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

  • * Corresponding author: Hongbin Liang

    * Corresponding author: Hongbin Liang 
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  • 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.

    Mathematics Subject Classification: Primary: 94A55; Secondary: 94B05.


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  • Table 1.  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
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