# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020031

## New classes of strictly optimal low hit zone frequency hopping sequence sets

 1 School of Computer Science, Sichuan Normal University, Chengdu, Sichuan 610066, China 2 School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

* Corresponding author: Hongyu Han

Received  February 2018 Revised  March 2019 Published  November 2019

Fund Project: This work is supported in part by the National Science Foundation of China under Grants 61701331, 61801401, and in part by the Project of Sichuan Education Department under Grant 18ZB0496

Low hit zone frequency hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable in quasi-synchronous frequency hopping multiple-access systems. An LHZ FHS set is considered to be strictly optimal when it has optimal partial Hamming correlation for all correlation windows. In this study, an interleaved construction of new sets of strictly optimal LHZ FHSs is proposed. Strictly optimal LHZ FHS sets with new and flexible parameters are obtained by selecting suitable known optimal FHSs and appropriate shift sequences.

Citation: Hongyu Han, Sheng Zhang. New classes of strictly optimal low hit zone frequency hopping sequence sets. Advances in Mathematics of Communications, doi: 10.3934/amc.2020031
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##### References:
 [1] "Specification of the Bluetooth SysemsCore," Bluetooth Special Interest Group (SIG), 2003. Available from: http://www.bluetooth.org. Google Scholar [2] H. Cai, Z. C. Zhou, Y. Yang and X. H. Tang, A new construction of frequency-hopping sequences with optimal partial Hamming correlation, IEEE Trans. Inform. Theory, 60 (2014), 5782-5790.  doi: 10.1109/TIT.2014.2332996.  Google Scholar [3] H. H. Chen, "The Next Generation CDMA Technologies", John Wiley & Sons, London, 2007. doi: 10.1002/9780470022962.  Google Scholar [4] J. H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by Cartesian product, IEEE Trans. Inform. Theory, 59 (2013), 726-732.  doi: 10.1109/TIT.2012.2213065.  Google Scholar [5] J. H. Chung, Y. K. Han and K. Yang, New classes of optimal frequency-hopping sequences by interleaving techniques, IEEE Trans. Inform. Theory, 55 (2009), 5783-5791.  doi: 10.1109/TIT.2009.2032742.  Google Scholar [6] C. Ding, R. Fuji-Hara, Y. Fujiwara, M. Jimbo and M. Mishima, Sets of frequency hopping sequences: Bounds and optimal constructions, IEEE Trans. Inform. Theory, 55 (2009), 3297-3304.  doi: 10.1109/TIT.2009.2021366.  Google Scholar [7] Y. C. Eun, S. Y. Jin, Y. P. Hong and H. Y. Song, Frequency hopping sequences with optimal partial autocorrelation properties, IEEE Trans. Inform. Theory, 50 (2004), 2438-2442.  doi: 10.1109/TIT.2004.834792.  Google Scholar [8] P. Z. Fan and M. Darnell, "Sequence Design for Communications Applications," John Wiley & Sons, London, 1996. Google Scholar [9] R. D. Gaudenzi, C. Elia and R. Viola, Bandlimited quasi-synchronous CDMA: A novel satellite access technique for mobile and personal communication systems, IEEE J. Sel. Areas Commun., 10 (1992), 328-343.  doi: 10.1109/49.126984.  Google Scholar [10] G. Ge, Y. Miao and Z. Yao, Optimal frequency hopping sequences: Auto- and cross-correlation properties, IEEE Trans. Inform. Theory, 55 (2009), 867-879.  doi: 10.1109/TIT.2008.2009856.  Google Scholar [11] G. Gong, Theory and applications of $q$-ary interleaved sequences, IEEE Trans. Inform. Theory, 41 (1995), 400-411.  doi: 10.1109/18.370141.  Google Scholar [12] H. Y. Han, D. Y. Peng and X. Liu, On low-hit-zone frequency-hopping sequence sets with optimal partial Hamming correlation, in Sequences and Their Applications - SETA 2014, Lecture Notes in Comput. Sci., 8665, Springer, Cham, 2014,293-304. doi: 10.1007/978-3-319-12325-7_25.  Google Scholar [13] H. Y. Han, D. Y. Peng and U. Parampalli, New sets of optimal low-hit-zone frequency-hopping sequences based on $m$-sequences, Cryptogr. Commun., 9 (2017), 511-522.  doi: 10.1007/s12095-016-0192-7.  Google Scholar [14] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Information Theory, 20 (1974), 90-94.  doi: 10.1109/tit.1974.1055169.  Google Scholar [15] W. P. Ma and S. H. Sun, New designs of frequency hopping sequences with low hit zone, Des. Codes Cryptogr., 60 (2011), 145-153.  doi: 10.1007/s10623-010-9422-8.  Google Scholar [16] X. H. Niu, D. Y. Peng, F. Liu and X. Liu, Lower bounds on the maximum partial correlations of frequency hopping sequence set with low hit zone, IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 93-A (2010), 2227-2231.  doi: 10.1587/transfun.E93.A.2227.  Google Scholar [17] X. H. Niu, D. Y. Peng and Z. C. Zhou, Frequency/time hopping sequence sets with optimal partial Hamming correlation properties, Sci. China Inf. Sci., 55 (2012), 2207-2215.  doi: 10.1007/s11432-012-4620-9.  Google Scholar [18] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal frequency hopping sequences with low hit zone, Adv. Math. Commun., 7 (2013), 293-310.  doi: 10.3934/amc.2013.7.293.  Google Scholar [19] 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.  Google Scholar [20] 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), 208-218.  doi: 10.1007/s11432-006-0208-6.  Google Scholar [21] M. K. Simon, J. K. Omura, R. A. Scholtz and B. K. Levitt, "Spread Spectrum Communications Handbook," McGraw-Hill, New York, NY, 2001. Google Scholar [22] C. Y. Wang, D. Y. Peng, H. Y. Han and L. M. N. Zhou, New sets of low-hit-zone frequency-hopping sequence with optimal maximum periodic partial Hamming correlation, Sci. China Inf. Sci., 58 (2015), 1-15.  doi: 10.1109/TIT.2016.2551225.  Google Scholar [23] X. N. Wang and P. Z. Fan, A class of frequency hopping sequences with no hit zone, in Proc. of the 4th International Conference on Parallel and Distributed Computing, Applications and Technologies, 2003,896-898. doi: 10.1109/PDCAT.2003.1236444.  Google Scholar [24] S. Zhang, J. Zhang, W. Zheng and H. So, Widely-linear complex-valued estimated-input LMS algorithm for bias-compensated adaptive filtering with noisy measurements, IEEE Trans. Signal Process., 67 (2019), 3592-3605.  doi: 10.1109/TSP.2019.2919412.  Google Scholar [25] X. Y. Zeng, H. Cai, X. H. Tang and Y. Yang, A class of optimal frequency hopping sequences with new parameters, IEEE Trans. Inform. Theory, 58 (2012), 4899-4907.  doi: 10.1109/TIT.2012.2195771.  Google Scholar [26] 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. Inform. Theory, 54 (2008), 4267-4273.  doi: 10.1109/TIT.2008.928256.  Google Scholar [27] Z. C. Zhou, X. H. Tang, X. H. Niu and P. Udaya, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inform. Theory, 58 (2012), 453-458.  doi: 10.1109/TIT.2011.2167126.  Google Scholar
Maximum partial Hamming correlations of $\mathcal{P}$ for the correlation window length $L = 8$ in Example 1
New sets of strictly optimal LHZ FHSs
 Based on individual FHSs with strictly optimal partial Hamming autocorrelation $L_c$ $(RN, I, l, v-1, R\alpha)$ Maximum partial Hamming correlation for the correlation window length $L$ Constraints [2] $g$ $(Reg, I, g, v\!-\!1, Re)$ $\left\lceil\frac{L}{g}\right\rceil$ $R\geq2$, $Iv=N$, $\gcd(s, N)=1$, $\gcd((v\!+\!1)s^{-1}(\textrm{mod}\ N), L_c)\!=\!1$, $(v+1)Rs^{-1}\equiv1$(mod $N$), $R=s$(mod $v$) [7] $q\!+\!1$ $(R(q^2\!-\!1), I, q, v\!-\!1, R(q-1))$ $\left\lceil\frac{L}{q+1}\right\rceil$ [27] $T$ $(R(q^n-1), I, q^{n-1}, v-1, R(q-1))$ $\left\lceil\frac{L}{T}\right\rceil$ $g$ is any odd integer with the prime factor decomposition $g=p_1^{m_1}p_2^{m_2}\cdots p_k^{m_k}$; $e>1$, $e|gcd(p_1-1, p_2-1, \cdots, p_k-1)$; $q$ is a prime power and $T=\frac{q^n-1}{q-1}$.
 Based on individual FHSs with strictly optimal partial Hamming autocorrelation $L_c$ $(RN, I, l, v-1, R\alpha)$ Maximum partial Hamming correlation for the correlation window length $L$ Constraints [2] $g$ $(Reg, I, g, v\!-\!1, Re)$ $\left\lceil\frac{L}{g}\right\rceil$ $R\geq2$, $Iv=N$, $\gcd(s, N)=1$, $\gcd((v\!+\!1)s^{-1}(\textrm{mod}\ N), L_c)\!=\!1$, $(v+1)Rs^{-1}\equiv1$(mod $N$), $R=s$(mod $v$) [7] $q\!+\!1$ $(R(q^2\!-\!1), I, q, v\!-\!1, R(q-1))$ $\left\lceil\frac{L}{q+1}\right\rceil$ [27] $T$ $(R(q^n-1), I, q^{n-1}, v-1, R(q-1))$ $\left\lceil\frac{L}{T}\right\rceil$ $g$ is any odd integer with the prime factor decomposition $g=p_1^{m_1}p_2^{m_2}\cdots p_k^{m_k}$; $e>1$, $e|gcd(p_1-1, p_2-1, \cdots, p_k-1)$; $q$ is a prime power and $T=\frac{q^n-1}{q-1}$.
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