Golay complementary sets with large zero odd-periodic correlation zones

Tinghua Hu; Yang Yang; Zhengchun Zhou

doi:10.3934/amc.2020040

Article Contents

2021, Volume 15, Issue 1: 23-33. Doi: 10.3934/amc.2020040

This issue Previous Article A construction of

$ p $

-ary linear codes with two or three weights Next Article On Hadamard full propelinear codes with associated group

$ C_{2t}\times C_2 $

Golay complementary sets with large zero odd-periodic correlation zones

1.
School of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China
2.
State Key Laboratory of Cryptology, P. O. Box 5159, Beijing, 100878, China

Corresponding author: Yang Yang
Corresponding author: Yang Yang

Received: January 31, 2019

Revised: May 31, 2019

Early access: November 2019

Published: February 2021

This work was supported in part by the National Science Foundation of China under Grants 61771016, 61661146003 and 11571285

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Abstract

Golay complementary sets (GCSs) are widely used in different communication systems, i.e., GCSs could be used in OFDM systems to control peak-to-mean envelope power ratio (PMEPR). In this paper, inspired by the work on GCSs with large zero correlation zone given by Chen et al in 2018, we investigate the relationship between GCSs and zero odd-periodic correlation zone (ZOCZ) sequence sets, and present GCSs with flexible sequence set sizes, sequence lengths, large ZOCZ and low PMEPR. Those proposed sequences could be applied in OFDM system for synchronization.

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Mathematics Subject Classification: Primary: 94A05; Secondary: 60G35.

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Table 1. Comparison of GCSs/GCPs with Large ZCZ/ZOCZ Property

Parameters	Constraints	Ref.
$ (2,2^m,2^{m-2}) $ Golay-ZCZ	$ g_m\in \{\frac{H}{2},0\} $, $ H\equiv0\; \mathrm{mod}\; 2 $	[10,11]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZCZ	$ \begin{array}{l} g_m\in \{0,\frac{H}{2}\},H\equiv 0\; \mathrm{mod}\; 2\pi(1)=m \end{array} $	[2]
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZCZ	$ \begin{array}{l}g_m\in \{0,\frac{H}{2}\}, H\equiv0\; \mathrm{mod}\; 2, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $	[2]
$ (2,2^m,2^{\pi(2)-1}) $ Golay-ZOCZ	$ \begin{array}{l} c_m\in \{\frac{H}{4},\frac{3H}{4}\},H\equiv0\; \mathrm{mod}\; 4, \pi(1)=m \end{array} $	Thm. 1
$ (2^k,2^m,2^{\pi_1(2)-1}) $ Golay-ZOCZ	$ \begin{array}{l}g_m\in \{\frac{H}{4},\frac{3H}{4}\}, H\equiv0\; \mathrm{mod}\; 4, \pi_\alpha(1)=m-\alpha+1, \;\mbox{ for }\; 1\le\alpha\le k\end{array} $	Thm. 2

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