A lower bound on the average Hamming correlation of frequency-hopping sequence sets

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February 2015, 9(1): 55-62. doi: 10.3934/amc.2015.9.55

A lower bound on the average Hamming correlation of frequency-hopping sequence sets

Aixian Zhang ^1, , Zhengchun Zhou ^2, and Keqin Feng ^3,

1.	Department of Mathematical Sciences, Xi'an University of Technology, Xi'an, Shanxi 710048, China
2.	School of Mathematics, Southwest Jiaotong University, Chengdu, 610031
3.	Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

Received May 2014 Revised August 2014 Published February 2015

Abstract
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The average Hamming correlation is an important indicator of frequency-hopping sequences (FHSs) which measures the average performance of FHSs employed in practical frequency-hopping multiple access (FHMA) communication systems. In this paper, a lower bound on average Hamming auto- and cross correlations of an FHS set is derived. It generalizes and improves the lower bound proposed recently by Peng, Niu and Tang. A simple necessary and sufficient condition for an FHS set to meet the new bound is given. Based on this condition, two classes of FHS sets whose average Hamming correlations reach the proposed bound are introduced.

Keywords: average correlation., Hamming correlation, Frequency-hopping sequences.

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

Citation: Aixian Zhang, Zhengchun Zhou, Keqin Feng. A lower bound on the average Hamming correlation of frequency-hopping sequence sets. Advances in Mathematics of Communications, 2015, 9 (1) : 55-62. doi: 10.3934/amc.2015.9.55

References:

[1]	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.
[2]	J. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum hamming correlations, IET Commun., 6 (2013), 2048-2053.
[3]	C. Ding, R. Fuji-Hara, Y. Fujiwara, M. 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.
[4]	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.
[5]	Y. K. 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.
[6]	F. Liu, D. Peng, Z. Zhou and X. Tang, A new frequency-hopping sequence set based upon generalized cyclotomy, Des. Codes Crypt., 69 (2013), 247-259. doi: 10.1007/s10623-012-9652-z.
[7]	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), 2149-2154. doi: 10.1109/TIT.2004.833362.
[8]	D. Peng, X. Niu and X. Tang, Average Hamming correlation for the cubic polynomial hopping sequences, IET Commun., 4 (2010), 1775-1786. doi: 10.1049/iet-com.2009.0783.
[9]	D. V. Sarwate, Reed-Solomon codes and the design of sequences for spread-spectrum multiple-access communications, in Reed-Solomon Codes and Their Applications (eds. S.B. Wicker and V.K. Bharagava), IEEE Press, Piscataway, 1994.
[10]	M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, 2002.
[11]	X. Zeng, H. Cai, X. 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.

show all references

References:

[1]	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.
[2]	J. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum hamming correlations, IET Commun., 6 (2013), 2048-2053.
[3]	C. Ding, R. Fuji-Hara, Y. Fujiwara, M. 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.
[4]	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.
[5]	Y. K. 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.
[6]	F. Liu, D. Peng, Z. Zhou and X. Tang, A new frequency-hopping sequence set based upon generalized cyclotomy, Des. Codes Crypt., 69 (2013), 247-259. doi: 10.1007/s10623-012-9652-z.
[7]	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), 2149-2154. doi: 10.1109/TIT.2004.833362.
[8]	D. Peng, X. Niu and X. Tang, Average Hamming correlation for the cubic polynomial hopping sequences, IET Commun., 4 (2010), 1775-1786. doi: 10.1049/iet-com.2009.0783.
[9]	D. V. Sarwate, Reed-Solomon codes and the design of sequences for spread-spectrum multiple-access communications, in Reed-Solomon Codes and Their Applications (eds. S.B. Wicker and V.K. Bharagava), IEEE Press, Piscataway, 1994.
[10]	M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook, McGraw-Hill, 2002.
[11]	X. Zeng, H. Cai, X. 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.

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