New constructions of optimal frequency hopping sequences with new parameters

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February 2013, 7(1): 91-101. doi: 10.3934/amc.2013.7.91

New constructions of optimal frequency hopping sequences with new parameters

Fang Liu ^1, , Daiyuan Peng ^2, , Zhengchun Zhou ^3, and Xiaohu Tang ^2,

1.	The Thirtieth Research Institute, China Electronic Technology Group Corporation, Chengdu, China
2.	School of Mobile Communications, Southwest Jiaotong University, Chengdu, 610031, China, China
3.	School of Mathematics, Southwest Jiaotong University, Chengdu, 610031

Received June 2012 Published January 2013

Abstract
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In this paper, three constructions of frequency hopping sequences (FHSs) are proposed using a new generalized cyclotomy with respect to $\textbf{Z}_{p^n}$, where $p$ is an odd prime and $n$ is a positive integer. Based on some basic properties of the new generalized cyclotomy, it is shown that all the constructed FHSs are optimal with respect to the well-known Lempel-Greenberger bound. Furthermore, these FHSs have new parameters which are not reported in the literature.

Keywords: Lempel-Greenberger bound., Generalized cyclotomy, Hamming autocorrelation, frequency-hopping sequences.

Mathematics Subject Classification: Primary: 94A05, 94B60; Secondary: 05B1.

Citation: 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

References:

[1]	T. M. Apostol, "Introduction to Analytic Number Theory,'' Springer-Verlag, New York, 1976.
[2]	W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inform. Theory, 51 (2005), 1139-1141.
[3]	J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inform. Theory, 56 (2010), 1685-1693.
[4]	C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inform. Theory, 44 (1998), 1699-1702.
[5]	C. Ding, R. Fuji-Hara, and Y. Fujiwara, Sets of frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366.
[6]	C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.
[7]	C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_1^{e_1}\cdots p_t^{e_t}$, IEEE Trans. Inform. Theory, 45 (1999), 467-474.
[8]	C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545.
[9]	C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.
[10]	P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency-hopping CDMA systems, IEEE Trans. Inform. Theory, 4 (2005), 2836-2842.
[11]	R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach, IEEE Trans. Inform. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783.
[12]	G. Ge, Y. Miao and Z. H. Yao, Optimal frequency hopping sequences: auto- and cross-correlation properties, IEEE Trans. Inform. Theory, 55 (2008), 867-879.
[13]	Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences, IEEE Trans. Inform. Theory, 55 (2009), 4279-4285.
[14]	J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping, IEEE J. Select. Areas Commun., 8 (1990), 819-822.
[15]	A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94. doi: 10.1109/TIT.1974.1055169.
[16]	D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross-correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362.
[17]	P. Udaya and M. N. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inform. Theory, IT-44 (1998), 1492-1503.
[18]	A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
[19]	M. Z. Win and R. A. Scholtz, Ultra-Wide Bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications, IEEE Trans. Commun., 58 (2002), 679-691.

show all references

References:

[1]	T. M. Apostol, "Introduction to Analytic Number Theory,'' Springer-Verlag, New York, 1976.
[2]	W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inform. Theory, 51 (2005), 1139-1141.
[3]	J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters, IEEE Trans. Inform. Theory, 56 (2010), 1685-1693.
[4]	C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two, IEEE Trans. Inform. Theory, 44 (1998), 1699-1702.
[5]	C. Ding, R. Fuji-Hara, and Y. Fujiwara, Sets of frequency hopping sequences: bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304. doi: 10.1109/TIT.2009.2021366.
[6]	C. Ding and T. Helleseth, New generalized cyclotomy and its applications, Finite Fields Appl., 4 (1998), 140-166.
[7]	C. Ding and T. Helleseth, Generalized cyclotomic codes of length $p_1^{e_1}\cdots p_t^{e_t}$, IEEE Trans. Inform. Theory, 45 (1999), 467-474.
[8]	C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 53 (2007), 2606-2610. doi: 10.1109/TIT.2007.899545.
[9]	C. Ding and J. Yin, Sets of optimal frequency-hopping sequences, IEEE Trans. Inform. Theory, 54 (2008), 3741-3745.
[10]	P. Z. Fan, M. H. Lee and D. Y. Peng, New family of hopping sequences for time/frequency-hopping CDMA systems, IEEE Trans. Inform. Theory, 4 (2005), 2836-2842.
[11]	R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach, IEEE Trans. Inform. Theory, 50 (2004), 2408-2420. doi: 10.1109/TIT.2004.834783.
[12]	G. Ge, Y. Miao and Z. H. Yao, Optimal frequency hopping sequences: auto- and cross-correlation properties, IEEE Trans. Inform. Theory, 55 (2008), 867-879.
[13]	Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences, IEEE Trans. Inform. Theory, 55 (2009), 4279-4285.
[14]	J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping, IEEE J. Select. Areas Commun., 8 (1990), 819-822.
[15]	A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties, IEEE Trans. Inform. Theory, 20 (1974), 90-94. doi: 10.1109/TIT.1974.1055169.
[16]	D. Y. Peng and P. Z. Fan, Lower bounds on the Hamming auto- and cross-correlations of frequency-hopping sequences, IEEE Trans. Inform. Theory, 50 (2004), 2149-2154. doi: 10.1109/TIT.2004.833362.
[17]	P. Udaya and M. N. Siddiqi, Optimal large linear complexity frequency hopping patterns derived from polynomial residue class rings, IEEE Trans. Inform. Theory, IT-44 (1998), 1492-1503.
[18]	A. L. Whiteman, A family of difference sets, Illinois J. Math., 6 (1962), 107-121.
[19]	M. Z. Win and R. A. Scholtz, Ultra-Wide Bandwidth time-hopping spread-spectrum impulse radio for wireless multiple-access communications, IEEE Trans. Commun., 58 (2002), 679-691.

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