# American Institute of Mathematical Sciences

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

Received  July 2021 Revised  November 2021 Early access January 2022

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.

Citation: Xiujie Zhang, Xianhua Niu, Xin Tan. Constructions of optimal low hit zone frequency hopping sequence sets with large family size. Advances in Mathematics of Communications, doi: 10.3934/amc.2021071
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##### References:
The Maximum Periodic Hamming Correlation of $S$
The Maximum Periodic Hamming Correlation of $S$
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] $(lN, q$, $M\left[\omega-(x-1)(l-1)\right]$, $l-1$, $lH_m)$ $x\ne1, 02l$, $xl-x2$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.1 $(lN, q, Ml+nM(\omega-{xl}+x-1), l-n, {lH}_m)$ $02, \omega>2l, M=\lceil\frac{N}{\omega}\rceil$, $02$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. 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] $\big(q^k(q^m-1), q^k, nM(\omega-1), q^k-n, q^m\big)$ $1\leq k\leq m$, $02q^k$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.1 $(q^m-1, q^k, Tq^k, L_H, q^{m-k})$ $m\ge1$, $02l$, $M=\lceil\frac{q^m-1}{l\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.2 $(\frac{q^m-1}{l}, q^k, T, L_H, \frac{q^{m-k}-1}{l})$ $m\ge1, 0\frac{2e'}{l}$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. 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 $y>1$.
 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] $(lN, q$, $M\left[\omega-(x-1)(l-1)\right]$, $l-1$, $lH_m)$ $x\ne1, 02l$, $xl-x2$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Thm.1 $(lN, q, Ml+nM(\omega-{xl}+x-1), l-n, {lH}_m)$ $02, \omega>2l, M=\lceil\frac{N}{\omega}\rceil$, $02$, $\omega>2l, M=\lceil\frac{N}{\omega}\rceil$. 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] $\big(q^k(q^m-1), q^k, nM(\omega-1), q^k-n, q^m\big)$ $1\leq k\leq m$, $02q^k$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.1 $(q^m-1, q^k, Tq^k, L_H, q^{m-k})$ $m\ge1$, $02l$, $M=\lceil\frac{q^m-1}{l\omega}\rceil$. Optimal Not optimal family size Inequivalent Cor.2 $(\frac{q^m-1}{l}, q^k, T, L_H, \frac{q^{m-k}-1}{l})$ $m\ge1, 0\frac{2e'}{l}$, $M=\lceil\frac{q^m-1}{\omega}\rceil$. 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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