# American Institute of Mathematical Sciences

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

## New constructions of optimal frequency hopping sequences with new parameters

 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

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.
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, (1976). [2] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inform. Theory, 51 (2005), 1139. [3] J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inform. Theory, 56 (2010), 1685. [4] C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two,, IEEE Trans. Inform. Theory, 44 (1998), 1699. [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. doi: 10.1109/TIT.2009.2021366. [6] C. Ding and T. Helleseth, New generalized cyclotomy and its applications,, Finite Fields Appl., 4 (1998), 140. [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. [8] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 53 (2007), 2606. 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. [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. [11] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach,, IEEE Trans. Inform. Theory, 50 (2004), 2408. 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. [13] Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences,, IEEE Trans. Inform. Theory, 55 (2009), 4279. [14] J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping,, IEEE J. Select. Areas Commun., 8 (1990), 819. [15] A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. 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. 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. [18] A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107. [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.

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##### References:
 [1] T. M. Apostol, "Introduction to Analytic Number Theory,'', Springer-Verlag, (1976). [2] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy,, IEEE Trans. Inform. Theory, 51 (2005), 1139. [3] J. H. Chung, and K. Yang, Optimal frequency-hopping sequences with new parameters,, IEEE Trans. Inform. Theory, 56 (2010), 1685. [4] C. Ding, Autocorrelation values of generalized cyclotomic sequences of order two,, IEEE Trans. Inform. Theory, 44 (1998), 1699. [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. doi: 10.1109/TIT.2009.2021366. [6] C. Ding and T. Helleseth, New generalized cyclotomy and its applications,, Finite Fields Appl., 4 (1998), 140. [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. [8] C. Ding, M. J. Moisio and J. Yuan, Algebraic constructions of optimal frequency-hopping sequences,, IEEE Trans. Inform. Theory, 53 (2007), 2606. 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. [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. [11] R. Fuji-Hara, Y. Miao and M. Mishima, Optimal frequency hopping sequences: a combinatorial approach,, IEEE Trans. Inform. Theory, 50 (2004), 2408. 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. [13] Y. K. Han and K. Yang, On the Sidelnikov sequences as frequency-hopping sequences,, IEEE Trans. Inform. Theory, 55 (2009), 4279. [14] J. J. Komo and S. C. Liu, Maximal length sequences for frequency hopping,, IEEE J. Select. Areas Commun., 8 (1990), 819. [15] A. Lempel and H. Greenberger, Families of sequences with optimal hamming correlation properties,, IEEE Trans. Inform. Theory, 20 (1974), 90. 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. 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. [18] A. L. Whiteman, A family of difference sets,, Illinois J. Math., 6 (1962), 107. [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.
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