Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

Yixin Ren; Chenyu Hou; Huaning Liu; Yixin Ren; Chenyu Hou; Huaning Liu

doi:10.3934/era.2023232

Electronic Research Archive

2023, Volume 31, Issue 8: 4549-4556. doi: 10.3934/era.2023232

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Research article Special Issues

Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

Research Center for Number Theory and Its Applications, School of Mathematics, Northwest University, Xi'an 710127, China

Received: 03 April 2023 Revised: 24 May 2023 Accepted: 30 May 2023 Published: 19 June 2023

Sequences with optimal autocorrelation properties play an important role in wireless communication, radar and cryptography. Interleaving is a very important method in constructing the optimal autocorrelation sequence. Tang and Gong gave three different constructions of interleaved sequences (generalized GMW sequences, twin prime sequences and Legendre sequences). Su et al. constructed a series of sequences with optimal autocorrelation magnitude via interleaving Ding-Helleseth-Lam sequences. In this paper we further study the correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences.

Keywords:

Citation: Yixin Ren, Chenyu Hou, Huaning Liu. Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences[J]. Electronic Research Archive, 2023, 31(8): 4549-4556. doi: 10.3934/era.2023232

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Abstract

1. Introduction and results

1.1. Optimal autocorrelation sequence

Given a binary sequence $v = \left(v(0), v(1), \cdots, v(N-1)\right)\in\{0, 1\}^N$ of period $N$ , $\mathbb{Z}_N$ is the integer ring of modulo $N$ , the set $C_{v} = \left\{t\in\mathbb{Z}_N: \ v(t) = 1\right\}$ is called the support of $v$ . Let $\tau\in\{1, \cdots, N-1\}$ , the autocorrelation function $R_{v}(\tau)$ of $v$ is defined by

$R_{v}(\tau) = \sum\limits_{i = 0}^{N-1}(-1)^{v(i)+v(i+\tau)},$

where $i+\tau$ is operating on modulo $N$ . Easy to verity

$R_{v}(\tau) = N-4\left|(C_{v}+\tau)\cap C_{v}\right|.$

Thus $R_{v}(\tau)\equiv N \ (\bmod\ 4)$ . In particular, $R_{v}(\tau)\equiv 0 \ (\bmod\ 4)$ for $N\equiv 0 \ (\bmod\ 4)$ .

If $R_{v}(\tau)\in\{0, -4\}$ or $R_{v}(\tau)\in\{0, +4\}$ where $\tau\in\{1, \cdots, N-1\}$ , then $v$ is referred to as a sequence with optimal autocorrelation value. If $R_{v}(\tau)\in\{0, \pm 4\}$ where $\tau\in\{1, \cdots, N-1\}$ , then $v$ is referred to as a sequence with optimal autocorrelation magnitude. Sequences with optimal autocorrelation properties play an important role in wireless communication, radar and cryptography^[1], so they have always been the focus of scholars' research (refer to ^{[2,3,4,5,6,7,8,9,10,11]}).

1.2. Interleaved method

Gong^[12] introduced powerful interleaved method to design sequences. The key idea of this method is to obtain long sequences with good correlation from shorter ones. Let

$v_k = \left(v_k(0), v_k(1), \cdots, v_k(N-1)\right)$

be a sequence of period $N$ , where $0\leq k\leq M-1$ . Define the $N\times M$ matrix $U = (U_{ij})$ as follows

$U = \left[ \begin{array}{cccc} v_{0}(0) & v_{1}(0) & \cdots & v_{M-1}(0) \\ v_{0}(1) & v_{1}(1) & \cdots & v_{M-1}(1) \\ \vdots & \vdots & & \vdots \\ v_{0}(N-1) & v_{1}(N-1) & \cdots & v_{M-1}(N-1) \end{array} \right].$

An interleaved sequence $u = (u(t))$ of period $MN$ is defined by

$u_{iM+j} = U_{ij} = v_{j}(i), \qquad 0\leq i < N, \ 0\leq j < M.$

For convenience we write

$u = I(v_0, v_1, \cdots, v_{M-1}).$

Interleaving is a very important method in constructing optimal autocorrelation sequences. The autocorrelation of the sequence $u$ is determined by the autocorrelation and cross-correlation of the sequences $v_0, v_1, \cdots, v_{M-1}$ .

1.3. Interleaved Legendre sequences

Tang and Gong^[10] gave three different constructions of interleaved sequences with the following structure:

$\begin{eqnarray} u = I\left(a_0+b(0), \ L^{d+\eta}(a_1)+b(1), \ L^{2d}(a_2)+b(2), \ L^{3d+\eta}(a_3)+b(3)\right), \end{eqnarray}$

(1.1)

where $\eta$ is an integer with $0\leq \eta < N$ , $d$ is an integer with $4d\equiv 1 \ (\bmod\ N)$ , $b = (b(0), b(1), b(2), b(3))$ is a binary perfect sequence, $a_i$ , $i = 0, 1, 2, 3$ , are sequences of period $N$ taken from three different pairs of sequences (generalized GMW sequences, twin prime sequences and Legendre sequences).

Let $N = p$ be an odd prime. The Legendre sequence $l = \left(l(0), l(1), \cdots, l(p-1)\right)$ of period $p$ is defined by

$l(i) = \left\{\begin{array}{ll} 0 \ \hbox{ or }\ 1, & \hbox{if }\ i = 0, \\ 1, & \hbox{if } i {\text{ is quadratic residue modulo }} p , \\ 0, & \hbox{if } i {\text{ is quadratic non-residue modulo }} p . \end{array}\right.$

In particular, $l$ is called the first type Legendre sequence if $l(0) = 1$ otherwise the second type Legendre sequence. Let $l$ and $l'$ be the first type and the second type of Legendre sequence of period $p$ , respectively.

Tang and Gong^[10] showed that the constructions of interleaved Legendre sequence have optimal autocorrelation magnitude. Li and Tang^[6] studied the linear complexity of binary sequences given in ^[10].

1.4. Interleaved Ding-Helleseth-Lam sequences

Let $p = 4f+1$ be an odd prime, where $f$ is a positive integer, $\alpha$ is a generator of $\mathbb{F}_p$ . Define sets

$D_i = \left\{\alpha^{i+4j}: \ 0\leq j\leq f-1\right\}, \qquad i = 0, 1, 2, 3.$

Then $D_i$ , $i = 0, 1, 2, 3$ are called the cyclotomic classes of order $4$ with respect to $\mathbb{F}_p$ . It is very easy to verity that $D_i$ , $i = 0, 1, 2, 3$ constitute a partition of $\mathbb{F}_p^{*}$ .

Ding et al.^[13] constructed some binary sequences based on the cyclotomic classes of order $4$ of $\mathbb{F}_p$ , and proved that these sequences are three-level autocorrelation.

Proposition 1.1. Let $p = 4f+1 = x^2+4y^2$ be an odd prime, where $f, x, y$ are integers. Let $s_1, s_2, s_3, s_4$ be binary sequences of period $p$ with supports $D_0\cup D_1$ , $D_0\cup D_3$ , $D_1\cup D_2$ and $D_2\cup D_3$ , respectively. Then $R_{s_i}(\tau)\in\{1, -3\}$ for all $1\leq \tau < p$ if and only if $f$ is odd and $y = \pm 1$ .

Let $a_0, a_1, a_2, a_3$ be four binary sequences of period $p$ , $b = (b(0), b(1), b(2), b(3))$ be a binary sequence of length $4$ . Construct a binary sequence $u = u(t)$ of length $4p$ as follows:

$\begin{eqnarray} u = I\left(a_0+b(0), \ L^d(a_1)+b(1), \ L^{2d}(a_2)+b(2), \ L^{3d}(a_3)+b(3)\right), \end{eqnarray}$

(1.2)

where $d$ is a integer with $4d\equiv 1 \ (\bmod\ p)$ , $L$ is a cyclic left shift operator, when $c = (c(0), c(1), \cdots, c(N-1))$ , $L(c) = (c(1), c(2), \cdots, c(N-1), c(0))$ .

Su et al.^[8] constructed a series sequences with optimal autocorrelation magnitude of period $4p$ via interleaving Ding-Helleseth-Lam sequences.

Proposition 1.2. Let $p = 4f+1 = x^2+4y^2$ be an odd prime, where $x$ is an integer, $y = \pm 1$ , $f$ is odd. Let $b(0) = b(2)$ , $b(1) = b(3)$ , and $(a_0, a_1, a_2, a_3)$ be chosen from

$\begin{eqnarray} &&(s_3, s_2, s_1, s_1), \quad (s_2, s_3, s_1, s_1), \quad (s_1, s_4, s_2, s_2), \quad (s_4, s_1, s_2, s_2), \\ &&(s_1, s_4, s_3, s_3), \quad (s_4, s_1, s_3, s_3), \quad (s_3, s_2, s_4, s_4), \quad (s_2, s_3, s_4, s_4). \end{eqnarray}$

(1.3)

Then the binary sequence $u$ by $(1.2)$ has $R_{u}(\tau)\in\{0, \pm 4\}$ for all $1\leq \tau < 4p$ .

Later Fan^[14] showed that the above sequences have large linear complexity. Sun et al.^[15] proved that the sequences have good $2$ -adic complexity.

1.5. Results of this paper

This paper will further study the fourth-order correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences. Our results are as follows.

Theorem 1.1. Let $p > 2$ be a prime, $b = (b(0), b(1), b(2), b(3))$ be a binary perfect sequence, $\eta$ be an integer with $0 < \eta < p$ and let the binary sequence $u$ be generated by $(1.1)$ . For $n\in \{0, 1, \cdots, 4p-1\}$ we have

$\begin{eqnarray*} &&(-1)^{u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p}} \\ && = \left\{\begin{array}{ll} (-1)^{a_0(p-\eta)+a_1(0)+a_2(p-\eta)+a_3(0)+1}, & n\equiv -4\eta \ (\bmod\ p), \\ (-1)^{a_0(0)+a_1(\eta)+a_2(0)+a_3(\eta)+1}, & n\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n\not\equiv -4\eta \ (\bmod\ p), \\ -1, & n\not\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n\not\equiv -4\eta \ (\bmod\ p). \end{array} \right. \end{eqnarray*}$

Theorem 1.2. Let $p = 4f+1$ be an odd prime, $b(0) = b(2)$ , $b(1) = b(3)$ and let $(a_0, a_1, a_2, a_3)$ be chosen from (1.3). Let the binary sequence $u$ be defined as in $(1.2)$ . For $n\in \{0, 1, \cdots, 4p-1\}$ we have

$(-1)^{u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p}} = \left\{\begin{array}{ll} +1, & n\equiv 0 \ (\bmod\ p), \\ -1, & n\not\equiv 0 \ (\bmod\ p). \end{array} \right.$

Our results show that the fourth-order correlation for interleaved Legendre sequences and Ding-Helleseth-Lam sequences $u$ we have $(-1)^{u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p}} = -1$ which the number of $n$ is at least $4p-5$ for all $n\in \{0, 1, \cdots, 4p-1\}$ . This means that there are at least four subsequences that do not satisfy lower mutual interference. Therefore, researchers should pay careful attention to these four subsequences when consider their application in communication systems.

2. Proof of Theorem 1.1

Write

$\begin{eqnarray*} u& = &I\left(v_0, \ v_1, \ v_2, \ v_3\right) \\ & = &I\left(a_0+b(0), \ L^{d+\eta}(a_1)+b(1), \ L^{2d}(a_2)+b(2), \ L^{3d+\eta}(a_3)+b(3)\right). \end{eqnarray*}$

Then

$\begin{eqnarray} (-1)^{v_0(n)}& = &(-1)^{b(0)}(-1)^{a_0(n)}, \end{eqnarray}$

(2.1)

$\begin{eqnarray} (-1)^{v_1(n)}& = &(-1)^{b(1)}(-1)^{a_1(n+d+\eta)}, \end{eqnarray}$

(2.2)

$\begin{eqnarray} (-1)^{v_2(n)}& = &(-1)^{b(2)}(-1)^{a_2(n+2d)}, \end{eqnarray}$

(2.3)

$\begin{eqnarray} (-1)^{v_3(n)}& = &(-1)^{b(3)}(-1)^{a_3(n+3d+\eta)}. \end{eqnarray}$

(2.4)

Case Ⅰ: $p\equiv 1\ (\bmod\ 4)$ . We have

$\begin{eqnarray*} p = 4\cdot\frac{p-1}{4}+1, \quad 2p = 4\cdot\frac{2(p-1)}{4}+2, \quad 3p = 4\cdot\frac{3(p-1)}{4}+3, \quad d = \frac{3p+1}{4}. \end{eqnarray*}$

Write $n = 4i+j$ , where $0\leq i\leq p-1$ and $0\leq j\leq 3$ . Hence,

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} \\ && = u_{4i+j}+u_{4i+j+4\cdot\frac{p-1}{4}+1}+u_{4i+j+4\cdot\frac{2(p-1)}{4}+2} +u_{4i+j+4\cdot\frac{3(p-1)}{4}+3}. \end{eqnarray}$

(2.5)

First we consider the case of $j = 1$ . By (2.1)–(2.5) we get

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} \\ && = u_{4i+1}+u_{4(i+\frac{p-1}{4})+2}+u_{4(i+\frac{2(p-1)}{4})+3} +u_{4(i+\frac{3(p-1)}{4})+4} \\ && = v_1(i)+v_2\left(i+\frac{p-1}{4}\right)+v_3\left(i+\frac{2(p-1)}{4}\right) +v_0\left(i+\frac{3p+1}{4}\right) \\ && = a_1\left(i+d+\eta\right)+b(1) +a_2\left(i+\frac{p-1}{4}+2d\right)+b(2) \\ &&\qquad+a_3\left(i+\frac{2(p-1)}{4}+3d+\eta\right)+b(3) +a_0\left(i+\frac{3p+1}{4}\right)+b(0) \\ && = b(0)+b(1)+b(2)+b(3)+a_0(i+d)+a_1(i+d+\eta) \\ &&\qquad+a_2(i+d)+a_3(i+d+\eta). \end{eqnarray}$

(2.6)

For all $j\in \{0, 1, 2, 3\}$ we can also have

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} = b(0)+b(1)+b(2)+b(3) \\ &&\qquad+a_0(i+jd)+a_1(i+jd+\eta)+a_2(i+jd)+a_3(i+jd+\eta). \end{eqnarray}$

(2.7)

Noting that $4d \equiv 1 \ (\bmod\ p)$ and $n = 4i+j$ , thus we get

$\begin{eqnarray*} &&i+jd\equiv 0 \ (\bmod\ p) \Longleftrightarrow 4i+j\equiv 0 \ (\bmod\ p) \Longleftrightarrow n\equiv 0 \ (\bmod\ p), \\ &&i+jd+\eta\equiv 0 \ (\bmod\ p) \Longleftrightarrow 4i+j+4\eta\equiv 0 \ (\bmod\ p) \Longleftrightarrow n+4\eta\equiv 0 \ (\bmod\ p). \end{eqnarray*}$

Therefore

$\begin{eqnarray*} &&(-1)^{u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p}} = (-1)^{a_0(i+jd)+a_1(i+jd+\eta)+a_2(i+jd)+a_3(i+jd+\eta)}\\ && = \left\{\begin{array}{ll} (-1)^{a_0(0)+a_1(0)+a_2(0)+a_3(0)+1}, & n\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n+4\eta\equiv 0 \ (\bmod\ p), \\ (-1)^{a_0(p-\eta)+a_1(0)+a_2(p-\eta)+a_3(0)+1}, & n\not\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n+4\eta\equiv 0 \ (\bmod\ p), \\ (-1)^{a_0(0)+a_1(\eta)+a_2(0)+a_3(\eta)+1}, & n\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n+4\eta\not\equiv 0 \ (\bmod\ p), \\ -1, & n\not\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n+4\eta\not\equiv 0 \ (\bmod\ p), \end{array} \right.\\ && = \left\{\begin{array}{ll} (-1)^{a_0(p-\eta)+a_1(0)+a_2(p-\eta)+a_3(0)+1}, & n\equiv -4\eta \ (\bmod\ p), \\ (-1)^{a_0(0)+a_1(\eta)+a_2(0)+a_3(\eta)+1}, & n\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n\not\equiv -4\eta \ (\bmod\ p), \\ -1, & n\not\equiv 0 \ (\bmod\ p) \ \hbox{ and }\ n\not\equiv -4\eta \ (\bmod\ p). \end{array} \right. \end{eqnarray*}$

Case Ⅱ: $p\equiv 3\ (\bmod\ 4)$ . We have

$\begin{eqnarray*} p = 4\cdot\frac{p-3}{4}+3, \quad 2p = 4\cdot\frac{2(p-1)}{4}+2, \quad 3p = 4\cdot\frac{3p-1}{4}+1, \quad d = \frac{p+1}{4}. \end{eqnarray*}$

Similarly we can get

This proves Theorem 1.1.

3. Proof of Theorem 1.2

Write

$\begin{eqnarray*} u& = &I\left(v_0, \ v_1, \ v_2, \ v_3\right) \\ & = &I\left(a_0+b(0), \ L^{d}(a_1)+b(1), \ L^{2d}(a_2)+b(2), \ L^{3d}(a_3)+b(3)\right). \end{eqnarray*}$

Then

$\begin{eqnarray} (-1)^{v_0(n)}& = &(-1)^{b(0)}(-1)^{a_0(n)}, \end{eqnarray}$

(3.1)

$\begin{eqnarray} (-1)^{v_1(n)}& = &(-1)^{b(1)}(-1)^{a_1(n+d)}, \end{eqnarray}$

(3.2)

$\begin{eqnarray} (-1)^{v_2(n)}& = &(-1)^{b(2)}(-1)^{a_2(n+2d)}, \end{eqnarray}$

(3.3)

$\begin{eqnarray} (-1)^{v_3(n)}& = &(-1)^{b(3)}(-1)^{a_3(n+3d)}. \end{eqnarray}$

(3.4)

Let $b(0) = b(2)$ and $b(1) = b(3)$ . Noting that $p\equiv 1\ (\bmod\ 4)$ , thus we have

$\begin{eqnarray*} p = 4\cdot\frac{p-1}{4}+1, \quad 2p = 4\cdot\frac{2(p-1)}{4}+2, \quad 3p = 4\cdot\frac{3(p-1)}{4}+3, \quad d = \frac{3p+1}{4}. \end{eqnarray*}$

Write $n = 4i+j$ , where $0\leq i\leq p-1$ and $0\leq j\leq 3$ . Hence,

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} \\ && = u_{4i+j}+u_{4i+j+4\cdot\frac{p-1}{4}+1}+u_{4i+j+4\cdot\frac{2(p-1)}{4}+2} +u_{4i+j+4\cdot\frac{3(p-1)}{4}+3}. \end{eqnarray}$

(3.5)

For the case of $j = 1$ , by (3.1)–(3.5) we get

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} \\ && = u_{4i+1}+u_{4(i+\frac{p-1}{4})+2}+u_{4(i+\frac{2(p-1)}{4})+3} +u_{4(i+\frac{3(p-1)}{4})+4} \\ && = v_1(i)+v_2\left(i+\frac{p-1}{4}\right)+v_3\left(i+\frac{2(p-1)}{4}\right) +v_0\left(i+\frac{3p+1}{4}\right) \\ && = a_1\left(i+d\right)+b(1) +a_2\left(i+\frac{p-1}{4}+2d\right)+b(2) \\ &&\qquad+a_3\left(i+\frac{2(p-1)}{4}+3d\right)+b(3) +a_0\left(i+\frac{3p+1}{4}\right)+b(0) \\ && = b(0)+b(1)+b(2)+b(3)+a_0(i+d)+a_1(i+d)+a_2(i+d)+a_3(i+d). \end{eqnarray}$

(3.6)

For all $j\in \{0, 1, 2, 3\}$ we can also have

$\begin{eqnarray} && u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p} \\ && = b(0)+b(1)+b(2)+b(3)+a_0(i+jd)+a_1(i+jd)+a_2(i+jd)+a_3(i+jd). \end{eqnarray}$

(3.7)

Clearly,

$i+jd\equiv 0 \ (\bmod\ p) \Longleftrightarrow 4i+j\equiv 0 \ (\bmod\ p) \Longleftrightarrow n\equiv 0 \ (\bmod\ p).$

Let $(a_0, a_1, a_2, a_3)$ be chosen from

$\begin{eqnarray*} &&(s_3, s_2, s_1, s_1), \qquad (s_2, s_3, s_1, s_1), \qquad (s_1, s_4, s_2, s_2), \qquad (s_4, s_1, s_2, s_2), \\ &&(s_1, s_4, s_3, s_3), \qquad (s_4, s_1, s_3, s_3), \qquad (s_3, s_2, s_4, s_4), \qquad (s_2, s_3, s_4, s_4). \end{eqnarray*}$

From (3.7) we get

$\begin{eqnarray*} (-1)^{u_{n}+u_{n+p}+u_{n+2p}+u_{n+3p}} & = &(-1)^{a_0(i+jd)+a_1(i+jd)+a_2(i+jd)+a_3(i+jd)}\\ & = &\left\{\begin{array}{ll} +1, & n\equiv 0 \ (\bmod\ p), \\ -1, & n\not\equiv 0 \ (\bmod\ p). \end{array} \right. \end{eqnarray*}$

This completes the proof of Theorem 1.2.

Use of AI tools declaration

The authors have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper is supported by National Natural Science Foundation of China under Grant No. 12071368, Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 22JSY017), and Northwest University Graduate Innovation Program Funds (No. CX2023066).

Conflict of interest

The authors declare there is no conflicts of interest.

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沈阳化工大学材料科学与工程学院沈阳 110142

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Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

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Abstract

1. Introduction and results

1.1. Optimal autocorrelation sequence

1.2. Interleaved method

1.3. Interleaved Legendre sequences

1.4. Interleaved Ding-Helleseth-Lam sequences

1.5. Results of this paper

2. Proof of Theorem 1.1

3. Proof of Theorem 1.2

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Correlation properties of interleaved Legendre sequences and Ding-Helleseth-Lam sequences

Related Papers:

Abstract

1. Introduction and results

1.1. Optimal autocorrelation sequence

1.2. Interleaved method

1.3. Interleaved Legendre sequences

1.4. Interleaved Ding-Helleseth-Lam sequences

1.5. Results of this paper

2. Proof of Theorem 1.1

3. Proof of Theorem 1.2

Use of AI tools declaration

Acknowledgments

Conflict of interest

References

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