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

Wenjuan Yin; Can Xiang; Fang-Wei Fu

doi:10.3934/amc.2020110

Article Contents

2022, Volume 16, Issue 2: 249-267. Doi: 10.3934/amc.2020110

This issue Previous Article Further results on 2-uniform states arising from irredundant orthogonal arrays Next Article New self-dual codes of length 68 from a

$2 \times 2$ block matrix construction and group rings

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

1.
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China
2.
College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, China

^* Corresponding author: Can Xiang
^* Corresponding author: Can Xiang

Received: March 2020

Revised: July 2020

Early access: September 2020

Published: May 2022

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Abstract

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).

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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
$\Big(k_1L_1, M_1N_1, q, Z_1-1, W, \left\lceil\frac{W}{L}\right\rceil\Big)$	$gcd(Z_1+1, L)=1, k_1(Z_1+1)\equiv 1\; (mod\; L_1)$ , $k_1\equiv1\; (mod\; Z_1), M_1Z_1=L_1$	[13]
$\Big(L_1L_2, p_2, p_1p_2, \min\{L_1, L_2\}-1, W; \left\lceil\frac{W}{T_1L_2}\right\rceil\Big)$	$gcd(L_1, L_2)=1, p_1(\frac{L_1}{T_1}-1+\eta)(\min\{L_1, L_2\}p_2-1)=$ $L_1L_2(\min\{L_1, L_2\}-p_1), 0 < \eta\leq 1$ $sl=q^n-1, 2\leq r_0 < \frac{(q^n-2)q}{q-1}, r_0\equiv t\; mod\; l$ ,	[24]
$\Big(r_0(q^n-1), s, q, l-1, \omega\frac{q^{n}-1}{q-1}; \omega\frac{q^{n-1}-1}{q-1}\Big)$	$(l+1)r_0t^{-1}\equiv1\; (mod\; (q^n-1)), gcd(t, q^n-1)=1$ $gcd((l+1)t^{-1}\; mod\; (q^n-1), \frac{q^n-1}{q-1})=1, 1\leq \omega\leq(q-1)r_0$	[10]
$\left((q_1-1)(q_2^n-1), q_1, q_1q_2^{n-1}, q_2^n-2, W;\left\lceil\frac{W(q_2-1)}{(q_2^n-1)(q_1-1)}\right\rceil\right)$	$q_1, q_2$ are two different prime powers satisfying $gcd(q_1-1, q_2^n-1)=1, q_1 >q_2^n$	[28]
$\Big(lcm(q-1, tv), q, qv, \min\{q-1, tv\}-1, W; \left\lceil\frac{W}{v(q-1)}\right\rceil\Big)$	$gcd(q-1, v)=1, gcd(q-1, t) >1, tv < q-1$ , or $tv >q-1$ and $t < \frac{gcd(q-1, t)(q^2-q-1)}{qv-v-2}$	[28]
$\Big(lcm(q-1, p(p^n-1)), qp^{n-1}, qp^n, \min\{q-1, p(p^n-1)\}-1,$	$gcd(q-1, p^n-1)=1, q-1 >p(p^n-1)$ , 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 $l\|(q-1), gcd(l, n)=1$ ,	[9]
$\left(\frac{q^n-1}{l}, \frac{q^n-1}{T}, q^k, \frac{T}{d'}-1, s\frac{q^n-1}{q-1}; s\frac{q^{n-k}-1}{q-1}\right)$	$T\|(q^n-1), T\nmid l, gcd(T, l)=d', m\|n$ , $1\leq k \leq m$ 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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