August  2019, 13(3): 373-391. doi: 10.3934/amc.2019024

More cyclotomic constructions of optimal frequency-hopping sequences

1. 

Department of Mathematics and physics, Nanjing Institute of Technology, Nanjing 211167, China

2. 

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

3. 

Key Laboratory of Mathematics and Interdisciplinary Sciences, Guangdong Higher Education Institutes, Guangzhou University, Guangzhou 510006, China

4. 

State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China

Received  May 2017 Published  April 2019

In this paper, some general properties of the Zeng-Cai-Tang-Yang cyclotomy are studied. As its applications, two constructions of frequency-hopping sequences (FHSs) and two constructions of FHS sets are presented, where the length of sequences can be any odd integer larger than 3. The FHSs and FHS sets generated by our construction are (near-) optimal with respect to the Lempel–Greenberger bound and Peng–Fan bound, respectively. By choosing appropriate indexes and index sets, a lot of (near-) optimal FHSs and FHS sets can be obtained by our construction. Furthermore, some of them have new parameters which are not covered in the literature.

Citation: 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
References:
[1]

T. M. Apostol, Introduction to Analytic Number Theory, New York, NY, USA: Springer-Verlag, 1976. Google Scholar

[2]

H. CaiX. ZengT. HellesethX. Tang and Y. Yang, A new construction of zero-difference balanced functions and its applications, IEEE Trans. Inf. Theory, 59 (2013), 5008-5015. doi: 10.1109/TIT.2013.2255114. Google Scholar

[3]

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

[4]

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

[5]

J. ChungY. 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. Google Scholar

[6]

J. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. Google Scholar

[7]

J. Chung and K. Yang, $k$-fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317. doi: 10.1109/TIT.2011.2112235. Google Scholar

[8]

J. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum Hamming correlations, IET Commun., 6 (2012), 2048-2053. Google Scholar

[9]

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

[10]

C. Ding, D. Y. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography, Singapore: World Scientific, 1996. doi: 10.1142/9789812779380. Google Scholar

[11]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166. doi: 10.1006/ffta.1998.0207. Google Scholar

[12]

C. DingM. Miosio 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

[13]

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

[14]

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

[15]

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. Google Scholar

[16]

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

[17]

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

[18]

Y. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences, IEEE Trans. Inf. Theory, 55 (2009), 4279-4285. doi: 10.1109/TIT.2009.2025569. Google Scholar

[19]

P. Kumar, Frequency-hopping code sequence designs having large linear span, IEEE Trans. Inf. Theory, 34 (1988), 146-151. doi: 10.1109/18.2616. Google Scholar

[20]

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. 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]

W. RenF. Fu and Z. Zhou, New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Cryptogr., 72 (2014), 423-434. doi: 10.1007/s10623-012-9774-3. Google Scholar

[23]

S. XuX. Cao and G. Xu, Recursive construction of optimal frequency-hopping sequences sets, IET Commun., 10 (2016), 1080-1086. Google Scholar

[24]

S. XuX. Cao and G. Xu, Optimal frequency-hopping sequence sets based on cyclotomy, Int. J. Found. Comput. S., 27 (2016), 443-462. doi: 10.1142/S012905411650009X. Google Scholar

[25]

Y. YangX. TangU. Parampalli and D. Peng, New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inf. Theory, 57 (2011), 7605-7613. doi: 10.1109/TIT.2011.2162571. Google Scholar

[26]

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. Google Scholar

[27]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248. doi: 10.1109/TIT.2013.2237754. Google Scholar

[28]

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

show all references

References:
[1]

T. M. Apostol, Introduction to Analytic Number Theory, New York, NY, USA: Springer-Verlag, 1976. Google Scholar

[2]

H. CaiX. ZengT. HellesethX. Tang and Y. Yang, A new construction of zero-difference balanced functions and its applications, IEEE Trans. Inf. Theory, 59 (2013), 5008-5015. doi: 10.1109/TIT.2013.2255114. Google Scholar

[3]

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

[4]

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

[5]

J. ChungY. 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. Google Scholar

[6]

J. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inf. Theory, 56 (2010), 1685-1693. doi: 10.1109/TIT.2010.2040888. Google Scholar

[7]

J. Chung and K. Yang, $k$-fold cyclotomy and its application to frequency-hopping sequences, IEEE Trans. Inf. Theory, 57 (2011), 2306-2317. doi: 10.1109/TIT.2011.2112235. Google Scholar

[8]

J. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum Hamming correlations, IET Commun., 6 (2012), 2048-2053. Google Scholar

[9]

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

[10]

C. Ding, D. Y. Pei and A. Salomaa, Chinese Remainder Theorem: Applications in Computing, Coding, Cryptography, Singapore: World Scientific, 1996. doi: 10.1142/9789812779380. Google Scholar

[11]

C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166. doi: 10.1006/ffta.1998.0207. Google Scholar

[12]

C. DingM. Miosio 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

[13]

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

[14]

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

[15]

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. Google Scholar

[16]

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

[17]

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

[18]

Y. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences, IEEE Trans. Inf. Theory, 55 (2009), 4279-4285. doi: 10.1109/TIT.2009.2025569. Google Scholar

[19]

P. Kumar, Frequency-hopping code sequence designs having large linear span, IEEE Trans. Inf. Theory, 34 (1988), 146-151. doi: 10.1109/18.2616. Google Scholar

[20]

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. 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]

W. RenF. Fu and Z. Zhou, New sets of frequency-hopping sequences with optimal Hamming correlation, Des. Codes Cryptogr., 72 (2014), 423-434. doi: 10.1007/s10623-012-9774-3. Google Scholar

[23]

S. XuX. Cao and G. Xu, Recursive construction of optimal frequency-hopping sequences sets, IET Commun., 10 (2016), 1080-1086. Google Scholar

[24]

S. XuX. Cao and G. Xu, Optimal frequency-hopping sequence sets based on cyclotomy, Int. J. Found. Comput. S., 27 (2016), 443-462. doi: 10.1142/S012905411650009X. Google Scholar

[25]

Y. YangX. TangU. Parampalli and D. Peng, New bound on frequency hopping sequence sets and its optimal constructions, IEEE Trans. Inf. Theory, 57 (2011), 7605-7613. doi: 10.1109/TIT.2011.2162571. Google Scholar

[26]

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. Google Scholar

[27]

X. ZengH. CaiX. Tang and Y. Yang, Optimal frequency hopping sequences of odd length, IEEE Trans. Inf. Theory, 59 (2013), 3237-3248. doi: 10.1109/TIT.2013.2237754. Google Scholar

[28]

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

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