Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation

M. Mossa Al-Sawalha; Khalil Hadi Hakami; Mohammad Alqudah; Qasem M. Tawhari; Hussain Gissy; M. Mossa Al-Sawalha; Khalil Hadi Hakami; Mohammad Alqudah; Qasem M. Tawhari; Hussain Gissy

doi:10.3934/math.2025324

AIMS Mathematics

2025, Volume 10, Issue 3: 7099-7126. doi: 10.3934/math.2025324

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Research article

Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Jazan University, P.O. Box 2097, Jazan 45142, Kingdom of Saudi Arabia
3.
Department of Basic Sciences, School of Electrical Engineering & Information Technology, German Jordanian University, Amman 11180, Jordan

Received: 03 February 2025 Revised: 14 March 2025 Accepted: 21 March 2025 Published: 27 March 2025
MSC : 34A08, 35A15, 35A23

In this paper, we investigate the fractional damped Burgers' equation using two efficient analytical approaches: the Laplace least squares residual power series method and the Laplace least squares variational iteration method. These techniques integrate the Laplace transform with the least squares residual power series and least squares variational iteration methods, providing highly accurate solutions for nonlinear fractional differential equations. The fractional derivatives are considered in the sense of the Caputo operator, allowing for a more realistic description of physical phenomena with memory effects. Comparative studies with exact and numerical solutions demonstrate the reliability and accuracy of the results. The proposed methodologies provide a powerful framework for solving nonlinear fractional models in fluid dynamics, shock wave theory, and applied sciences.

Keywords:

Citation: M. Mossa Al-Sawalha, Khalil Hadi Hakami, Mohammad Alqudah, Qasem M. Tawhari, Hussain Gissy. Novel Laplace-integrated least square methods for solving the fractional nonlinear damped Burgers' equation[J]. AIMS Mathematics, 2025, 10(3): 7099-7126. doi: 10.3934/math.2025324

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Abstract

1. Introduction

The linear complexity and the $k$ -error linear complexity are important cryptographic characteristics of stream cipher sequences. The linear complexity of an $N$ -periodic sequence $s = \{s_u\}^\infty_{u = 0}$ , denoted by $LC(s)$ , is defined as the length of the shortest linear feedback shift register (LFSR) that generates it ^[1]. With the Berlekamp-Massey (B-M) algorithm ^[2], if $LC(s)\geq N/2$ , then $s$ is regarded as a good sequence with respect to its linear complexity. For an integer $k\geq0$ , the $k$ -error linear complexity $LC_k(s)$ is the smallest linear complexity that can be obtained by changing at most $k$ terms of $s$ in the first period and periodically continued ^[3]. The cryptographic background of the $k$ -error linear complexity is that some key streams with large linear complexity can be approximated by some sequences with much lower linear complexity ^[2]. For a sequence to be cryptographically strong, its linear complexity should be large enough, and its $k$ -error linear complexity should be close to the linear complexity.

The relationship between the linear complexity and the DFT of the sequence was given by Blahut in ^[4]. Let $m$ be the order of 2 modulo an odd number $N$ . For a primitive $N$ -th root $\beta\in\mathbb{F}_{2^m}$ of unity, the DFT of $s$ is defined by

$\begin{equation} \rho_i = \sum\limits_{u = 0}^{N-1}s_u\beta^{-iu}\mbox{, }0\leq i\leq N-1. \end{equation}$

(1.1)

Then

$\begin{equation} LC(s) = W_H(\rho_0, \;\rho_1, \;\cdots, \;\rho_{N-1}), \end{equation}$

(1.2)

where $W_H(A)$ is the hamming weight of the sequence $A$ . The polynomial

$\begin{equation} G(X) = \sum\limits_{i = 0}^{N-1}\rho_iX^i \in \mathbb{F}_{2^m}[X] \end{equation}$

(1.3)

is called the Mattson-Solomon polynomial (M-S polynomial) of $s$ ^[5]. It can be deduced from Eqs (1.2)and (1.3) that the linear complexity of $s$ is equal to the number of the nonzero terms of $G(X)$ , namely

$\begin{equation} LC(s) = |G(X)|. \end{equation}$

(1.4)

By the inverse DFT,

$\begin{equation} s_u = \sum\limits_{i = 0}^{N-1}\rho_i\beta^{iu} = G(\beta^u)\mbox{, }0\leq u\leq N-1. \end{equation}$

(1.5)

There are many studies about two-prime generators. In 1997–1998, Ding calculated the linear complexity and the autocorrelation values of binary Whiteman generalized cyclotomic sequences of order two ^[6,7]. In 2013, Li defined a new generalized cyclotomic sequence of order two of length $pq$ , which is based on Whiteman generalized cyclotomic classes, and calculated its linear complexity ^[8]. In 2015, Wei determined the $k$ -error linear complexity of Legendre sequences for $k = 1, 2$ ^[9]. In 2018, Hofer and Winterhof studied the 2-adic complexity of the two-prime generator of period $pq$ ^[10]. Alecu and Sălăgean transformed the optimisation problem of finding the $k$ -error linear complexity of a sequence into an optimisation problem in the DFT domain, by using Blahut's theorem in the same year ^[11]. In 2019, in terms of the DFT, Chen and Wu discussed the $k$ -error linear complexity for Legendre, Ding-Helleseth-Lam, and Hall's sextic residue sequences of odd prime period $p$ ^[12]. In 2020, Zhou and Liu presented a type of binary sequences based on a general two-prime generalized cyclotomy, and derived their minimal polynomial and linear complexity ^[13]. In 2021, the autocorrelation distribution and the 2-adic complexity of generalized cyclotomic binary sequences of order 2 with period $pq$ were determined by Jing ^[14].

This paper is organized as follows. Firstly, we present some preliminaries about Whiteman generalized cyclotomic classes and the linear complexity in Section 2. In Section 3, we give main results about the linear complexity of Whiteman generalized cyclotomic sequences of order two. In Section 4, we give the 1-error linear complexity of these sequences. At last, we conclude this paper in Section 5.

2. Preliminaries

Let $p$ and $q$ be two distinct odd primes with $\mbox{gcd}(p-1, q-1) = 2$ , and $N = pq$ , $e = (p-1)(q-1)/2$ . By the Chinese Remainder Theorem, there is a fixed common primitive root $g$ of both $p$ and $q$ such that $\mbox{ord}_N(g) = e$ . Let $x$ be an integer satisfying

$\begin{equation} x = g(\bmod p)\mbox{, }x = 1(\bmod q).\nonumber \end{equation}$

Then the set

$\begin{equation} D_i = \{g^sx^i \bmod N:s = 0, 1, \cdots, e-1\}\nonumber \end{equation}$

for $i = 0, 1$ is called a Whiteman generalized cyclotomic class of order two ^[15].

As pointed out in ^[14], the unit group of the ring $Z_N$ is

$\begin{align*} Z_N^*& = \{a(\mbox{mod }N):\mbox{gcd}(a, N) = 1\}\nonumber\\ & = \{ip+jq(\mbox{mod }N):1\leq i\leq q-1\mbox{, }1\leq j\leq p-1\}.\nonumber \end{align*}$

Let $P = \{p, 2p, \cdots, (q-1)p\}$ , $Q = \{q, 2q, \cdots, (p-1)q\}$ and $R = \{0\}$ . Then $Z_N = Z_N^*\cup P\cup Q \cup R$ . The sequence $s(a, b, c) = \{s_u\}^\infty_{u = 0}$ over $\mathbb{F}_2$ is defined by

$\begin{align*} s_u = \begin{cases} c, &\mbox{if }\;u = 0, \\ a, &\mbox{if }\;u\in P, \\ b, &\mbox{if }\;u\in Q, \\ \frac{1}{2}(1-(\frac{u}{p})(\frac{u}{q})), &\mbox{if }\;u\in Z_N^*, \\ \end{cases} \end{align*}$

where $(\frac{\cdot}{\cdot})$ denotes the Legendre symbol and $a, b, c\in \mathbb{F}_2$ ^[14].

Lemma 2.1. ^[7] $-1\in D_1$ , if $|p-q|/2$ is odd; and $-1\in D_0$ , if $|p-q|/2$ is even.

Lemma 2.2. ^[6]

$\begin{array}{l} {{\mathit{(1)} \;If \; p\equiv\pm 1(\bmod 8) , \;q\equiv\pm 1(\bmod 8) \; or \;p\equiv\pm 3(\bmod 8) , \;q\equiv\pm 3(\bmod 8) , \;then \; 2\in D_0 .}} \\ {{\mathit{(2)}\;If \; p\equiv\pm 1(\bmod 8) ,\; q\equiv\pm 3(\bmod 8) \; or \; p\equiv\pm 3(\bmod 8) , \;q\equiv\pm 1(\bmod 8) , \;then \;2\in D_1 .}} \end{array}$

Lemma 2.3. ^[6] (1) If $a\in P$ , then $aP = P$ and $aQ = R$ .

(2) If $a\in Q$ , then $aP = R$ and $aQ = Q$ .

(3) If $a\in D_i$ , then $aP = P$ , $aQ = Q$ , and $aD_j = D_{(i+j)\bmod 2}$ , where $i, j = 0, 1$ .

It was shown in ^[6] that, for a primitive $N$ -th root $\beta\in\mathbb{F}_{2^m}$ of unity, we have

$\sum\limits_{i\in P}\beta^i = 1, \;\sum\limits_{i\in Q}\beta^i = 1,$

and

$\begin{equation} \sum\limits_{i\in D_0}\beta^i+\sum\limits_{i\in D_1}\beta^i+\sum\limits_{i\in P}\beta^i+\sum\limits_{i\in Q}\beta^i = 1. \end{equation}$

(2.1)

Lemma 2.4. ^[6]

$\begin{align*} \sum\limits_{u\in D_j}\beta^{iu} = \begin{cases} \frac{p-1}{2}(\bmod 2), &\mathit{\mbox{if}}\;i\in P, \\ \frac{q-1}{2}(\bmod 2), &\mathit{\mbox{if}}\;i\in Q. \end{cases} \end{align*}$

Actually, if $p\equiv -1(\bmod 8)$ or $p\equiv 3(\bmod 8)$ , then $(p-1)/2 = 1$ ; if $p\equiv 1(\bmod 8)$ or $p\equiv -3(\bmod 8)$ , then $(p-1)/2 = 0$ . By symmetry, if $q\equiv -1(\bmod 8)$ or $q\equiv 3(\bmod 8)$ , then $(q-1)/2 = 1$ ; if $q\equiv 1(\bmod 8)$ or $q\equiv -3(\bmod 8)$ , then $(q-1)/2 = 0$ .

Lemma 2.5. Define

$D_i(X) = \sum\limits_{u\in D_i}X^u\in \mathbb{F}_2[X], \mathit{\mbox{}}i = 0, 1.$

Then for $\beta$ , we have $D_0(\beta) = 0$ and $D_1(\beta) = 1$ if $2\in D_0$ ; $D_0(\beta) = \omega$ and $D_1(\beta) = 1+\omega$ if $2\in D_1$ , where $\omega\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

Proof. (1) If $2\in D_0$ , by Lemma 2.3 we have

$[D_i(\beta)]^2 = D_i(\beta^2) = \sum\limits_{2u\in D_i}\beta^{2u} = D_i(\beta)\in\mathbb{F}_2.$

(2) If $2\in D_1$ , by Lemma 2.3 we have

$\begin{align*} &[D_i(\beta)]^2 = D_i(\beta^2) = \sum\limits_{2u\in D_{i+1}}\beta^{2u} = D_{i+1}(\beta), \\ &[D_i(\beta)]^4 = [D_i(\beta)^2]^2 = [D_{i+1}(\beta)]^2 = D_{i+1}(\beta^2) = \sum\limits_{2u\in D_i}\beta^{2u} = D_i(\beta). \end{align*}$

Hence $D_i(\beta)\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

And by Eq (2.1), we have $D_0(\beta)\neq D_1(\beta)$ and $D_0(\beta)+D_1(\beta) = 1$ . Assume that $D_0(\beta) = 0$ , $D_1(\beta) = 1$ for $2\in D_0$ , and $D_0(\beta) = \omega$ , $D_1(\beta) = 1+\omega$ for $2\in D_1$ , where $\omega\in\mathbb{F}_4\setminus\mathbb{F}_2$ .

3. The linear complexity of $s(a, b, c)$

Let $LC(s(a, b, c))$ be the linear complexity of $s(a, b, c)$ , and the other symbols be the same as before.

Theorem 3.1. Let $p\equiv v(\bmod 8)$ and $q\equiv w(\bmod 8)$ , where $v, w = \pm1, \pm3$ . Then the linear complexity of $s(a, b, c)$ respect to different values of $p$ and $q$ is as shown as Table 1.

Table 1. The linear complexity of

$s(a, b, c)$ .

	$s(0, 0, 0)$	$s(0, 0, 1)$	$s(0, 1, 0)$	$s(0, 1, 1)$	$s(1, 0, 0)$	$s(1, 0, 1)$	$s(1, 1, 0))$	$s(1, 1, 1)$
$(-1, -3)$ or $(3, 1)$	$pq-p$	$pq-q+1$	$pq-1$	$pq-p-q+2$	$pq-p-q+1$	$pq$	$pq-q$	$pq-p+1$
$(-1, 3)$ or $(3, -1)$	$pq-1$	$pq-p-q+2$	$pq-p$	$pq-q+1$	$pq-q$	$pq-p+1$	$pq-p-q+1$	$pq$
$(-1, 1)$ or $(3, -3)$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$
$(-1, -1)$ or $(3, 3)$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$
$(-3, -1)$ or $(1, 3)$	$pq-q$	$pq-p+1$	$pq-p-q+1$	$pq$	$pq-1$	$pq-p-q+2$	$pq-p$	$pq-q+1$
$(1, -1)$ or $(-3, 3)$	$\frac{pq+p-q-1}{2}$	$\frac{pq-p+q+1}{2}$	$\frac{pq-p-q+1}{2}$	$\frac{pq+p+q-1}{2}$	$\frac{pq+p+q-3}{2}$	$\frac{pq-p-q+3}{2}$	$\frac{pq-p+q-1}{2}$	$\frac{pq+p-q+1}{2}$

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Proof. We provide the process of calculating $LC(s(0, 0, 0))$ when $v = -1$ and $w = -3$ , and can prove other cases in a similar way.

By Lemmas 2.1–2.3 and Eq (1.1), we have $-1\in D_1$ , $2\in D_1$ , then

$\begin{align*} \rho_i = \sum\limits_{u = 0}^{N-1}s_u\beta^{-iu} = \sum\limits_{u\in D_1}\beta^{-iu} = \sum\limits_{u\in D_0}\beta^{iu}, \end{align*}$

and $\rho_0 = 0$ . By Eq (1.3), we have

$\begin{align*} G(X)& = \sum\limits_{i = 0}^{N-1}\rho_iX^i = \sum\limits_{i\in D_0}\rho_iX^i+\sum\limits_{i\in D_1}\rho_iX^i+\sum\limits_{i\in P}\rho_iX^i+\sum\limits_{i\in Q}\rho_iX^i+\rho_0\\ & = \sum\limits_{i\in D_0}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in D_1}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in P}\sum\limits_{u\in D_0}\beta^{iu}X^i+\sum\limits_{i\in Q}\sum\limits_{u\in D_0}\beta^{iu}X^i.\\ \end{align*}$

Let $t = iu$ . Then by Lemmas 2.3–2.5, we have

$\begin{align*} G(X)& = \sum\limits_{i\in D_0}\sum\limits_{t\in D_0}\beta^tX^i+\sum\limits_{i\in D_1}\sum\limits_{t\in D_1}\beta^tX^i+\sum\limits_{i\in P}\frac{p-1}{2}X^i+\sum\limits_{i\in Q}\frac{q-1}{2}X^i\\ & = D_0(\beta)D_0(X)+D_1(\beta)D_1(X)+\sum\limits_{i\in P}X^i\\ & = \omega D_0(X)+(1+\omega)D_1(X)+\sum\limits_{i\in P}X^i. \end{align*}$

By Eq (1.4) we can get the linear complexity of $s(0, 0, 0)$ as

$\begin{align*} LC(s(0, 0, 0)) = |G(X)| = pq-p. \end{align*}$

Actually, the linear complexity of $s(1, 0, 0)$ was studied by Ding in ^[6] with its minimal polynomial.

4. The 1-error linear complexity of $s(a, b, c)$

Let $LC_k(s(a, b, c))$ be the $k$ -error linear complexity of $s(a, b, c)$ , $\tilde{s} = \{\tilde{s}_u\}^\infty_{u = 0}$ be the new sequence obtained by changing at most $k$ terms of $s$ , that $\tilde{s} = s+e$ , where $e = \{e_u\}^\infty_{u = 0}$ is an error sequence of period $N$ . Ding has provided in ^[2] that, the $k$ -error linear complexity of a sequence can be expressed as

$\begin{equation} LC_k(s) = \mbox{min}_{W_H(e)\leq k}\{LC(s+e)\}. \end{equation}$

(4.1)

It is clearly that $LC_0(s) = LC(s)$ and

$\begin{equation} N\geq LC_0(s)\geq LC_1(s)\geq\cdots\geq LC_l(s) = 0, \nonumber \end{equation}$

where $l = W_H(s)$ .

Let $G(X)$ , $G_k(X)$ and $\tilde{G}(X)$ be the M-S polynomials of $s$ , $e$ and $\tilde{s}$ respectively. Note that

$\begin{equation} G(X) = \sum\limits_{i = 0}^{N-1}\rho_iX^i, \mbox{ }G_k(X) = \sum\limits_{i = 0}^{N-1}\eta_iX^i, \mbox{ }\tilde{G}(X) = \sum\limits_{i = 0}^{N-1}\xi_iX^i, \end{equation}$

(4.2)

where $\rho_i$ , $\eta_i$ and $\xi_i$ are the DFTs of $s$ , $e$ and $\tilde{s}$ respectively. By Eqs (1.5), (4.1) and (4.2), we have $\tilde{G}(X) = G(X)+G_k(X)$ , then

$\begin{equation} \xi_i = \rho_i+\eta_i. \end{equation}$

(4.3)

Assume that $e_{u_0} = 1$ for $0\leq u_0\leq N-1$ and $e_u = 0$ for $u\neq u_0$ in the first period of $e$ . Then the DFT of $e$ is

$\begin{equation} \eta_i = \sum\limits_{u = 0}^{N-1}e_u\beta^{-iu} = \beta^{-iu_0}\mbox{, }0\leq i\leq N-1.\nonumber \end{equation}$

Specially, if $u_0 = 0$ , then $\eta_i = 1$ for all $0\leq i\leq N-1$ ; otherwise, $\eta_0 = 1$ and the order of $\eta_i$ is $N$ for $1\leq i\leq N-1$ .

Theorem 4.1. Let $p\equiv v(\bmod 8)$ and $q\equiv w(\bmod 8)$ , where $v, w = \pm1, \pm3$ , and the other symbols be the same as before. Then the 1-error linear complexity of $s(a, b, c)$ is as shown as Table 2.

Table 2. The 1-error linear complexity of

$s(a, b, c)$ .

	$s(0, 0, 0)$ and $s(0, 0, 1)$	$s(0, 1, 0)$ and $s(0, 1, 1)$	$s(1, 0, 0)$ and $s(1, 0, 1)$	$s(1, 1, 0))$ and $s(1, 1, 1)$
$(-1, -3)$ or $(3, 1)$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .	$pq-p-q+2$	$pq-p-q+1$	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .
$(-1, 3)$ or $(3, -1)$	$pq-p-q+2$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .	$pq-p-q+1$
$(-1, 1)$ or $(3, -3)$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .	$\frac{pq-p-q+3}{2}$	$\frac{pq-p-q+1}{2}$	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .
$(-1, -1)$ or $(3, 3)$	$\frac{pq-p-q+3}{2}$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .	$\frac{pq-p-q+1}{2}$
$(-3, -1)$ or $(1, 3)$	(1) $pq-p+1$ , if $p>q$ ; (2) $pq-q$ , if $p<q$ .	$pq-p-q+1$	$pq-p-q+2$	(1) $pq-p$ , if $p>q$ ; (2) $pq-q+1$ , if $p<q$ .
$(1, -1)$ or $(-3, 3)$	(1) $\frac{pq-p+q+1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q-1}{2}$ , if $p<q$ .	$\frac{pq-p-q+1}{2}$	$\frac{pq-p-q+3}{2}$	(1) $\frac{pq-p+q-1}{2}$ , if $p>q$ ; (2) $\frac{pq+p-q+1}{2}$ , if $p<q$ .

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Proof. We consider the case $v = -1, w = -3$ for $LC_1(s(0, 0, 0))$ . By Lemmas 2.1–2.5 and Eq (1.1), we have $-1\in D_1$ , $2\in D_1$ and

$\begin{align*} \xi_i = \rho_i+\eta_i = \sum\limits_{u\in D_0}\beta^{iu}+\beta^{-iu_0} = \begin{cases} \omega+\beta^{-iu_0}, &\mbox{if }\;i\in D_0, \\ 1+\omega+\beta^{-iu_0}, &\mbox{if }\;i\in D_1, \\ 1+\beta^{-iu_0}, &\mbox{if }\;i\in P, \\ \beta^{-iu_0}, &\mbox{if }\;i\in Q, \\ 1, &\mbox{if }\;i = 0.\\ \end{cases} \end{align*}$

Then by Eq (4.2), we can get

$\begin{align*} \tilde{G}(X) = \sum\limits_{i = 0}^{N-1}\xi_iX^i = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in P}\left(1+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1. \end{align*}$

According to Lemma 2.3, we can get the following results.

(1) If $u_0 = 0$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+1\right)X^i+\sum\limits_{i\in D_1}\omega X^i+\sum\limits_{i\in Q}X^i+1, \\ &|\tilde{G}(X)| = pq-q+1. \end{align*}$

(2) If $u_0\in Q$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1, \\ &|\tilde{G}(X)| = pq-q+1. \end{align*}$

(3) If $u_0\in D_0$ or $u_0\in D_1$ or $u_0\in P$ , then

$\begin{align*} &\tilde{G}(X) = \sum\limits_{i\in D_0}\left(\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in D_1}\left(1+\omega+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in P}\left(1+\beta^{-iu_0}\right)X^i+\sum\limits_{i\in Q}\beta^{-iu_0}X^i+1, \\ &|\tilde{G}(X)| = pq. \end{align*}$

Compare the results of Cases (1)–(3) with $LC(s(0, 0, 0)) = pq-p$ . If $p > q$ , then $pq-p < pq-q+1 < pq$ ; if $p < q$ , then $pq-q+1 < pq-p < pq$ . Hence

$\begin{align*} LC_1(s(0, 0, 0)) = \begin{cases} pq-p, &\mbox{if }p > q, \\ pq-q+1, &\mbox{if }p < q. \end{cases} \end{align*}$

Similarly we can prove the other cases for $LC_1(s(a, b, c))$ .

All results of $LC(s(a, b, c))$ and $LC_1(s(a, b, c))$ in Sections 3 and 4 have been tested by MAGMA program.

5. Conclusions

The purpose of this paper is to determine the linear complexity and the 1-error linear complexity of $s(a, b, c)$ . In most of the cases, $s(a, b, c)$ possesses high linear complexity, namely $LC(s(a, b, c)) > N/2$ , consequently has decent stability against 1-bit error. Notice that the linear complexity of some of the sequences above is close to $N/2$ . Then the sequences can be selected to construct cyclic codes by their minimal generating polynomials with the method introduced by Ding ^[16].

Acknowledgments

This work was supported by Fundamental Research Funds for the Central Universities (No. 20CX05012A), the Major Scientific and Technological Projects of CNPC under Grant (No. ZD2019-183-008), the National Natural Science Foundation of China (Nos. 61902429, 11775306) and Shandong Provincial Natural Science Foundation of China (ZR2019MF070).

Conflict of interest

The authors declare that they have no conflicts of interest.

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