# American Institute of Mathematical Sciences

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

 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

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.
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.  doi: 10.1109/TIT.2004.842708.  Google Scholar [2] J. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum hamming correlations,, IET Commun., 6 (2013), 2048.   Google Scholar [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.  doi: 10.1109/TIT.2009.2021366.  Google Scholar [4] 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 [5] Y. K. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences,, IEEE Trans. Inf. Theory, 55 (2009), 4279.  doi: 10.1109/TIT.2009.2025569.  Google Scholar [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.  doi: 10.1007/s10623-012-9652-z.  Google Scholar [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.  doi: 10.1109/TIT.2004.833362.  Google Scholar [8] D. Peng, X. Niu and X. Tang, Average Hamming correlation for the cubic polynomial hopping sequences,, IET Commun., 4 (2010), 1775.  doi: 10.1049/iet-com.2009.0783.  Google Scholar [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), (1994).   Google Scholar [10] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook,, McGraw-Hill, (2002).   Google Scholar [11] X. Zeng, H. Cai, X. Tang and Y. Yang, Optimal frequency hopping sequences of odd length,, IEEE Trans. Inf. Theory, 59 (2013), 3237.  doi: 10.1109/TIT.2013.2237754.  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. Chung and K. Yang, New frequency-hopping sequence sets with optimal average and good maximum hamming correlations,, IET Commun., 6 (2013), 2048.   Google Scholar [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.  doi: 10.1109/TIT.2009.2021366.  Google Scholar [4] 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 [5] Y. K. Han and K. Yang, On the Sidel'nikov sequences as frequency-hopping sequences,, IEEE Trans. Inf. Theory, 55 (2009), 4279.  doi: 10.1109/TIT.2009.2025569.  Google Scholar [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.  doi: 10.1007/s10623-012-9652-z.  Google Scholar [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.  doi: 10.1109/TIT.2004.833362.  Google Scholar [8] D. Peng, X. Niu and X. Tang, Average Hamming correlation for the cubic polynomial hopping sequences,, IET Commun., 4 (2010), 1775.  doi: 10.1049/iet-com.2009.0783.  Google Scholar [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), (1994).   Google Scholar [10] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, Spread Spectrum Communications Handbook,, McGraw-Hill, (2002).   Google Scholar [11] X. Zeng, H. Cai, X. Tang and Y. Yang, Optimal frequency hopping sequences of odd length,, IEEE Trans. Inf. Theory, 59 (2013), 3237.  doi: 10.1109/TIT.2013.2237754.  Google Scholar
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