# 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
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##### References:
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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