1)}{(q_2^n-1)(q_1-1)}\right\rceil\right) $ $ q_1, q_2 $ are two different prime powers satisfying
$ gcd(q_1-1, q_2^n-1)=1, q_1 >q_2^n $		 [28]
$ \Big(lcm(q-1, tv), q, qv, \min\{q-1, tv\}-1, W; \left\lceil\frac{W}{v(q-1)}\right\rceil\Big) $ $ gcd(q-1, v)=1, gcd(q-1, t) >1, tv < q-1 $,
or $ tv >q-1 $ and $ t < \frac{gcd(q-1, t)(q^2-q-1)}{qv-v-2} $ 		[28]
$ \Big(lcm(q-1, p(p^n-1)), qp^{n-1}, qp^n, \min\{q-1, p(p^n-1)\}-1, $ $ gcd(q-1, p^n-1)=1, q-1 >p(p^n-1) $, or $ q-1 < p(p^n-1) $ [28]
$ W; \left\lceil\frac{W}{(p^n-1)(q-1)}\right\rceil\Big) $ and $ gcd(q-1, p) >\frac{p^n(p+1)(q-1)+q(q-2-p)+1}{q^2p^{n-1}-qp^{n-1}-1} $
$ \left((p^m-1)p, p^m, p^m, p^m-2, W;\left\lceil\frac{W}{p^m-1}\right\rceil\right) $ $ p $ is a prime, $ m\geq 2 $ [14]
$ \left(2\rho, 2, \rho, \rho-1, W;\left\lceil\frac{W}{\varrho}\right\rceil\right) $ $ \rho $ is an even integer, [14]
$ \Big(\frac{q^n-1}{l}, q-1, q^k, \frac{q^n-1}{q-1}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\Big) $ $ l|(q-1), gcd(l, n)=1 $,
$ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer
$ l|(q-1), gcd(l, n)=1 $,		 [9]
$ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^k, \frac{T}{d'}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\right) $ $ T|(q^n-1), T\nmid l, gcd(T, l)=d', m|n $,
$ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer		 Theorem 3.1
$ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^{n-1}, T-1, W; \left\lceil\frac{W(q-1)}{q^n-1}\right\rceil\right) $ $ l|(q-1), gcd(l, n)=1 $,
$ T|(q^n-1), T\nmid l $ and $ gcd(T, l)=1 $		 Theorem 3.2
$ \left(\frac{q^n-1}{q-1}, \frac{q^n-1}{T}, q^{n-1}, \frac{T}{d'}-1, W;1\right) $ $ T|(q^n-1), T\nmid (q-1) $
and $ d'=gcd(T, q-1) $		 Theorem 3.3
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parameters$ (L, M, r, Z_H, W;H_{pmz}) $ Constraints Reference
$ \Big(k_1L_1, M_1N_1, q, Z_1-1, W, \left\lceil\frac{W}{L}\right\rceil\Big) $ $ gcd(Z_1+1, L)=1, k_1(Z_1+1)\equiv 1\; (mod\; L_1) $,
$ k_1\equiv1\; (mod\; Z_1), M_1Z_1=L_1 $		 [13]
$ \Big(L_1L_2, p_2, p_1p_2, \min\{L_1, L_2\}-1, W; \left\lceil\frac{W}{T_1L_2}\right\rceil\Big) $ $ gcd(L_1, L_2)=1, p_1(\frac{L_1}{T_1}-1+\eta)(\min\{L_1, L_2\}p_2-1)= $
$ L_1L_2(\min\{L_1, L_2\}-p_1), 0 < \eta\leq 1 $
$ sl=q^n-1, 2\leq r_0 < \frac{(q^n-2)q}{q-1}, r_0\equiv t\; mod\; l $,		 [24]
$ \Big(r_0(q^n-1), s, q, l-1, \omega\frac{q^{n}-1}{q-1}; \omega\frac{q^{n-1}-1}{q-1}\Big) $ $ (l+1)r_0t^{-1}\equiv1\; (mod\; (q^n-1)), gcd(t, q^n-1)=1 $
$ gcd((l+1)t^{-1}\; mod\; (q^n-1), \frac{q^n-1}{q-1})=1, 1\leq \omega\leq(q-1)r_0 $		 [10]
$ \left((q_1-1)(q_2^n-1), q_1, q_1q_2^{n-1}, q_2^n-2, W;\left\lceil\frac{W(q_2-1)}{(q_2^n-1)(q_1-1)}\right\rceil\right) $ $ q_1, q_2 $ are two different prime powers satisfying
$ gcd(q_1-1, q_2^n-1)=1, q_1 >q_2^n $		 [28]
$ \Big(lcm(q-1, tv), q, qv, \min\{q-1, tv\}-1, W; \left\lceil\frac{W}{v(q-1)}\right\rceil\Big) $ $ gcd(q-1, v)=1, gcd(q-1, t) >1, tv < q-1 $,
or $ tv >q-1 $ and $ t < \frac{gcd(q-1, t)(q^2-q-1)}{qv-v-2} $ 		[28]
$ \Big(lcm(q-1, p(p^n-1)), qp^{n-1}, qp^n, \min\{q-1, p(p^n-1)\}-1, $ $ gcd(q-1, p^n-1)=1, q-1 >p(p^n-1) $, or $ q-1 < p(p^n-1) $ [28]
$ W; \left\lceil\frac{W}{(p^n-1)(q-1)}\right\rceil\Big) $ and $ gcd(q-1, p) >\frac{p^n(p+1)(q-1)+q(q-2-p)+1}{q^2p^{n-1}-qp^{n-1}-1} $
$ \left((p^m-1)p, p^m, p^m, p^m-2, W;\left\lceil\frac{W}{p^m-1}\right\rceil\right) $ $ p $ is a prime, $ m\geq 2 $ [14]
$ \left(2\rho, 2, \rho, \rho-1, W;\left\lceil\frac{W}{\varrho}\right\rceil\right) $ $ \rho $ is an even integer, [14]
$ \Big(\frac{q^n-1}{l}, q-1, q^k, \frac{q^n-1}{q-1}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\Big) $ $ l|(q-1), gcd(l, n)=1 $,
$ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer
$ l|(q-1), gcd(l, n)=1 $,		 [9]
$ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^k, \frac{T}{d'}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\right) $ $ T|(q^n-1), T\nmid l, gcd(T, l)=d', m|n $,
$ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer		 Theorem 3.1
$ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^{n-1}, T-1, W; \left\lceil\frac{W(q-1)}{q^n-1}\right\rceil\right) $ $ l|(q-1), gcd(l, n)=1 $,
$ T|(q^n-1), T\nmid l $ and $ gcd(T, l)=1 $		 Theorem 3.2
$ \left(\frac{q^n-1}{q-1}, \frac{q^n-1}{T}, q^{n-1}, \frac{T}{d'}-1, W;1\right) $ $ T|(q^n-1), T\nmid (q-1) $
and $ d'=gcd(T, q-1) $		 Theorem 3.3
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