• Previous Article
    Generalized Hamming weights of toric codes over hypersimplices and squarefree affine evaluation codes
  • AMC Home
  • This Issue
  • Next Article
    Z-complementary pairs with flexible lengths and large zero odd-periodic correlation zones
doi: 10.3934/amc.2020110
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

Two constructions of low-hit-zone frequency-hopping sequence sets

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).

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. DingM. 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. DingR. Fuji-HaraY. FujiwaraM. 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. DingY. 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. GeY. 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. HuS. ShaoG. 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. HanS. ZhangL. 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. HanD. PengU. ParampalliZ. 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. HanD. 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. LiuD. 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. NiuD. PengF. 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. NiuD. 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. NiuD. 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. NiuH. 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. PengP. 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. WangD. PengH. 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. WangD. 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. WangD. PengX. 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. ZhouD. PengH. LiangC. 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. ZhouD. PengH. LiangC. 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. ZhouX. TangD. 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. ZhouX. TangX. 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. DingM. 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. DingR. Fuji-HaraY. FujiwaraM. 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. DingY. 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. GeY. 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. HuS. ShaoG. 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. HanS. ZhangL. 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. HanD. PengU. ParampalliZ. 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. HanD. 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. LiuD. 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. NiuD. PengF. 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. NiuD. 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. NiuD. 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. NiuH. 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. PengP. 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. WangD. PengH. 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. WangD. 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. WangD. PengX. 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. ZhouD. PengH. LiangC. 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. ZhouD. PengH. LiangC. 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. ZhouX. TangD. 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. ZhouX. TangX. 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
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
[1]

Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004

[2]

Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

[3]

Hongyu Han, Sheng Zhang. New classes of strictly optimal low hit zone frequency hopping sequence sets. Advances in Mathematics of Communications, 2020, 14 (4) : 579-589. doi: 10.3934/amc.2020031

[4]

Xianhua Niu, Daiyuan Peng, Zhengchun Zhou. New classes of optimal frequency hopping sequences with low hit zone. Advances in Mathematics of Communications, 2013, 7 (3) : 293-310. doi: 10.3934/amc.2013.7.293

[5]

Xing Liu, Daiyuan Peng. Frequency hopping sequences with optimal aperiodic Hamming correlation by interleaving techniques. Advances in Mathematics of Communications, 2017, 11 (1) : 151-159. doi: 10.3934/amc.2017009

[6]

Wenli Ren, Feng Wang. A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020131

[7]

Xing Liu, Daiyuan Peng. Sets of frequency hopping sequences under aperiodic Hamming correlation: Upper bound and optimal constructions. Advances in Mathematics of Communications, 2014, 8 (3) : 359-373. doi: 10.3934/amc.2014.8.359

[8]

Shanding Xu, Xiwang Cao, Jiafu Mi, Chunming Tang. More cyclotomic constructions of optimal frequency-hopping sequences. Advances in Mathematics of Communications, 2019, 13 (3) : 373-391. doi: 10.3934/amc.2019024

[9]

Yang Yang, Xiaohu Tang, Guang Gong. New almost perfect, odd perfect, and perfect sequences from difference balanced functions with d-form property. Advances in Mathematics of Communications, 2017, 11 (1) : 67-76. doi: 10.3934/amc.2017002

[10]

Jingjun Bao. New families of strictly optimal frequency hopping sequence sets. Advances in Mathematics of Communications, 2018, 12 (2) : 387-413. doi: 10.3934/amc.2018024

[11]

Hua Liang, Wenbing Chen, Jinquan Luo, Yuansheng Tang. A new nonbinary sequence family with low correlation and large size. Advances in Mathematics of Communications, 2017, 11 (4) : 671-691. doi: 10.3934/amc.2017049

[12]

Zhenyu Zhang, Lijia Ge, Fanxin Zeng, Guixin Xuan. Zero correlation zone sequence set with inter-group orthogonal and inter-subgroup complementary properties. Advances in Mathematics of Communications, 2015, 9 (1) : 9-21. doi: 10.3934/amc.2015.9.9

[13]

Ming Su, Arne Winterhof. Hamming correlation of higher order. Advances in Mathematics of Communications, 2018, 12 (3) : 505-513. doi: 10.3934/amc.2018029

[14]

Nian Li, Xiaohu Tang, Tor Helleseth. A class of quaternary sequences with low correlation. Advances in Mathematics of Communications, 2015, 9 (2) : 199-210. doi: 10.3934/amc.2015.9.199

[15]

Fang Liu, Daiyuan Peng, Zhengchun Zhou, Xiaohu Tang. New constructions of optimal frequency hopping sequences with new parameters. Advances in Mathematics of Communications, 2013, 7 (1) : 91-101. doi: 10.3934/amc.2013.7.91

[16]

Qi Wang, Yue Zhou. Sets of zero-difference balanced functions and their applications. Advances in Mathematics of Communications, 2014, 8 (1) : 83-101. doi: 10.3934/amc.2014.8.83

[17]

Wei-Wen Hu. Integer-valued Alexis sequences with large zero correlation zone. Advances in Mathematics of Communications, 2017, 11 (3) : 445-452. doi: 10.3934/amc.2017037

[18]

Yu Zhou. On the distribution of auto-correlation value of balanced Boolean functions. Advances in Mathematics of Communications, 2013, 7 (3) : 335-347. doi: 10.3934/amc.2013.7.335

[19]

Zilong Wang, Guang Gong. Correlation of binary sequence families derived from the multiplicative characters of finite fields. Advances in Mathematics of Communications, 2013, 7 (4) : 475-484. doi: 10.3934/amc.2013.7.475

[20]

Ferruh Özbudak, Eda Tekin. Correlation distribution of a sequence family generalizing some sequences of Trachtenberg. Advances in Mathematics of Communications, 2021, 15 (4) : 647-662. doi: 10.3934/amc.2020087

2020 Impact Factor: 0.935

Article outline

Figures and Tables

[Back to Top]