# American Institute of Mathematical Sciences

August  2013, 7(3): 293-310. doi: 10.3934/amc.2013.7.293

## New classes of optimal frequency hopping sequences with low hit zone

 1 School of Mathematics and Computer Engineering, The Key Laboratory of Network Intelligent Information Processing, Xihua University, Chengdu, Sichuan 610039, China 2 School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 3 School of Mathematics, Southwest Jiaotong University, Chengdu, 610031

Received  July 2012 Revised  February 2013 Published  July 2013

In this paper, a new design of frequency hopping sequences (FHSs) sets with low hit zone (LHZ) is presented based on interleaving technique. The key idea of the new design is to use short FHSs with good Hamming correlation together with certain appropriate shift sequences to construct a set of long FHSs with LHZ. By the new design, new sets of FHSs meeting the Peng-Fan-Lee bound are obtained. It is shown that all the sequences in the proposed FHS sets are shift distinct. The proposed FHS sets are suitable for quasi-synchronous frequency hopping code division multiple access systems to eliminate multiple-access interference.
Citation: 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
##### References:
 [1] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inf. Theory, 51 (2005), 1139. doi: 10.1109/TIT.2004.842708. Google Scholar [2] J. H. Chung, Y. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving technique,, IEEE Trans. Inf. Theory, 55 (2009), 5783. doi: 10.1109/TIT.2009.2032742. Google Scholar [3] J. H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inf. Theory, 56 (2010), 1685. doi: 10.1109/TIT.2010.2040888. Google Scholar [4] C. Ding, R. Fuji-Hara, Y. Fujiwara, et al., Sets of frequency hopping sequences: Bounds and optimal constructions,, IEEE Trans. Inf. Theory, 55 (2009), 3297. doi: 10.1109/TIT.2009.2021366. Google Scholar [5] C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation,, IEEE Trans. Inf. Theory, 47 (2001), 428. doi: 10.1109/18.904555. Google Scholar [6] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency hopping sequences,, IEEE Trans. Inf. Theory, 53 (2007), 2606. doi: 10.1109/TIT.2007.899545. Google Scholar [7] C. Ding and J. Yin, Sets of optimal frequency-hopping sequences,, IEEE Trans. Inf. Theory, 54 (2008), 3741. doi: 10.1109/TIT.2008.926410. Google Scholar [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'', RSP-John Wiley Sons Inc., (1996). Google Scholar [9] P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency hopping CDMA systems,, IEEE Trans. Wir. Commun., 4 (2005), 2836. Google Scholar [10] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach,, IEEE Trans. Inf. Theory, 50 (2004), 2408. doi: 10.1109/TIT.2004.834783. Google Scholar [11] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency hopping sequences,, J. Combin. Theory Ser. A, 113 (2006), 1699. doi: 10.1016/j.jcta.2006.03.019. Google Scholar [12] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: auto- and crosscorrelation properties,, IEEE Trans. Inf. Theory, 55 (2009), 867. doi: 10.1109/TIT.2008.2009856. Google Scholar [13] G. Gong, Theory and applications of q-ary interleaved sequences,, IEEE Trans. Inf. Theory, 41 (1995), 400. doi: 10.1109/18.370141. Google Scholar [14] G. Gong, New designs for signal sets with low cross correlation, balance property and large linear span: GF(p) case,, IEEE Trans. Inf. Theory, 48 (2002), 2847. doi: 10.1109/TIT.2002.804044. Google Scholar [15] S. Hong, C. Seol and K. Cheun, Performance of soft decision decoded synchronous FHSS multiple access networks using MFSK modulation under rayleigh fading,, IEEE Trans. Commun., 59 (2011), 1066. Google Scholar [16] H. D. Jia, D. Yuan, D. Y. Peng, et al., On a general class of quadratic hopping sequences,, Sci. China Ser. F, 12 (2008), 2101. doi: 10.1007/s11432-008-0136-8. Google Scholar [17] N. R. Lanka, S. A. Patnaik and R. A. Harjani, Frequency-hopped quadrature frequency synthesizer in 0.13-$\mu$m technology,, IEEE J. Solid-State Circuits, 46 (2011), 1. Google Scholar [18] A. Lempel and H. Greenberger, Families of sequence with optimal Hamming correlation properties,, IEEE Trans. Inf. Theory, 20 (1974), 90. Google Scholar [19] W. P. Ma and S. H. Sun, New designs of frequency hopping sequences with low hit zone,, Des. Codes Crypt., 60 (2010), 145. doi: 10.1007/s10623-010-9422-8. Google Scholar [20] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal LHZ FHS with new parameters,, in, (2011), 10. Google Scholar [21] D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency hopping sequences,, IEEE Trans. Inf. Theory, 50 (2004), 2149. doi: 10.1109/TIT.2004.833362. Google Scholar [22] D. Y. Peng, P. Z. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone,, Sci. China Ser. F, 49 (2006), 1. doi: 10.1007/s11432-006-0208-6. Google Scholar [23] H. Shao and N. Beaulieu, Direct sequence and time-hopping sequence designs for narrow band interference mitigation in impulse radio UWB systems,, IEEE Trans. Commun., 59 (2011), 1957. Google Scholar [24] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, "Spread Spectrum Communications Handbook,'', McGraw-Hill, (1994). Google Scholar [25] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inf. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar [26] P. Udaya and X. Tang, Low correlation zone sequences from interleaved construction,, IEICE Trans. Fund., 93-A (2010), 2220. Google Scholar [27] X. N. Wang and P. Z. Fan, A class of frequency hopping sequences with no hit zone,, in, (2003), 896. Google Scholar [28] W. X. Ye and P. Z. Fan, Two classes of frequency hopping sequences with no-hit zone,, in, (2003), 304. Google Scholar [29] W. X. Ye and P. Z. Fan, Construction of frequency hopping sequences with no hit zone,, J. Electronics (China), 24 (2007), 305. doi: 10.1007/s11767-005-0202-y. Google Scholar [30] W. X. Ye, P. Z. Fan and E. M. Gabidulin, Construction of non-repeating frequency-hopping sequences with no-hit zone,, Electronics Letters, 42 (2006), 681. doi: 10.1049/el:20060775. Google Scholar [31] Q. Zeng, H. S. Li, Z. H. Zhang, et al., A frequency-hopping based communication infrastructure for wireless metering in smart grid,, in, (2011), 23. Google Scholar [32] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect sequences,, IEICE Trans. Fund., 91 (2008), 3691. Google Scholar [33] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique,, IEEE Trans. Inf. Theory, 54 (2008), 4267. doi: 10.1109/TIT.2008.928256. Google Scholar

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##### References:
 [1] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inf. Theory, 51 (2005), 1139. doi: 10.1109/TIT.2004.842708. Google Scholar [2] J. H. Chung, Y. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving technique,, IEEE Trans. Inf. Theory, 55 (2009), 5783. doi: 10.1109/TIT.2009.2032742. Google Scholar [3] J. H. Chung and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inf. Theory, 56 (2010), 1685. doi: 10.1109/TIT.2010.2040888. Google Scholar [4] C. Ding, R. Fuji-Hara, Y. Fujiwara, et al., Sets of frequency hopping sequences: Bounds and optimal constructions,, IEEE Trans. Inf. Theory, 55 (2009), 3297. doi: 10.1109/TIT.2009.2021366. Google Scholar [5] C. Ding, T. Helleseth and H. Martinsen, New families of binary sequences with optimal three-level autocorrelation,, IEEE Trans. Inf. Theory, 47 (2001), 428. doi: 10.1109/18.904555. Google Scholar [6] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency hopping sequences,, IEEE Trans. Inf. Theory, 53 (2007), 2606. doi: 10.1109/TIT.2007.899545. Google Scholar [7] C. Ding and J. Yin, Sets of optimal frequency-hopping sequences,, IEEE Trans. Inf. Theory, 54 (2008), 3741. doi: 10.1109/TIT.2008.926410. Google Scholar [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications,'', RSP-John Wiley Sons Inc., (1996). Google Scholar [9] P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency hopping CDMA systems,, IEEE Trans. Wir. Commun., 4 (2005), 2836. Google Scholar [10] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: A combinatorial approach,, IEEE Trans. Inf. Theory, 50 (2004), 2408. doi: 10.1109/TIT.2004.834783. Google Scholar [11] G. Ge, R. Fuji-Hara and Y. Miao, Further combinatorial constructions for optimal frequency hopping sequences,, J. Combin. Theory Ser. A, 113 (2006), 1699. doi: 10.1016/j.jcta.2006.03.019. Google Scholar [12] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: auto- and crosscorrelation properties,, IEEE Trans. Inf. Theory, 55 (2009), 867. doi: 10.1109/TIT.2008.2009856. Google Scholar [13] G. Gong, Theory and applications of q-ary interleaved sequences,, IEEE Trans. Inf. Theory, 41 (1995), 400. doi: 10.1109/18.370141. Google Scholar [14] G. Gong, New designs for signal sets with low cross correlation, balance property and large linear span: GF(p) case,, IEEE Trans. Inf. Theory, 48 (2002), 2847. doi: 10.1109/TIT.2002.804044. Google Scholar [15] S. Hong, C. Seol and K. Cheun, Performance of soft decision decoded synchronous FHSS multiple access networks using MFSK modulation under rayleigh fading,, IEEE Trans. Commun., 59 (2011), 1066. Google Scholar [16] H. D. Jia, D. Yuan, D. Y. Peng, et al., On a general class of quadratic hopping sequences,, Sci. China Ser. F, 12 (2008), 2101. doi: 10.1007/s11432-008-0136-8. Google Scholar [17] N. R. Lanka, S. A. Patnaik and R. A. Harjani, Frequency-hopped quadrature frequency synthesizer in 0.13-$\mu$m technology,, IEEE J. Solid-State Circuits, 46 (2011), 1. Google Scholar [18] A. Lempel and H. Greenberger, Families of sequence with optimal Hamming correlation properties,, IEEE Trans. Inf. Theory, 20 (1974), 90. Google Scholar [19] W. P. Ma and S. H. Sun, New designs of frequency hopping sequences with low hit zone,, Des. Codes Crypt., 60 (2010), 145. doi: 10.1007/s10623-010-9422-8. Google Scholar [20] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal LHZ FHS with new parameters,, in, (2011), 10. Google Scholar [21] D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross correlations of frequency hopping sequences,, IEEE Trans. Inf. Theory, 50 (2004), 2149. doi: 10.1109/TIT.2004.833362. Google Scholar [22] D. Y. Peng, P. Z. Fan and M. H. Lee, Lower bounds on the periodic Hamming correlations of frequency hopping sequences with low hit zone,, Sci. China Ser. F, 49 (2006), 1. doi: 10.1007/s11432-006-0208-6. Google Scholar [23] H. Shao and N. Beaulieu, Direct sequence and time-hopping sequence designs for narrow band interference mitigation in impulse radio UWB systems,, IEEE Trans. Commun., 59 (2011), 1957. Google Scholar [24] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, "Spread Spectrum Communications Handbook,'', McGraw-Hill, (1994). Google Scholar [25] P. Udaya and M. U. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings,, IEEE Trans. Inf. Theory, 44 (1998), 1492. doi: 10.1109/18.681324. Google Scholar [26] P. Udaya and X. Tang, Low correlation zone sequences from interleaved construction,, IEICE Trans. Fund., 93-A (2010), 2220. Google Scholar [27] X. N. Wang and P. Z. Fan, A class of frequency hopping sequences with no hit zone,, in, (2003), 896. Google Scholar [28] W. X. Ye and P. Z. Fan, Two classes of frequency hopping sequences with no-hit zone,, in, (2003), 304. Google Scholar [29] W. X. Ye and P. Z. Fan, Construction of frequency hopping sequences with no hit zone,, J. Electronics (China), 24 (2007), 305. doi: 10.1007/s11767-005-0202-y. Google Scholar [30] W. X. Ye, P. Z. Fan and E. M. Gabidulin, Construction of non-repeating frequency-hopping sequences with no-hit zone,, Electronics Letters, 42 (2006), 681. doi: 10.1049/el:20060775. Google Scholar [31] Q. Zeng, H. S. Li, Z. H. Zhang, et al., A frequency-hopping based communication infrastructure for wireless metering in smart grid,, in, (2011), 23. Google Scholar [32] Z. C. Zhou, Z. Pan and X. H. Tang, New families of optimal zero correlation zone sequences based on interleaved technique and perfect sequences,, IEICE Trans. Fund., 91 (2008), 3691. Google Scholar [33] Z. C. Zhou, X. H. Tang and G. Gong, A new class of sequences with zero or low correlation zone based on interleaving technique,, IEEE Trans. Inf. Theory, 54 (2008), 4267. doi: 10.1109/TIT.2008.928256. Google Scholar
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