On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $

Lin Yi; Xiangyong Zeng; Zhimin Sun; Shasha Zhang

doi:10.3934/amc.2021019

Article Contents

2023, Volume 17, Issue 3: 733-756. Doi: 10.3934/amc.2021019

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On the linear complexity and autocorrelation of generalized cyclotomic binary sequences with period $ 4p^n $

Hubei Key Laboratory of Applied Mathematics, Faculty of Mathematics and Statistics, Hubei University, Wuhan 430062, China

^* Corresponding author: Xiangyong Zeng
^* Corresponding author: Xiangyong Zeng

Received: January 2021

Revised: April 2021

Early access: June 2021

Published: June 2023

The work was supported by Application foundation frontier project of Wuhan Science and Technology Bureau under Grant 2020010601012189, and by the National Nature Science Foundation of China (NSFC) under Grant 62072161. The work of Sun was supported by the Natural Science Foundation of Hubei under Grant 2019CFB544. The work of Zhang was supported by the Research Foundation of Education Bureau of Hubei Province, China under Grant D2020104

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Abstract

In this paper, a new class of generalized cyclotomic binary sequences with period $ 4p^n $ is proposed. These sequences are almost balanced, and the explicit formulas of their linear complexity and autocorrelation are presented.

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Mathematics Subject Classification: Primary: 94A55; Secondary: 94A60.

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Table 1. The autocorrelation distribution of the binary sequence of period $ 108 $

$ R_{S^{(d,e)}}(\tau) $	$ \tau $
$ -68 $	$ 1, 11, 13, 23, 25, 35, 37, 47, 49, 59, 61, 71, 73, 83, 85, 95, 97,107 $
$ 32 $	$ 2, 10, 14, 22, 26, 34, 38, 46, 50, 58, 62, 70, 74, 82, 86, 94, 98,106 $
$ -4 $	$ 3, 9, 33, 39, 69, 75, 99,105 $
$ -32 $	$ 4, 8, 16, 20, 28, 32, 40, 44, 52, 56, 64, 68, 76, 80, 88, 92,100,104 $
$ 68 $	$ 5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91,101,103 $
$ -80 $	$ 6, 30, 42, 66, 78,102 $
$ 80 $	$ 12, 24, 48, 60, 84, 96 $
$ 4 $	$ 15, 21, 45, 51, 57, 63, 87, 93,99 $
$ -96 $	$ 18, 90 $
$ 0 $	$ 27, 81 $
$ 96 $	$ 36, 72 $
$ -104 $	$ 54 $

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