A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets

Wenli Ren; Feng Wang

doi:10.3934/amc.2020131

Article Contents

2023, Volume 17, Issue 2: 342-352. Doi: 10.3934/amc.2020131

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A new class of optimal wide-gap one-coincidence frequency-hopping sequence sets

Wenli Ren^, and
Feng Wang

School of Mathematics and Big Data, Dezhou University, Dezhou, 253023, China

^* Corresponding author: Wenli Ren
^* Corresponding author: Wenli Ren

Received: July 2020

Revised: November 2020

Early access: January 2021

Published: April 2023

This research is supported in part by the Natural Science Foundation of Shandong Province (Grant Nos. ZR2018LA001, ZR2018LA003) and in part by the Program of Science and Technology of Shandong Province (Grant Nos. J16LI58, J18KB099)

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Abstract

In this paper, we propose a new class of optimal one-coincidence FHS (OC-FHS) sets with respect to the Peng-Fan bounds, including prime sequence sets and HMC sequence sets as special cases. Thereafter, through investigating their properties, we determine all of the FHS distances in the OC-FHS set. Finally, for a given positive integer, we also propose a new class of wide-gap one-coincidence FHS (WG-OC-FHS) sets where the FHS gap is larger than the given positive integer. Moreover, such a WG-OC-FHS set is optimal with respect to the WG-Lempel-Greenberger bound and the WG-Peng-Fan bounds simultaneously.

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

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Table 1. The optimal OC-FHS set $\mathcal{B}^w$ with FHS distances $d(B_k^w)$

$\mathcal{B}^w$	Frequencies	$d(B_k^w)$
$B_1^7$	$(21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31)$	7
$B_2^7$	$(42, 56, 70, 67, 64, 61, 58, 55, 52, 49, 63, 60, 57, 54, 51, 48, 45)$	3
$B_3^7$	$(46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59)$	4
$B_4^7$	$(50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56)$	6
$B_5^7$	$(54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 48, 49, 50, 51, 52, 53)$	1
$B_6^7$	$(41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50)$	8
$B_7^7$	$(45, 60, 58, 56, 54, 52, 67, 65, 63, 61, 59, 57, 55, 53, 51, 49, 47)$	2
$B_8^7$	$(66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61)$	5
$B_9^7$	$(36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41)$	5
$B_{10}^{7}$	$(57, 59, 61, 63, 65, 67, 52, 54, 56, 58, 60, 45, 47, 49, 51, 53, 55)$	2
$B_{11}^7$	$(61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52)$	8
$B_{12}^7$	$(48, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49)$	1
$B_{13}^7$	$(52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46)$	6
$B_{14}^7$	$(56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43)$	4
$B_{15}^7$	$(60, 63, 49, 52, 55, 58, 61, 64, 67, 70, 56, 42, 45, 48, 51, 54, 57)$	3
$B_{16}^7$	$(81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71)$	7

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Table 2. The optimal WG-OC-FHS set $\mathcal{C}^7$ with the FHS gap $D = 3$

$\mathcal{C}^7$	Frequencies	$d(B_k^w)$
$B_1^7$	$(21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 81, 71, 61, 51, 41, 31)$	7
$B_3^7$	$(46, 50, 54, 58, 62, 66, 53, 57, 61, 65, 69, 56, 43, 47, 51, 55, 59)$	4
$B_4^7$	$(50, 61, 72, 66, 60, 54, 65, 59, 53, 47, 58, 52, 46, 40, 51, 62, 56)$	6
$B_6^7$	$(41, 49, 57, 48, 56, 64, 55, 63, 71, 62, 70, 61, 52, 60, 51, 42, 50)$	8
$B_8^7$	$(66, 71, 76, 64, 69, 57, 62, 50, 55, 43, 48, 36, 41, 46, 51, 56, 61)$	5
$B_9^7$	$(36, 48, 43, 55, 50, 62, 57, 69, 64, 76, 71, 66, 61, 56, 51, 46, 41)$	5
$B_{11}^7$	$(61, 70, 62, 71, 63, 55, 64, 56, 48, 57, 49, 41, 50, 42, 51, 60, 52)$	8
$B_{13}^7$	$(52, 58, 47, 53, 59, 65, 54, 60, 66, 72, 61, 50, 56, 62, 51, 40, 46)$	6
$B_{14}^7$	$(56, 69, 65, 61, 57, 53, 66, 62, 58, 54, 50, 46, 59, 55, 51, 47, 43)$	4
$B_{16}^7$	$(81, 91, 84, 77, 70, 63, 56, 49, 42, 35, 28, 21, 31, 41, 51, 61, 71)$	7

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