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Constructions of optimal low hit zone frequency hopping sequence sets with large family size

  • * Corresponding author: Xianhua Niu

    * Corresponding author: Xianhua Niu 
Abstract Full Text(HTML) Figure(2) / Table(1) Related Papers Cited by
  • Frequency hopping sequences with low hit zone is significant for application in quasi synchronous multiple-access systems. In this paper, we obtained two constructions of optimal frequency hopping sequence sets with low hit zone based on interleaving techniques. The presented low hit zone frequency hopping sequence sets are with new and flexible parameters and large family size which can meet the needs of the practical applications. Moreover, all the sequences in the proposed sets are cyclically inequivalent. Some low hit zone frequency hopping sequence sets constructed in literatures are included in our family. The proposed frequency hopping sequence sets with low hit zone are contributed for quasi-synchronous frequency hopping multiple access system to reduce or eliminate multiple-access interference.

    Mathematics Subject Classification: Primary: 94A55, 94A05; Secondary: 94B60.

    Citation:

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  • Figure 1.  The Maximum Periodic Hamming Correlation of $ S $

    Figure 2.  The Maximum Periodic Hamming Correlation of $ S $

    Table 1.  SOME OPTIMAL LHZ FHS SETS WITH OPTIMAL HAMMING CORRELATION

    Parameters
    $(N, q, l, L_H, H_m)$
    Constraints According to the bound(3) According to the bound(4) Cyclical equivalence Ref.
    $(TN, q, Ml, \omega-1, TH_m)$ $M=\lceil \frac{N}{\omega}\rceil$, $gcd(l, N)=1$,
    $T=\lambda\omega+1$, $\lambda\ge1, T < lN.$
    Optimal Not optimal family size Inequivalent [9]
    $(TN, q, M, \omega-1, TH_a)$ $M\omega=N, T\ge2$,
    $gcd(s, N)=1.$
    Optimal Not optimal family size Inequivalent [10]
    $(lN, q, M, \omega l-1, lH_m)$ $M=\lceil \frac{N}{\omega}\rceil, gcd(l, N)=1.$ Optimal Not optimal family size Inequivalent [12]
    $(lN, q$, $M\left[\omega-(x-1)(l-1)\right]$, $l-1$, $lH_m)$ $x\ne1, 0<x<\frac{\omega}{l}-1$,
    $M=\lceil \frac{N}{\omega}\rceil, \omega>2l$,
    $xl-x<r<\omega.$
    Optimal Not optimal family size Inequivalent [6]
    $(lN, q, nM(\omega-1), l-n, {lH}_m)$ $0<n\leq\lceil\frac{l}{2}\rceil, l>2$,
    $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$.
    Optimal Not optimal family size Inequivalent Thm.1
    $(lN, q, Ml+nM(\omega-{xl}+x-1), l-n, {lH}_m)$ $0<n\leq\lceil\frac{l}{2}\rceil, n\ne1$,
    $l>2, \omega>2l, M=\lceil\frac{N}{\omega}\rceil$,
    $0<x<\frac{\omega-1}{l-1}, x\ne1$.
    Optimal Not optimal family size Inequivalent Thm.3
    $(lN, q, M\omega, l-2, lH_m)$ $x=1, n=1, l>2$,
    $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$.
    Optimal Not optimal family size Inequivalent Thm.3
    $(s(q^m-1), q, l, L_H, s(q^{m-1}-1))$ $m\ge1, q^m-1=(L_H+1)l$,
    $gcd(s, q^m-1)=1, s\leq l$.
    Optimal Not optimal family size Equivalent [8]
    $(p^2(q-1), pq, pq, min\left\{p^2-1, q-2\right\}, p)$ $gcd(p, q-1)=1, 2p\leq q-1.$ Optimal Not optimal family size Inequivalent [1]
    $(p^2(p^2-1), p^2, p, p^2-2, p(p-1))$ $gcd(p^2, p^2-1)=1.$ Optimal Not optimal family size Inequivalent [21]
    $\big(q^k(q^m-1), q^k, nM(\omega-1), q^k-n, q^m\big)$ $1\leq k\leq m$, $0<n\leq\lceil\frac{q^k}{2}\rceil$,
    $\omega>2q^k$, $M=\lceil\frac{q^m-1}{\omega}\rceil$.
    Optimal Not optimal family size Inequivalent Cor.1
    $(q^m-1, q^k, Tq^k, L_H, q^{m-k})$ $m\ge1$, $0<k\leq m$,
    $q^m-1=T(L_H+1)$.
    Optimal Not optimal family size Inequivalent [20]
    $(q^m-1, q^k, nM(\omega-1), l-n, q^{m-k}-1)$ $m\ge1, 0<k\leq m$,
    $l|(q-1)$, $gcd(l, m)=1$,
    $0<n\leq\lceil\frac{l}{2}\rceil$, $\omega>2l$,
    $M=\lceil\frac{q^m-1}{l\omega}\rceil$.
    Optimal Not optimal family size Inequivalent Cor.2
    $(\frac{q^m-1}{l}, q^k, T, L_H, \frac{q^{m-k}-1}{l})$ $m\ge1, 0<k\leq m$,
    $l|(q-1), gcd(l, m)=1$,
    $q^m-1=T(L_H+1)$.
    Optimal Not optimal family size Equivalent [20]
    $(\frac{q^m-1}{l}, q^k, \frac{q^m-1}{T}, \frac{T}{d'}-1, \frac{q^{m-k}-1}{l})$ $m\ge1, 0<k\leq m$,
    $l|(q-1)$, $gcd(l, m)=1$,
    $T\mid q^m-1$, $T\nmid l$,
    $gcd(T, l)=d'$.
    Optimal Not optimal family size Equivalent [19]
    $(\frac{q-1}{l}, q, \frac{q^m-1}{\omega}, \omega-1, m-1)$ $\omega|\frac{q-1}{l}$, $\omega\ne 1$,
    $1\leq m<min\left\{i:i|\frac{q-1}{l}\right\}$.
    Not Optimal optimal family size Inequivalent [7]
    $\big(e'\frac{(q^m-1)}{l}, \frac{e'}{l}+1, nM(\omega-1), \frac{e'}{l}-n, e'q^k\big)$ $1\leq k\leq m-1, l|(q-1)$,
    $l\ge2$, $e'=q^{m-k}-1$,
    $0<n\leq\lceil\frac{e'}{2l}\rceil$, $\omega>\frac{2e'}{l}$,
    $M=\lceil\frac{q^m-1}{\omega}\rceil$.
    Optimal Not optimal family size Inequivalent Cor.3
    $p$ is a prime, $q$ is a prime power and $lpf(y)$ denotes the least prime factor of an integer $y>1$.
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