# American Institute of Mathematical Sciences

doi: 10.3934/amc.2020040

## 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

Received  February 2019 Revised  June 2019 Published  November 2019

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

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.

Citation: Tinghua Hu, Yang Yang, Zhengchun Zhou. Golay complementary sets with large zero odd-periodic correlation zones. Advances in Mathematics of Communications, doi: 10.3934/amc.2020040
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##### References:
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
 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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