doi: 10.3934/amc.2022024
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Construction of three classes of strictly optimal frequency-hopping sequence sets

1. 

Key Laboratory of Electromagnetic Space Information, CAS, School of Cyber Science and Technology, University of Science and Technology of China, Hefei, Anhui 230027, China

2. 

Wu Wen-Tsun Key Laboratory of Mathematics, School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui 230027, China

3. 

Beijing Electronic Science and Technology Institute, Beijing 100070, China

*Corresponding author: Yi Ouyang

Received  October 2021 Revised  January 2022 Early access April 2022

Fund Project: Research is partially supported by Anhui Initiative in Quantum Information Technologies (Grant No. AHY150200) and NSFC (Grant No. 11571328)

In this paper, we construct three classes of strictly optimal frequency-hopping sequence (FHS) sets with respect to partial Hamming correlation and family size. The first and second classes are based on the trace map, the third class is based on a generic construction.

Citation: Xianhong Xie, Yi Ouyang, Honggang Hu, Ming Mao. Construction of three classes of strictly optimal frequency-hopping sequence sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2022024
References:
[1]

J. Bao, New families of strictly frequency hopping sequence sets, Adv. Math. Commun., 12 (2018), 387-413.  doi: 10.3934/amc.2018024.

[2]

J. Bao and L. Ji, Frequency hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 62 (2016), 3768-3783.  doi: 10.1109/TIT.2016.2551225.

[3]

H. CaiY. YangZ. Zhou and X. Tang, Strictly optimal frequency-hopping sequence sets with optimal family sizes, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093.  doi: 10.1109/TIT.2015.2512859.

[4]

H. CaiZ. ZhouY. Yang and X. Tang, A new construction of frequency-hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 60 (2014), 5782-5790.  doi: 10.1109/TIT.2014.2332996.

[5]

B. ChenL. LinS. Ling and H. Liu, Three new classes of optimal frequency-hopping sequence sets, Des. Codes Cryptogr., 83 (2017), 219-232.  doi: 10.1007/s10623-016-0220-9.

[6]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.

[7]

J.-H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791.  doi: 10.1109/TIT.2009.2032742.

[8]

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.

[9]

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.

[10]

Y. C. EunS. Y. JinY. P. Hong and H. Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inf. Theory, 50 (2004), 2438-2442.  doi: 10.1109/TIT.2004.834792.

[11]

C. FanH. Cai and X. Tang, A combinatorial construction for strictly optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 62 (2016), 4769-4774.  doi: 10.1109/TIT.2016.2556710.

[12]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.

[13]

G. GeR. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718.  doi: 10.1016/j.jcta.2006.03.019.

[14]

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.

[15]

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.

[16]

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.

[17]

H. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inf. Theory, 56 (2010), 1702-1713.  doi: 10.1109/TIT.2010.2040887.

[18]

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.

[19]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94.  doi: 10.1109/tit.1974.1055169.

[20]

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

[21]

X. LiuL. Zhou and S. Li, A new method to construct strictly optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 65 (2019), 1828-1844.  doi: 10.1109/TIT.2018.2864154.

[22]

S. L. Ng and M. B. Paterson, Disjoint difference families and their applications, Des. Codes Cryptogr., 78 (2016), 103-127.  doi: 10.1007/s10623-015-0149-4.

[23]

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. Fund., E93-A (2010), 2227-2231. 

[24]

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.

[25]

A. Pott and Q. Wang, Some results on difference balanced functions, Arithmetic of Finite Fields, LNCS, Springer, 9061 (2015), 111-120.  doi: 10.1007/978-3-319-16277-5_6.

[26]

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

[27]

G. Solomn, Optimal frequency hopping sequences for multiple access, In Pro. Symp. Spread Spectr. Commun., (1973), 33–35.

[28]

H. Y. Song and S. W. Golomb, On the nonperiodic cyclic equivalence classes of Reed-Solomon codes, IEEE Trans. Inf. Theory, 39 (1993), 1431-1434. 

[29]

S. XuX. CaoJ. Gao and C. Tang, A kind of disjoint cyclic perfect Mendelsohn difference family and its applications in strictly optimal FHSs, IEICE Trans. Fund., E101-A (2018), 2338-2343. 

[30]

X. ZengH. CaiX. Tang and Y. Yang, A class of optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 58 (2012), 4899-4907.  doi: 10.1109/TIT.2012.2195771.

[31]

L. ZhouD. PengH. Han and H. Liang, Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation, Adv. Math. Commun., 12 (2018), 67-79.  doi: 10.3934/amc.2018004.

[32]

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.

[33]

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.

show all references

References:
[1]

J. Bao, New families of strictly frequency hopping sequence sets, Adv. Math. Commun., 12 (2018), 387-413.  doi: 10.3934/amc.2018024.

[2]

J. Bao and L. Ji, Frequency hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 62 (2016), 3768-3783.  doi: 10.1109/TIT.2016.2551225.

[3]

H. CaiY. YangZ. Zhou and X. Tang, Strictly optimal frequency-hopping sequence sets with optimal family sizes, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093.  doi: 10.1109/TIT.2015.2512859.

[4]

H. CaiZ. ZhouY. Yang and X. Tang, A new construction of frequency-hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inf. Theory, 60 (2014), 5782-5790.  doi: 10.1109/TIT.2014.2332996.

[5]

B. ChenL. LinS. Ling and H. Liu, Three new classes of optimal frequency-hopping sequence sets, Des. Codes Cryptogr., 83 (2017), 219-232.  doi: 10.1007/s10623-016-0220-9.

[6]

W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141.  doi: 10.1109/TIT.2004.842708.

[7]

J.-H. ChungY. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inf. Theory, 55 (2009), 5783-5791.  doi: 10.1109/TIT.2009.2032742.

[8]

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.

[9]

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.

[10]

Y. C. EunS. Y. JinY. P. Hong and H. Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inf. Theory, 50 (2004), 2438-2442.  doi: 10.1109/TIT.2004.834792.

[11]

C. FanH. Cai and X. Tang, A combinatorial construction for strictly optimal frequency-hopping sequences, IEEE Trans. Inf. Theory, 62 (2016), 4769-4774.  doi: 10.1109/TIT.2016.2556710.

[12]

R. Fuji-HaraY. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach, IEEE Trans. Inf. Theory, 50 (2004), 2408-2420.  doi: 10.1109/TIT.2004.834783.

[13]

G. GeR. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency-hopping sequences, J. Combin. Theory Ser. A, 113 (2006), 1699-1718.  doi: 10.1016/j.jcta.2006.03.019.

[14]

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.

[15]

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.

[16]

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.

[17]

H. Hu and G. Gong, New sets of zero or low correlation zone sequences via interleaving techniques, IEEE Trans. Inf. Theory, 56 (2010), 1702-1713.  doi: 10.1109/TIT.2010.2040887.

[18]

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.

[19]

A. Lempel and H. Greenberger, Families of sequences with optimal Hamming-correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94.  doi: 10.1109/tit.1974.1055169.

[20]

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

[21]

X. LiuL. Zhou and S. Li, A new method to construct strictly optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 65 (2019), 1828-1844.  doi: 10.1109/TIT.2018.2864154.

[22]

S. L. Ng and M. B. Paterson, Disjoint difference families and their applications, Des. Codes Cryptogr., 78 (2016), 103-127.  doi: 10.1007/s10623-015-0149-4.

[23]

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. Fund., E93-A (2010), 2227-2231. 

[24]

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.

[25]

A. Pott and Q. Wang, Some results on difference balanced functions, Arithmetic of Finite Fields, LNCS, Springer, 9061 (2015), 111-120.  doi: 10.1007/978-3-319-16277-5_6.

[26]

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

[27]

G. Solomn, Optimal frequency hopping sequences for multiple access, In Pro. Symp. Spread Spectr. Commun., (1973), 33–35.

[28]

H. Y. Song and S. W. Golomb, On the nonperiodic cyclic equivalence classes of Reed-Solomon codes, IEEE Trans. Inf. Theory, 39 (1993), 1431-1434. 

[29]

S. XuX. CaoJ. Gao and C. Tang, A kind of disjoint cyclic perfect Mendelsohn difference family and its applications in strictly optimal FHSs, IEICE Trans. Fund., E101-A (2018), 2338-2343. 

[30]

X. ZengH. CaiX. Tang and Y. Yang, A class of optimal frequency hopping sequences with new parameters, IEEE Trans. Inf. Theory, 58 (2012), 4899-4907.  doi: 10.1109/TIT.2012.2195771.

[31]

L. ZhouD. PengH. Han and H. Liang, Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation, Adv. Math. Commun., 12 (2018), 67-79.  doi: 10.3934/amc.2018004.

[32]

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.

[33]

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.

Table 1.  Known Strictly Optimal FHS Sets
Length Alphabet Size $ \mathcal{M}( \mathcal{F};L) $ Family Size Constraints Reference
$ \frac{p^m-1}{r} $ $ p^{m-1} $ $ \left\lceil\frac{L}{T}\right\rceil $ $ r $ $ \psi(x) $ is identity, $ r|p-1 $, $ \gcd(r, m)=1 $ [32]
$ p^{2m}-1 $ $ p^{m} $ $ \left\lceil\frac{L}{p^m+1}\right\rceil $ 1 [10,32]
$ p(p^m-1) $ $ p^m $ $ \left\lceil\frac{L}{p^m-1}\right\rceil $ $ p^{m-1} $ $ \phi(x) $ is identity [3]
$ \frac{p^m-1}{r} $ $ p^{m-1} $ $ \left\lceil\frac{L}{T}\right\rceil $ $ r $ $ \psi(x) $, $ r|p-1 $, $ \gcd(r, m)=1 $ Theorem 3.2 here
$ \frac{p^{2m}-1}{r} $ $ p^m $ $ \left\lceil\frac{L}{p^m+1}\right\rceil $ $ r $ $ f(x) $, $ r|p-1 $, $ \gcd(r, 2m)=1 $ Theorem 3.4 here
$ p(p^m-1) $ $ p^m $ $ \left\lceil\frac{L}{p^m-1}\right\rceil $ $ {p^{m-1}} $ $ \phi(x) $ Theorem 4.5 here
$ T=\frac{q-1}{p-1} $, $ q=p^m $ and $ \gcd(d, p-1)=1 $; $ \psi(x) $ is $ \mathbb{F}_p $-linear automorphism of $ \mathbb{F}_{q} $; $ \phi(x) $ can be found in Lemma 4.3; $ f: \mathbb{F}_{q^2}\rightarrow \mathbb{F}_q $ is $ d $-form difference-balanced function.
Length Alphabet Size $ \mathcal{M}( \mathcal{F};L) $ Family Size Constraints Reference
$ \frac{p^m-1}{r} $ $ p^{m-1} $ $ \left\lceil\frac{L}{T}\right\rceil $ $ r $ $ \psi(x) $ is identity, $ r|p-1 $, $ \gcd(r, m)=1 $ [32]
$ p^{2m}-1 $ $ p^{m} $ $ \left\lceil\frac{L}{p^m+1}\right\rceil $ 1 [10,32]
$ p(p^m-1) $ $ p^m $ $ \left\lceil\frac{L}{p^m-1}\right\rceil $ $ p^{m-1} $ $ \phi(x) $ is identity [3]
$ \frac{p^m-1}{r} $ $ p^{m-1} $ $ \left\lceil\frac{L}{T}\right\rceil $ $ r $ $ \psi(x) $, $ r|p-1 $, $ \gcd(r, m)=1 $ Theorem 3.2 here
$ \frac{p^{2m}-1}{r} $ $ p^m $ $ \left\lceil\frac{L}{p^m+1}\right\rceil $ $ r $ $ f(x) $, $ r|p-1 $, $ \gcd(r, 2m)=1 $ Theorem 3.4 here
$ p(p^m-1) $ $ p^m $ $ \left\lceil\frac{L}{p^m-1}\right\rceil $ $ {p^{m-1}} $ $ \phi(x) $ Theorem 4.5 here
$ T=\frac{q-1}{p-1} $, $ q=p^m $ and $ \gcd(d, p-1)=1 $; $ \psi(x) $ is $ \mathbb{F}_p $-linear automorphism of $ \mathbb{F}_{q} $; $ \phi(x) $ can be found in Lemma 4.3; $ f: \mathbb{F}_{q^2}\rightarrow \mathbb{F}_q $ is $ d $-form difference-balanced function.
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