# American Institute of Mathematical Sciences

February  2018, 12(1): 67-79. doi: 10.3934/amc.2018004

## Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation

 1 Provincial Key Laboratory of Information Coding and Transmission, Institute of Mobile Communications, Southwest Jiaotong University, Chengdu, Sichuan 610031, China 2 School of Transportation And Logistics, Southwest Jiaotong University, Chengdu, Sichuan 610031, China

* Corresponding author: Hongbin Liang

Received  March 2016 Revised  January 2017 Published  March 2018

In practice, when a frequency-hopping sequence (FHS) set is applied in a frequency-hopping multiple-access (FHMA) system, its periodic partial Hamming correlation (PPHC) rather than its periodic Hamming correlation (PHC) within the whole period is used to evaluate the system performance. Moreover, FHS sets with low hit zone (LHZ) can be well applied in quasi-synchronous (QS) FHMA systems in which some relative time delay among different users within a zone around the origin can be allowed. Therefore, it is very urgent to conduct research on LHZ FHS sets with optimal PPHC property in depth. In this paper, we first derive a new tighter lower bound on the maximum PPHC of an LHZ FHS set. Then we present a new class of optimal one-coincidence FHS sets. Finally we have a construction of LHZ FHS sets which can be optimal with respect to our new lower bound.

Citation: Limengnan Zhou, Daiyuan Peng, Hongyu Han, Hongbin Liang, Zheng Ma. Construction of optimal low-hit-zone frequency hopping sequence sets under periodic partial Hamming correlation. Advances in Mathematics of Communications, 2018, 12 (1) : 67-79. doi: 10.3934/amc.2018004
##### References:
 [1] D. Aifindel, Specification of the Bluetooth Systems-Core. The Bluetooth Special Interest Group (SIG), available online at http://www.bluetooth.com [2] H. Cai, Y. Yang, Z. C. Zhou and X. H. Tang, Strictly optimal frequency-hopping sequence sets with optimal family size, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093. [3] 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. Inf. Theory, 60 (2014), 5782-5790. [4] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. [5] J. H. Chung and K. Yang, Low-hit-zone frequency-hopping sequence sets with new parameters, in SETA 2012, 2012,202-211. [6] J. H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by Cartesian product, IEEE Trans. Inf. Theory, 59 (2013), 726-732. [7] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, Wiley, London, 1996. [8] S. W. Golomb and G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography and Radar, Cambridge Univ. Press, Cambridge, 2005. [9] H. Y. Han, D. Y. Peng and X. Liu, On low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation, in SETA 2014, 2014,293-304. [10] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [11] X. Liu, D. Y. Peng and H. Y. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Crypt., 73 (2013), 167-176. [12] W. P. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, IEICE Trans. Fund., 60 (2011), 145-153. [13] 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 Trans. Fund., E93-A (2010), 2227-2231. [14] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fund. Electr. Commun. Comp. Sci., 95 (2012), 1835-1842. [15] 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. [16] 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, 49 (2006), 208-218. [17] 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. [18] Z. C. Zhou, X. H. Tang and P. Udaya, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458.

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##### References:
 [1] D. Aifindel, Specification of the Bluetooth Systems-Core. The Bluetooth Special Interest Group (SIG), available online at http://www.bluetooth.com [2] H. Cai, Y. Yang, Z. C. Zhou and X. H. Tang, Strictly optimal frequency-hopping sequence sets with optimal family size, IEEE Trans. Inf. Theory, 62 (2016), 1087-1093. [3] 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. Inf. Theory, 60 (2014), 5782-5790. [4] W. Chu and C. J. Colbourn, Optimal frequency-hopping sequences via cyclotomy, IEEE Trans. Inf. Theory, 51 (2005), 1139-1141. [5] J. H. Chung and K. Yang, Low-hit-zone frequency-hopping sequence sets with new parameters, in SETA 2012, 2012,202-211. [6] J. H. Chung and K. Yang, New classes of optimal low-hit-zone frequency-hopping sequence sets by Cartesian product, IEEE Trans. Inf. Theory, 59 (2013), 726-732. [7] P. Z. Fan and M. Darnell, Sequence Design for Communications Applications, Research Studies Press, Wiley, London, 1996. [8] S. W. Golomb and G. Gong, Signal Design for Good Correlation: for Wireless Communication, Cryptography and Radar, Cambridge Univ. Press, Cambridge, 2005. [9] H. Y. Han, D. Y. Peng and X. Liu, On low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation, in SETA 2014, 2014,293-304. [10] A. Lempel and H. Greenberger, Families of sequences with optimal Hamming correlation properties, IEEE Trans. Inf. Theory, 20 (1974), 90-94. [11] X. Liu, D. Y. Peng and H. Y. Han, Low-hit-zone frequency hopping sequence sets with optimal partial Hamming correlation properties, Des. Codes Crypt., 73 (2013), 167-176. [12] W. P. Ma and S. Sun, New designs of frequency hopping sequences with low hit zone, IEICE Trans. Fund., 60 (2011), 145-153. [13] 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 Trans. Fund., E93-A (2010), 2227-2231. [14] X. H. Niu, D. Y. Peng and Z. C. Zhou, New classes of optimal low hit zone frequency hopping sequences with new parameters by interleaving technique, IEICE Trans. Fund. Electr. Commun. Comp. Sci., 95 (2012), 1835-1842. [15] 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. [16] 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, 49 (2006), 208-218. [17] 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. [18] Z. C. Zhou, X. H. Tang and P. Udaya, New classes of frequency-hopping sequences with optimal partial correlation, IEEE Trans. Inf. Theory, 58 (2012), 453-458.
Comparison of parameters of some LHZ FHS sets with optimal PPHC property
 Parameters $(L, N, r, W, L_{pz}, H_{pzm}(S;W))$ Constrains Ref. $(j_1L_1, k_1N_1, r_1, W_2, z_1-1, \Big\lceil\frac{W_2}{T_1}\Big\rceil)$ $k_1z_1=L_1$, $\gcd(z_1+1,T_1)=1$, $j_1(z_1+1)\equiv 1 (\mod L_1)$, $j_1=\lambda z_1+1$, $\lambda\geq 1$ [11] $(lL_2, N_2, r_2, W_3, L_2-1, \gamma)$ $l>0$ [9] $(L_3L_4, p_4, p_3p_4, W_6, \min\{L_3,L_4\}-1, \Big\lceil\frac{W_6}{T_3L_4}\Big\rceil)$ $\gcd(L_3,L_4)=1$, $p_3(\frac{L_3}{T_3}-1+\eta)(\min\{L_3,L_4\}p_4-1)=L_3L_4(\min\{L_3,L_4\}-p_3)$, $0<\eta\leq1$ [17] $(pq(q^m-1), pq^{m-1}, pq^m, W,$ $\min\{p,q(q^m-1)\}-1,$ $\Big\lceil\frac{W}{p(q^m-1)}\Big\rceil)$ $pq^{m+1}+p^2-pq-q-p^2q^{m-1}+1<0$ if $p=\min\{p,q(q^m-1)\}$ This paper
 Parameters $(L, N, r, W, L_{pz}, H_{pzm}(S;W))$ Constrains Ref. $(j_1L_1, k_1N_1, r_1, W_2, z_1-1, \Big\lceil\frac{W_2}{T_1}\Big\rceil)$ $k_1z_1=L_1$, $\gcd(z_1+1,T_1)=1$, $j_1(z_1+1)\equiv 1 (\mod L_1)$, $j_1=\lambda z_1+1$, $\lambda\geq 1$ [11] $(lL_2, N_2, r_2, W_3, L_2-1, \gamma)$ $l>0$ [9] $(L_3L_4, p_4, p_3p_4, W_6, \min\{L_3,L_4\}-1, \Big\lceil\frac{W_6}{T_3L_4}\Big\rceil)$ $\gcd(L_3,L_4)=1$, $p_3(\frac{L_3}{T_3}-1+\eta)(\min\{L_3,L_4\}p_4-1)=L_3L_4(\min\{L_3,L_4\}-p_3)$, $0<\eta\leq1$ [17] $(pq(q^m-1), pq^{m-1}, pq^m, W,$ $\min\{p,q(q^m-1)\}-1,$ $\Big\lceil\frac{W}{p(q^m-1)}\Big\rceil)$ $pq^{m+1}+p^2-pq-q-p^2q^{m-1}+1<0$ if $p=\min\{p,q(q^m-1)\}$ This paper

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