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

Xiujie Zhang; Xianhua Niu; Xin Tan

doi:10.3934/amc.2021071

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2022. Doi: 10.3934/amc.2021071

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

School of Computer and Software Engineering, Xihua University, Chengdu, Sichuan 610039, China

^* Corresponding author: Xianhua Niu
^* Corresponding author: Xianhua Niu

Received: June 30, 2021

Revised: October 31, 2021

Early access: January 2022

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Abstract

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.

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

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

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

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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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