New Optimal Quaternary Sequences with Period 2N from Interleaving Tang–Lindner Sequences

Abstract

In this paper, using the interleaving technique, we present a method for constructing M-ary sequences of length $4N$. We propose a new concept, referred to as the semi-interleaved sequence, based on some of the special cases of our construction. The period of these semi-interleaved sequences is $2N$, and their autocorrelations can be obtained in the same way as those of interleaved sequences. Applying the construction to certain known sequences, we obtain new quaternary sequences having period $2N$ where $N=4f+1$ is prime and f is an odd integer. The nontrivial autocorrelations of the new sequences are 2 and $-2$. From the autocorrelation distributions, we know that the new sequences cannot be obtained by previously known methods.

1. Introduction

Let $\mathit{a}={\left(a\left(t\right)\right)}_{N-1}^{t=0}$ and $\mathit{b}={\left(b\left(t\right)\right)}_{N-1}^{t=0}$ be two sequences of length N over the integer residue ring ${\mathit{Z}}_M=\{0,1,\dots ,M-1\}$. Then, $\mathit{a}$ is called a binary sequence if $M=2$, and a quaternary sequence if $M=4$. The cross-correlation function between $\mathit{a}$ and $\mathit{b}$ at the integer shift $\tau ,0\le \tau <N,$ is given by

$$R_{\mathit{a},\mathit{b}}\left(\tau \right)=\sum _{t=0}^{N-1}{\omega }^{a\left(t\right)-b(t+\tau )},$$

(1)

where $\omega =e^{{\displaystyle \frac{2\pi \sqrt{-1}}{M}}}$ is a complex primitive Mth root of unity, and the addition $t+\tau$ is performed modulo N. When the two sequences $\mathit{a}$ and $\mathit{b}$ are identical, the cross-correlation function is called the autocorrelation function, and is denoted by $R_{\mathit{a}}\left(\tau \right)$. For the case of length $N\equiv 2(mod4)$, a sequence $\mathit{a}$ is called optimal if $R_{\mathit{a}}\left(\tau \right)\in \{\pm 2\}$ [1].

For the sake of simplicity of implementation, binary and quaternary sequences are preferred for most applications such as digital communications, measurement, random number generation, radar ranging, and cryptography. In cryptography, sequences are employed to generate key streams in stream cipher encryption and, in code-division multiple access (CDMA) communication systems, they are needed to obtain accurate timing information of received signals. In some communication systems, the employed sequences are required to have out-of-phase autocorrelation values as low as possible in order to ensure message synchronization.

A number of constructions of balanced binary sequences with optimal autocorrelation property have been developed (see [2,3,4,5]). Ding, Helleseth, and Lam [6] constructed several classes of binary sequences of period $N\equiv 1$ (mod 4) with good autocorrelation. Ding and Tang [4] proved that some of these sequences form optimal pairs. The construction of balanced quaternary sequences with good autocorrelation has attracted much attention as of late (see the survey [5]). Tang and Lindner [7] constructed a class of quaternary cyclotomic sequences of period $N\equiv 1$ (mod 4) with good autocorrelation. Yang and Tang [8] proved that the these sequences form optimal pairs. In [9], Luke gave some balanced quaternary sequences of period $4N$, whose maximum out-of-phase autocorrelations were shown to be 4. In [1], Tang and Helleseth presented a simple but general transform method for constructing quaternary sequences of period $2N$ from quaternary sequences of odd period N. Quaternary sequences with period $2N$ were also discussed in [10,11,12].

The interleaving method proposed by Gong [13] is a powerful technique in sequence design. For example, using generalized GMW sequences, twin-prime sequences, and Legendre sequences, Tang and Gong [14] obtained a method for constructing binary sequences of period $4N$ with optimal autocorrelation. Recently, the authors in [15] presented a construction of binary sequences of period $4N$ via interleaving four suitable Ding–Helleseth–Lam sequences.

Motivated by the works mentioned above, we give constructions that are more general, which include the constructions mentioned above as special cases, and which also produce previously unknown M-ary sequences. In particular, we obtain quaternary sequences with period $2N$ by interleaving certain known sequences. The out-of-phase autocorrelations of the constructed quaternary sequences are 2 and $-2$.

2. Interleaved Sequences

2.1. Interleaved Sequences

Let $\{{\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1}\}$ be a set of T sequences of period N where ${\mathit{a}}_k=(a_k\left(0\right),a_k\left(1\right),\dots ,$$a_k(N-1))$ for $0\le k\le T-1$. An interleaved sequence $\mathit{u}=\left(u\right(t\left)\right)$ of length $TN$ is defined as follows:

$$u(iT+j)=a_j\left(i\right),0\le i<N,0\le j<T.$$

For simplicity, the interleaved sequence $\mathit{u}$ can be written as

$$\mathit{u}=I({\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1}),$$

where I is called the interleaving operator and ${\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1}$ are called the column sequences of $\mathit{u}$.

Let $\mathit{v}=I({\mathit{b}}_0,{\mathit{b}}_1,\dots ,{\mathit{b}}_{T-1})$ be another interleaved sequence of length $TN$, and let L be the left cyclical shift operator(for example, $\mathit{a}=\left(a\right(0),a(1),a(2),a(3),a(4\left)\right)$, then $L\left(\mathit{a}\right)=\left(a\right(1),a(2),a(3),a(4),a(0\left)\right)$ and $L^2\left(\mathit{a}\right)=(a\left(2\right),a\left(3\right),a\left(4\right),a\left(0\right),a\left(1\right))$). Then,

$$\begin{array}{c}\hfill L^{\tau }\left(\mathit{v}\right)=I(L^{{\tau }_1}\left({\mathit{b}}_{{\tau }_2}\right),\dots ,L^{{\tau }_1}\left({\mathit{b}}_{T-1}\right),L^{{\tau }_1+1}\left({\mathit{b}}_0\right),\dots ,L^{{\tau }_1+1}\left({\mathit{b}}_{{\tau }_2-1}\right)),\end{array}$$

where $\tau ={\tau }_{1}T+{\tau }_2$, $0\le {\tau }_1\le N-1$, and $0\le {\tau }_2\le T-1$. Then, the cross-correlation function between the interleaved sequences $\mathit{u}$ and $\mathit{v}$ at the shift $\tau$ can be obtained by summing the cross-correlation of column sequences in $\mathit{u}$ and $\mathit{v}$, i.e.,

$$R_{\mathit{u},\mathit{v}}\left(\tau \right)=\sum _{i=0}^{T-{\tau }_2-1}{R}_{{\mathit{a}}_i,{\mathit{b}}_{i+{\tau }_2}}\left({\tau }_1\right)+\sum _{i=T-{\tau }_2}^{T-1}{R}_{{\mathit{a}}_i,{\mathit{b}}_{i+{\tau }_2-T}}({\tau }_1+1).$$

(2)

2.2. Generic Construction of M-Ary Sequences with Period 4N

Let $N\equiv 3 (\mathrm{mod} 4)$, $\mathit{a}$, and $\mathit{b}$ be two binary ideal sequences of period N. Tang and Ding [1] presented a method for constructing binary sequences of length $4N$ with optimal autocorrelation as follows:

$$\mathit{u}=I(\mathit{a},L^{{\displaystyle \frac{N+1}{4}}}\left(\mathit{b}\right),L^{{\displaystyle \frac{N+1}{2}}}\left(\mathit{a}\right),L^{{\displaystyle \frac{3(N+1)}{4}}}\left(\mathit{b}\right)+1).$$

Here, we generalize the above construction.

Construction: Let $\mathit{a}=\left(a\right(0),a(1),\dots ,a(N-1\left)\right)$ and $\mathit{b}=\left(b\right(0),b(1),\dots ,b(N-1\left)\right)$ be two M-ary sequences of period N and $\mathit{e}=(e_0,e_1,e_2,e_3)$ be an M-ary sequence of length 4. An M-ary sequence $\mathit{u}=u\left(t\right)$ of length $4N$ can be constructed as

$$\mathit{u}=I(\mathit{a}+e_0,L^d\left(\mathit{b}\right)+e_1,L^{2d}\left(\mathit{a}\right)+e_2,L^{3d}\left(\mathit{b}\right)+e_3),$$

(3)

where d is some integer with $4d\equiv 1 (\mathrm{mod} N)$.

We have the following theorem.

Theorem 1. 

Let $\mathit{a}$ and $\mathit{b}$ be two M-ary sequences of period N. For all $0<\tau <4N$, write $\tau =4{\tau }_1+{\tau }_2$ where $(0<{\tau }_1<N$ and ${\tau }_2=0)$ or $(0<{\tau }_1<N$ and $0\le {\tau }_2<4)$. Then, the autocorrelation of a sequence $\mathit{u}$ constructed as in (3) is given by

$$R_{\mathit{u}}\left(\tau \right)=\left\{\begin{array}{cc}2(R_{\mathit{a}}\left({\tau }_1\right)+R_{\mathit{b}}\left({\tau }_1\right)),\hfill & \mathit{if}{\tau }_2=0,\hfill \\ ({\omega }^{e_0-e_1}+{\omega }^{e_2-e_3})R_{\mathit{a},\mathit{b}}({\tau }_1+d)+({\omega }^{e_1-e_2}+{\omega }^{e_3-e_0})R_{\mathit{b},\mathit{a}}({\tau }_1+d),\hfill & \mathit{if}{\tau }_2=1,\hfill \\ ({\omega }^{e_0-e_2}+{\omega }^{e_2-e_0})R_{\mathit{a}}({\tau }_1+2d)+({\omega }^{e_1-e_3}+{\omega }^{e_3-e_1})R_{\mathit{b}}({\tau }_1+2d),\hfill & \mathit{if}{\tau }_2=2,\hfill \\ ({\omega }^{e_0-e_3}+{\omega }^{e_2-e_1})R_{\mathit{a},\mathit{b}}({\tau }_1+3d)+({\omega }^{e_1-e_0}+{\omega }^{e_3-e_2})R_{\mathit{b},\mathit{a}}({\tau }_1+3d).\hfill & \mathit{if}{\tau }_2=3,\hfill \end{array}\right.$$

Proof. 

For the sake of simplicity, we set $\mathit{a}\mathit{b}\left[i\right]=\left\{\begin{array}{cc}\mathit{a},\hfill & \mathrm{if}i\mathrm{is}\mathrm{odd},\hfill \\ \mathit{b},\hfill & \mathrm{if}i\mathrm{is}\mathrm{even},\hfill \end{array}\right.$ and $\mathit{A}\mathit{B}\left[i\right]=\mathit{a}\mathit{b}\left[i\right]+e_i$; then, the sequence $\mathit{u}$ constructed in (3) can be presented as

$$\begin{array}{c}\hfill \mathit{u}=I(L^{0d}\mathit{A}\mathit{B}\left[0\right],L^{1d}\mathit{A}\mathit{B}\left[1\right],L^{2d}\mathit{A}\mathit{B}\left[2\right],L^{3d}\mathit{A}\mathit{B}\left[3\right]).\end{array}$$

Therefore, the autocorrelation of $\mathit{u}$ at shift $\tau =4{\tau }_1+{\tau }_2$ is given by

$$\begin{array}{cc}\hfill R_{\mathit{u}}\left(\tau \right)=& \sum _{i=0}^3{R}_{L^{id}\mathit{AB}\left[i\right],L^{(id+{\tau }_{2}d)-4d{1}_{i+{\tau }_2\ge 4}}\mathit{AB}[i+{\tau }_2]}({\tau }_1+1_{i+{\tau }_2\ge 4})\hfill \\ \hfill =& \sum _{i=0}^3{R}_{\mathit{AB}\left[i\right],\mathit{AB}[i+{\tau }_2]}({\tau }_1+1_{i+{\tau }_2\ge 4}+(id+{\tau }_{2}d)-4d{1}_{i+{\tau }_2\ge 4}-id)\hfill \\ \hfill =& \sum _{i=0}^3{R}_{\mathit{AB}\left[i\right],\mathit{AB}[i+{\tau }_2]}({\tau }_1+{\tau }_{2}d)\hfill \\ \hfill =& \sum _{i=0}^3{\omega }^{e_i-e_{i+{\tau }_2}}{R}_{\mathit{ab}\left[i\right],\mathit{ab}[i+{\tau }_2]}({\tau }_1+{\tau }_{2}d),\hfill \end{array}$$

This is our desired construction. □

2.3. Semi-Interleaved Sequences

We first look at the following example.

Example 1. 

Let $\mathit{a}=\left(a\right(0),a(1),a(2),a(3),a(4\left)\right)$ and $\mathit{b}=\left(b\right(0),b(1),b(2),b(3),b(4\left)\right)$ be two M-ary sequences with period 5, and $d=4$; then, the interleaved sequence

$$\begin{array}{c}\hfill \mathit{u}=I(\mathit{a},L^d\left(\mathit{b}\right),L^{2d}\left(\mathit{a}\right),L^{3d}\left(\mathit{b}\right))=\left(\begin{array}{cccc}a\left(0\right)& b\left(4\right)& a\left(3\right)& b\left(2\right)\\ a\left(1\right)& b\left(0\right)& a\left(4\right)& b\left(3\right)\\ a\left(2\right)& b\left(1\right)& a\left(0\right)& b\left(4\right)\\ a\left(3\right)& b\left(2\right)& a\left(1\right)& b\left(0\right)\\ a\left(4\right)& b\left(3\right)& a\left(2\right)& b\left(1\right)\end{array}\right).\end{array}$$

Clearly, the period of $\mathit{u}$ is $2\times 5$, not $4\times 5$.

Motivated by the above example. We propose the following concept.

Definition 1. 

Let T be a positive even integer, and $\{{\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1}\}$ a set of T sequences of period N. Then, the length of the interleaved sequence $\mathit{u}=I({\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1})=(u\left(0\right),u\left(1\right),\cdots ,u(TN-1))$ is $TN$. The sequence

$${\mathit{u}}_s=(u\left(0\right),u\left(1\right),\cdots ,u({\displaystyle \frac{TN}{2}}-1))$$

is called a semi-interleaved sequence of $I({\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1})$.

Using the construction of semi-interleaved sequences together with (2), we have the following.

Proposition 1. 

Let T be a positive, even integer, and $\{{\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1}\}$ a set of T sequences of period N. If the period of $\mathit{u}=I({\mathit{a}}_0,{\mathit{a}}_1,\dots ,{\mathit{a}}_{T-1})$ is ${\displaystyle \frac{TN}{2}}$, then for $0\le \tau \le {\displaystyle \frac{TN}{2}}-1$, the autocorrelation of the semi-interleaved sequence ${\mathit{u}}_s$ of $\mathit{u}$ is given by

$$R_{{\mathit{u}}_s}\left(\tau \right)=\sum _{i=0}^{T-{\tau }_2-1}{R}_{{\mathit{a}}_i,{\mathit{b}}_{i+{\tau }_2}}\left({\tau }_1\right)+\sum _{i=T-{\tau }_2}^{T-1}{R}_{{\mathit{a}}_i,{\mathit{b}}_{i+{\tau }_2-T}}({\tau }_1+1),$$

(4)

where $\tau ={\tau }_{1}T+{\tau }_2$, $0\le {\tau }_1\le N-1$ and $0\le {\tau }_2\le T-1$.

Proposition 2. 

Let $N=4f+1$, $d=3f+1$, $e_0=e_2$, and $e_1=e_3$. Then, the period of the interleaved sequence $\mathit{u}$ construction using (3) is $2N$.

Proof. 

For $N=4f+1$ and $d=3f+1$, we have $4d\equiv 1 (\mathrm{mod} N)$, $2d=6f+2\equiv 2f+1 (\mathrm{mod} N)$, and $3d=9f+3\equiv f+1 (\mathrm{mod} N)$. Using the notations in the proof of Theorem 1, then the construction in (3) becomes

$$u(4i+j)=\mathit{AB}\left[j\right](dj+i)=\mathit{AB}\left[j\right](dj+4di)=\mathit{AB}[4i+j]\left(d\right(4i+j\left)\right),$$

that is, $\mathit{u}\left(i\right)=\mathit{AB}\left[i\right]\left(di\right)$. Therefore,

$$u(i+2N)=\mathit{AB}[i+2N](di+2dN)=\mathit{AB}\left[i\right]\left(di\right)=u\left(i\right).$$

That is, the period of the sequence $\mathit{u}$ is $2N$. □

Corollary 1. 

Let $\mathit{a}$ and $\mathit{b}$ be two M-ary sequences of period N. For all $0<\tau <2N$, write $\tau =4{\tau }_1+{\tau }_2$ where $(0<{\tau }_1<N$ and ${\tau }_2=0)$ or $(0\le {\tau }_1<N$ and $0<{\tau }_2<4)$. If $e_0=e_2$ and $e_1=e_3$, then the autocorrelation of $\mathit{u}$ constructed by (3) is given by

$$R_{\mathit{u}}\left(\tau \right)=\left\{\begin{array}{cc}{R}_{\mathit{a}}\left({\tau }_1\right)+R_{\mathit{b}}\left({\tau }_1\right),\hfill & \mathit{if}{\tau }_2=0,\hfill \\ {\omega }^{e_0-e_1}{R}_{\mathit{a},\mathit{b}}({\tau }_1+d)+{\omega }^{e_1-e_0}{R}_{\mathit{b},\mathit{a}}({\tau }_1+d),\hfill & \mathit{if}{\tau }_2=1,\hfill \\ R_{\mathit{a}}({\tau }_1+2d)+R_{\mathit{b}}({\tau }_1+2d),\hfill & \mathit{if}{\tau }_2=2,\hfill \\ {\omega }^{e_0-e_1}{R}_{\mathit{a},\mathit{b}}({\tau }_1+3d)+{\omega }^{e_1-e_0}{R}_{\mathit{b},\mathit{a}}({\tau }_1+3d).\hfill & \mathit{if}{\tau }_2=3,\hfill \end{array}\right.$$

In particular, when M is even and $e_0=e_2=0$, $e_1=e_3={\displaystyle \frac{M}{2}}$, then

$$R_{\mathit{u}}\left(\tau \right)=\left\{\begin{array}{cc}{R}_{\mathit{a}}\left({\tau }_1\right)+R_{\mathit{b}}\left({\tau }_1\right),\hfill & \mathit{if}{\tau }_2=0,\hfill \\ -(R_{\mathit{a},\mathit{b}}({\tau }_1+d)+R_{\mathit{b},\mathit{a}}({\tau }_1+d)),\hfill & \mathit{if}{\tau }_2=1,\hfill \\ R_{\mathit{a}}({\tau }_1+2d)+R_{\mathit{b}}({\tau }_1+2d),\hfill & \mathit{if}{\tau }_2=2,\hfill \\ -(R_{\mathit{a},\mathit{b}}({\tau }_1+3d)+R_{\mathit{b},\mathit{a}}({\tau }_1+3d)),\hfill & \mathit{if}{\tau }_2=3.\hfill \end{array}\right.$$

Proof. 

Because the period of sequence $\mathit{u}$ constructed by (3) is $2N$, the results can be obtained by Theorem 1. □

2.4. New Quaternary Sequences

For $M=4$, let $N=4f+1$ be a prime once more, where f is a positive integer. Let $\alpha$ be a primitive element in a finite field $GF\left(N\right)$; let $D_0=\left\{{\alpha }^{4l}\right|l=0,1,\dots ,f-1\}$ and $D_i={\alpha }^i{D}_0$ for $i=1,2,3$.

Again, we have the quadratic partition $N=x^2+4y^2$, where $x\equiv 1 (\mathrm{mod} 4)$ and the sign of y is ambiguous and depends on the choice of $\alpha$.

Let $i=\sqrt{-1}$. Tang and Lindner [7] constructed the following sequences:

$$a_t=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}t\in D_0\cup \left\{0\right\},\hfill \\ 1,\hfill & \mathrm{if}t\in D_1,\hfill \\ 2,\hfill & \mathrm{if}t\in D_2,\hfill \\ 3,\hfill & \mathrm{if}t\in D_3,\hfill \end{array}\right.\mathrm{and}{b}_t=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}t\in D_0\cup \left\{0\right\},\hfill \\ 3,\hfill & \mathrm{if}t\in D_1,\hfill \\ 2,\hfill & \mathrm{if}t\in D_2,\hfill \\ 1,\hfill & \mathrm{if}t\in D_3.\hfill \end{array}\right.$$

(5)

When f is odd, the autocorrelation and cross-correlation of the sequences $\mathit{a}$ and $\mathit{b}$ assumes the following distribution

$$\begin{array}{c}\hfill R_{\mathit{a}}\left(\tau \right)=\left\{\begin{array}{cc}N,\hfill & \mathrm{if}\tau =0,\hfill \\ -1,\hfill & \mathrm{if}\tau \in D_0\cup D_2,\hfill \\ -1-2i,\hfill & \mathrm{if}\tau \in D_1,\hfill \\ -1+2i,\hfill & \mathrm{if}\tau \in D_3,\hfill \end{array}\right.\mathrm{and}{R}_{\mathit{b}}\left(\tau \right)=\left\{\begin{array}{cc}N,\hfill & \mathrm{if}\tau =0,\hfill \\ -1,\hfill & \mathrm{if}\tau \in D_0\cup D_2,\hfill \\ -1+2i,\hfill & \mathrm{if}\tau \in D_1,\hfill \\ -1-2i,\hfill & \mathrm{if}\tau \in D_3.\hfill \end{array}\right.\end{array}$$

Yang and Tang [8] proved that the cross-correlation of the sequences defined in (5) assumes the following distribution.

$$R_{\mathit{a},\mathit{b}}\left(\tau \right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\tau =0,\hfill \\ x-2yi,\hfill & \mathrm{if}\tau \in D_0\cup D_2,\hfill \\ -x+2yi,\hfill & \mathrm{if}\tau \in D_1\cup D_3.\hfill \end{array}\right.$$

Furthermore, they proved that such sequences form optimal pairs.

Lemma 1. 

When $N=4f+1=x^2+4y^2$ and f is odd, then

$$R_{\mathit{b},\mathit{a}}\left(\tau \right)=\left\{\begin{array}{cc}1,\hfill & \mathit{if}\tau =0,\hfill \\ x+2yi,\hfill & \mathit{if}\tau \in D_0\cup D_2,\hfill \\ -x-2yi,\hfill & \mathit{if}\tau \in D_1\cup D_3.\hfill \end{array}\right.$$

Proof. 

For $z\in \mathbb{C}$, let $\overline{z}$ denote its conjugate. Then,

$$R_{\mathit{b},\mathit{a}}\left(\tau \right)=\overline{R_{\mathit{a},\mathit{b}}(-\tau )}.$$

Clearly, for $\tau =0$, $R_{\mathit{b},\mathit{a}}\left(0\right)=1$.

Notice we have $-1={\alpha }^{{\displaystyle \frac{4f}{2}}}={\alpha }^{2f}$. Thus, $-1\in D_2$ since f is odd. For $0<\tau <q$, let $\tau \in D_k$. Then, $-\tau \in D_{k+2}$, whence $R_{\mathit{b},\mathit{a}}\left(\tau \right)=\overline{R_{\mathit{a},\mathit{b}}(-\tau )}$, and the result follows. □

For $M=4$, $\mathit{a}$ and $\mathit{b}$ are the two Tang–Lindner sequences. Applying Corollary 1 to Tang–Linder sequences, we have the following theorem.

Theorem 2. 

Let $\mathit{a}$ and $\mathit{b}$ be the two Tang–Lindner sequences of period N defined as in (5), where $N=4f+1=x^2+4y^2$ and f is odd, $\mathit{e}=\{0,2,0,2\}$ is a quaternary sequence, and ${\mathit{u}}_s$ is the semi-interleaved sequence constructed as in (3). Then, $R_{{\mathit{u}}_s}\left(\tau \right)\in \{\pm 2\}$ for all $1\le \tau <2N$ if and only if $x=\pm 1$.

Example 2. 

Again, let N = 37 and α = 2 be a primitive element of $GF\left(N\right)$. The following four cyclotomic classes can be derived

$$\begin{array}{cc}\hfill D_0=& \{1,7,9,10,12,16,26,33,34\},\hfill \\ \hfill D_1=& \{2,14,15,18,20,24,29,31,32\},\hfill \\ \hfill D_2=& \{3,4,11,21,25,27,28,30,36\},\hfill \\ \hfill D_3=& \{5,6,8,13,17,19,22,23,35\}.\hfill \end{array}$$

In this case, x = 1, y = 3, and f = 9. We generate two sequences using (5) as follows:

$$\begin{array}{cc}\hfill \mathit{a}=& (0,0,1,2,2,3,3,0,3,0,0,2,0,3,1,1,0,3,1,3,1,2,3,3,1,2,0,2,2,1,2,1,1,0,0,3,2),\hfill \\ \hfill \mathit{b}=& (0,0,3,2,2,1,1,0,1,0,0,2,0,1,3,3,0,1,3,1,3,2,1,1,3,2,0,2,2,3,2,3,3,0,0,1,2).\hfill \end{array}$$

Take $\mathit{e}=(0,2,0,2)$, and d = 28. Using the construction in (3), we have the following interleaved sequence:

$$\begin{array}{cc}\hfill \mathit{u}=& I(\mathit{a}+e_0,L^d\left(\mathit{b}\right)+e_1,L^{2d}\left(\mathit{a}\right)+e_2,L^{3d}\left(\mathit{b}\right)+e_3)\hfill \\ \hfill =& (0,0,3,2,0,1,1,0,1,0,2,2,2,1,3,3,2,1,3,1,3,2,1,1,3,2,2,2,0,3,0,3,3,0,\hfill \\ \hfill & 2,1,0,2,2,3,0,2,1,1,2,1,2,0,0,0,1,3,3,0,1,3,1,3,0,1,1,3,0,0,0,2,3,2,\hfill \\ \hfill & 3,3,2,0,1,2,0,0,3,2,0,1,1,0,1,0,2,2,2,1,3,3,2,1,3,1,3,2,1,1,3,2,2,2,\hfill \\ \hfill & 0,3,0,3,3,0,2,1,0,2,2,3,0,2,1,1,2,1,2,0,0,0,1,3,3,0,1,3,1,3,0,1,1,3,\hfill \\ \hfill & 0,0,0,2,3,2,3,3,2,0,1,2).\hfill \end{array}$$

Clearly, the period of $\mathit{u}$ is 74, and the semi-interleaved sequence ${\mathit{u}}_s$ is balanced. By computer experiment, the autocorrelation of ${\mathit{u}}_s$ is given by

$$\begin{array}{cc}\hfill {\left(R_{{\mathit{u}}_s}\left(\tau \right)\right)}_{\tau =1}^{73}=& (-2,-2,-2,-2,2,-2,-2,-2,-2,-2,-2,-2,2,-2,2,-2,2,-2,\hfill \\ \hfill & 2,-2,-2,-2,2,-2,-2,-2,-2,-2,2,-2,2,-2,-2,-2,2,-2,-2,\hfill \\ \hfill & -2,2,-2,-2,-2,2,-2,2,-2,-2,-2,-2,-2,2,-2,-2,-2,2,-2,\hfill \\ \hfill & 2,-2,2,-2,2,-2,-2,-2,-2,-2,-2,-2,2,-2,-2,-2,-2).\hfill \end{array}$$

3. Conclusions

In this paper, we present a method for constructing M-ary sequences of period $4N$ using the interleaved structure given in (3). Motivated by some of the special cases of the construction in (3), we propose the concept of the semi-interleaved sequence. In particular, we obtain quaternary sequences of period $2N$ with a semi-interleaved structure and the nontrivial autocorrelations of the new sequences are 2 and $-2$.