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Two constructions of low-hit-zone frequency-hopping sequence sets

  • * Corresponding author: Can Xiang

    * Corresponding author: Can Xiang 
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  • In this paper, we present two constructions of low-hit-zone frequen-cy-hopping sequence (LHZ FHS) sets. The constructions in this paper generalize the previous constructions based on $ m $-sequences and $ d $-form functions with difference-balanced property, and generate several classes of optimal LHZ FHS sets and LHZ FHS sets with optimal periodic partial Hamming correlation (PPHC).

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


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  • Table 1.  The parameters of some existing LHZ FHS sets with optimal PPHC properties and our new ones

    parameters$ (L, M, r, Z_H, W;H_{pmz}) $ Constraints Reference
    $ \Big(k_1L_1, M_1N_1, q, Z_1-1, W, \left\lceil\frac{W}{L}\right\rceil\Big) $ $ gcd(Z_1+1, L)=1, k_1(Z_1+1)\equiv 1\; (mod\; L_1) $,
    $ k_1\equiv1\; (mod\; Z_1), M_1Z_1=L_1 $
    $ \Big(L_1L_2, p_2, p_1p_2, \min\{L_1, L_2\}-1, W; \left\lceil\frac{W}{T_1L_2}\right\rceil\Big) $ $ gcd(L_1, L_2)=1, p_1(\frac{L_1}{T_1}-1+\eta)(\min\{L_1, L_2\}p_2-1)= $
    $ L_1L_2(\min\{L_1, L_2\}-p_1), 0 < \eta\leq 1 $
    $ sl=q^n-1, 2\leq r_0 < \frac{(q^n-2)q}{q-1}, r_0\equiv t\; mod\; l $,
    $ \Big(r_0(q^n-1), s, q, l-1, \omega\frac{q^{n}-1}{q-1}; \omega\frac{q^{n-1}-1}{q-1}\Big) $ $ (l+1)r_0t^{-1}\equiv1\; (mod\; (q^n-1)), gcd(t, q^n-1)=1 $
    $ gcd((l+1)t^{-1}\; mod\; (q^n-1), \frac{q^n-1}{q-1})=1, 1\leq \omega\leq(q-1)r_0 $
    $ \left((q_1-1)(q_2^n-1), q_1, q_1q_2^{n-1}, q_2^n-2, W;\left\lceil\frac{W(q_2-1)}{(q_2^n-1)(q_1-1)}\right\rceil\right) $ $ q_1, q_2 $ are two different prime powers satisfying
    $ gcd(q_1-1, q_2^n-1)=1, q_1 >q_2^n $
    $ \Big(lcm(q-1, tv), q, qv, \min\{q-1, tv\}-1, W; \left\lceil\frac{W}{v(q-1)}\right\rceil\Big) $ $ gcd(q-1, v)=1, gcd(q-1, t) >1, tv < q-1 $,
    or $ tv >q-1 $ and $ t < \frac{gcd(q-1, t)(q^2-q-1)}{qv-v-2} $
    $ \Big(lcm(q-1, p(p^n-1)), qp^{n-1}, qp^n, \min\{q-1, p(p^n-1)\}-1, $ $ gcd(q-1, p^n-1)=1, q-1 >p(p^n-1) $, or $ q-1 < p(p^n-1) $ [28]
    $ W; \left\lceil\frac{W}{(p^n-1)(q-1)}\right\rceil\Big) $ and $ gcd(q-1, p) >\frac{p^n(p+1)(q-1)+q(q-2-p)+1}{q^2p^{n-1}-qp^{n-1}-1} $
    $ \left((p^m-1)p, p^m, p^m, p^m-2, W;\left\lceil\frac{W}{p^m-1}\right\rceil\right) $ $ p $ is a prime, $ m\geq 2 $ [14]
    $ \left(2\rho, 2, \rho, \rho-1, W;\left\lceil\frac{W}{\varrho}\right\rceil\right) $ $ \rho $ is an even integer, [14]
    $ \Big(\frac{q^n-1}{l}, q-1, q^k, \frac{q^n-1}{q-1}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\Big) $ $ l|(q-1), gcd(l, n)=1 $,
    $ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer
    $ l|(q-1), gcd(l, n)=1 $,
    $ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^k, \frac{T}{d'}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\right) $ $ T|(q^n-1), T\nmid l, gcd(T, l)=d', m|n $,
    $ 1\leq k \leq m $ and $ s (1\leq s\leq \frac{q-1}{l}) $ is an integer
    Theorem 3.1
    $ \left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^{n-1}, T-1, W; \left\lceil\frac{W(q-1)}{q^n-1}\right\rceil\right) $ $ l|(q-1), gcd(l, n)=1 $,
    $ T|(q^n-1), T\nmid l $ and $ gcd(T, l)=1 $
    Theorem 3.2
    $ \left(\frac{q^n-1}{q-1}, \frac{q^n-1}{T}, q^{n-1}, \frac{T}{d'}-1, W;1\right) $ $ T|(q^n-1), T\nmid (q-1) $
    and $ d'=gcd(T, q-1) $
    Theorem 3.3
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