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

  • * Corresponding author: Hongyu Han

    * Corresponding author: Hongyu Han 

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

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  • 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.

    Mathematics Subject Classification: Primary: 94A05; Secondary: 94B60.


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  • Figure 1.  Maximum partial Hamming correlations of $ \mathcal{P} $ for the correlation window length $ L = 8 $ in Example 1

    Table 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 $T=\frac{q^n-1}{q-1}$.
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