New classes of strictly optimal low hit zone frequency hopping sequence sets

Hongyu Han; Sheng Zhang

doi:10.3934/amc.2020031

Article Contents

2020, Volume 14, Issue 4: 579-589. Doi: 10.3934/amc.2020031

This issue Previous Article Giophantus distinguishing attack is a low dimensional learning with errors problem Next Article Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs

New classes of strictly optimal low hit zone frequency hopping sequence sets

Hongyu Han^1, , and
Sheng Zhang^2,

1.
School of Computer Science, Sichuan Normal University, Chengdu, Sichuan 610066, China
2.
School of Information Science and Technology, Southwest Jiaotong University, Chengdu, Sichuan 611756, China

^* Corresponding author: Hongyu Han
^* Corresponding author: Hongyu Han

Received: February 2018

Revised: March 2019

Early access: November 2019

Published: November 2020

This work is supported in part by the National Science Foundation of China under Grants 61701331, 61801401, and in part by the Project of Sichuan Education Department under Grant 18ZB0496

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Abstract

Low hit zone frequency hopping sequences (LHZ FHSs) with favorable partial Hamming correlation properties are desirable in quasi-synchronous frequency hopping multiple-access systems. An LHZ FHS set is considered to be strictly optimal when it has optimal partial Hamming correlation for all correlation windows. In this study, an interleaved construction of new sets of strictly optimal LHZ FHSs is proposed. Strictly optimal LHZ FHS sets with new and flexible parameters are obtained by selecting suitable known optimal FHSs and appropriate shift sequences.

Keywords:

Frequency hopping sequences,
low hit zone,
partial Hamming correlation,
interleaved technique,
quasi-synchronous frequency hopping communication.

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

Citation:

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Figure 1. Maximum partial Hamming correlations of $ \mathcal{P} $ for the correlation window length $ L = 8 $ in Example 1

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Table 1. New sets of strictly optimal LHZ FHSs

Based on individual FHSs with strictly optimal partial Hamming autocorrelation	$L_c$	$(RN, I, l, v-1, R\alpha)$	Maximum partial Hamming correlation for the correlation window length $L$	Constraints
[2]	$g$	$(Reg, I, g, v\!-\!1, Re)$	$\left\lceil\frac{L}{g}\right\rceil$	$R\geq2$, $Iv=N$, $\gcd(s, N)=1$, $\gcd((v\!+\!1)s^{-1}(\textrm{mod}\ N), L_c)\!=\!1$, $(v+1)Rs^{-1}\equiv1$(mod $N$), $R=s$(mod $v$)
[7]	$q\!+\!1$	$(R(q^2\!-\!1), I, q, v\!-\!1, R(q-1))$	$\left\lceil\frac{L}{q+1}\right\rceil$
[27]	$T$	$(R(q^n-1), I, q^{n-1}, v-1, R(q-1))$	$\left\lceil\frac{L}{T}\right\rceil$
$g$ is any odd integer with the prime factor decomposition $g=p_1^{m_1}p_2^{m_2}\cdots p_k^{m_k}$; $e>1$, $e\|gcd(p_1-1, p_2-1, \cdots, p_k-1)$; $q$ is a prime power and $T=\frac{q^n-1}{q-1}$.

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