Constructions of Goethals–Seidel Sequences by Using k-Partition

Abstract

In this paper, we are devoted to finding Goethals–Seidel sequences by using k-partition, and based on the finite Parseval relation, the construction of Goethals–Seidel sequences could be transformed to the construction of the associated polynomials. Three different structures of Goethals–Seidel sequences will be presented. We first propose a method based on T-matrices directly to obtain a quad of Goethals–Seidel sequences. Next, by introducing the k-partition, we utilize two classes of 8-partitions to obtain a new class of polynomials still remaining the same (anti)symmetrical properties, with which a quad of Goethals–Seidel sequences could be constructed. Moreover, an adoption of the 4-partition together with a quad of four symmetrical sequences can also lead to a quad of Goethals–Seidel sequences.

1. Introduction

Hadamard matrices (HMs) are applied to many fields, such as cryptography, coding theory and signal processing, and many works are devoted to studying the properties and constructions of them. HMs $H_n$ of order n are the square matrices with entries $\pm 1$ that satisfy $H_n{H}_n^T=n{I}_n$, where $H_n^T$ is the transpose of $H_n$ and $I_n$ is the identity matrix. It is necessary that the order $n>2$ is always divisible by 4. There are many references about HMs and how to construct them, such as [1,2,3,4,5,6,7,8]. According to the definition of HMs, we consider them as block structures. Let an HM be $H_n=\left[{\left(H_{ij}\right)}_m\right]$, where n and m are the orders satisfying $n=km$ with $i,j=1,2,\cdots ,k$, and the block matrices $H_{ij}$ satisfy the following conditions:

$$\begin{array}{ccc}\left(\mathrm{i}\right)\hfill & {\displaystyle \sum _j}{H}_{i_{1}j}{H}_{i_{2}j}^T=n{I}_m,\hfill & i_1=i_2,\hfill \\ \left(\mathrm{ii}\right)\hfill & {\displaystyle \sum _j}{H}_{i_{1}j}{H}_{i_{2}j}^T=0,\hfill & i_1\ne i_2.\hfill \end{array}$$

(1)

One alternative method to obtain these matrix blocks is making use of the circulant matrices, such as Williamson array, Goethals–Seidel (GS) array, Kharaghani array, and so on [6,9].

In the earlier works on constructing HMs using circulant matrix blocks, it was found that some skew-HMs did not exist, even if for small orders, such as 36 [6]. To overcome this difficulty, Goethals and Seidel [10] designed the GS array

$$\begin{array}{c}\hfill G=\left(\begin{array}{cccc}A& BR& CR& DR\\ -BR& A& D^{T}R& -C^{T}R\\ -CR& -D^{T}R& A& B^{T}R\\ -DR& C^{T}R& -B^{T}R& A,\end{array}\right)\end{array}$$

(2)

where $A,B,C,D$ are four circulant matrices, and R denotes the back-diagonal identity matrix. Obviously the block matrices of GS array (2) satisfy the condition (1). Then, it is meaningful to investigate this array in the construction of HMs. In addition, compared with the Williamson array, the GS array has no requirements of symmetries, which will be relatively friendly to constructions. In this paper, we mainly focus on the construction of GS array and transform it into the construction of first rows of circulant matrices $A,B,C,D$ through the finite Parseval relation [11]. In particular, we call first rows of these circulant matrices as a quad of GS sequences.

Goethals and Seidel [10] studied the HMs of GS type of order $4n$, where they obtained a quad of sequences with order $n=9,13$, and leave $n=23$ the only unconstructed order less than 25. Later, Doković [12,13] searched them by computer and obtained the GS array of order $4n$, $n=37,43,49,\cdots ,163$. More orders of GS array about Doković’s results can be referred to [14,15,16,17,18,19]. In [20], Fletcher et al. found a GS array of order $4n=36$. Furthermore, in [21,22], a family of GS sequences with order $n=q+1$ was constructed by using the Parseval relation in Galois field $GF\left(q^2\right)$ theoretically, where $q\equiv 3\left(mod8\right)$ is a prime power. In addition, using T-matrices or T-sequences can also lead to GS sequences. In [23], Yang discovered that, if a quad of Williamson sequences of order n and T-sequences of order m exist, then a quad of GS sequences of order $mn$ also exists. Yang also presented some new infinite families of GS sequences by utilizing T-sequences and Williamson sequences in [24,25,26,27]. By the way, Williamson sequences can be considered as a special case of GS sequences, and Whiteman [11] obtained a quad of Williamson sequences of order $(q+1)/2$ for $q\equiv 1\left(mod4\right)$ being a prime power.

In the literature mentioned above, two sequences used in [11] are actually a 2-partition. Four sequences in [21,22] and Yang’s works are based on a 4-partition or the combination of 4-partitions. Thus, it is natural to study whether more GS sequences could be constructed by more partitions. In this paper, we mainly focus on the structures of GS sequences by using symmetrical and antisymmetrical k-partition, and eventually obtain three quads of GS sequences with different structures. There are two main reasons why we adopt symmetrical and antisymmetrical k-partition: it facilitates for us to reduce the range of searching k-partitions, and a k-partition with these properties is friendly to the construction. Then, the first method is based on the T-matrices, and a quad of GS sequences could be constructed directly, including the Williamson sequences as the special cases. The second way is utilizing two classes of 8-th partitions, where half are symmetrical and others are antisymmetrical, to generate a new class of polynomials. In this process, this class of polynomials would remain the same properties of symmetry and antisymmetry as two original classes of 8-partitions, and could lead to a quad of GS sequences too. Finally, we employ the 4-partition as well as a quad of Williamson sequences to obtain a quad of GS sequences. Since constructing Hadamard matrices of GS type are now transformed into finding GS sequences, we in this paper offer some alternative approaches, which include finding some appropriate 8-partitions or novel 4-partitions instead of T-matrix sequences.

The rest of this paper is organized as follows: In Section 2, we mainly introduce some notations and definitions used later. In Section 3, the definition of k-partition will be given, with which we rigorously prove that 8-partitions or the combination of 4-partition and Williamson sequences could lead to GS sequences. Some conclusions will be drawn in Section 4.

2. Preliminaries

2.1. Parseval Relation

Let $\mathit{a}=(a_0,a_1,\cdots ,a_{n-1})$ be a sequence, whose periodic autocorrelation function $R_{\mathit{a}}\left(\tau \right)$ is defined as

$$R_{\mathit{a}}\left(\tau \right)=\sum _{i=0}^{n-1}{a}_i{\overline{a}}_{i+\tau },\tau =0,1,\cdots ,n-1,$$

(3)

where ${\overline{a}}_i$ is the conjugate of $a_i$, and the subscript $i+\tau$ is evaluated modulo-n. The polynomial

$${\mathsf{\Phi }}_{\mathit{a}}\left(\xi \right)=a_0+a_1\xi +a_2{\xi }^2+\cdots +a_{n-1}{\xi }^{n-1}$$

(4)

is called the associated polynomial of sequence $\mathit{a}$, where $\xi$ is the n-th root of unity. The finite Parseval relation [11] (also the Wiener–Khinchin theorem [28,29]) between $R_{\mathit{a}}\left(\tau \right)$ and ${\mathsf{\Phi }}_{\mathit{a}}\left(\xi \right)$ is presented in the following identity:

$$R_{\mathit{a}}\left(\tau \right)=\frac{1}{n}\sum _{j=0}^{n-1}{\parallel {\mathsf{\Phi }}_{\mathit{a}}\left({\xi }^j\right)\parallel }^2{\xi }^{j\tau },\tau =0,1,\cdots ,n-1,$$

(5)

with its inverse form

$$\parallel {\mathsf{\Phi }}_{\mathit{a}}\left({\xi }^j\right){\parallel }^2={\mathsf{\Phi }}_{\mathit{a}}\left({\xi }^j\right)\overline{{\mathsf{\Phi }}_{\mathit{a}}\left({\xi }^j\right)}=\sum _{\tau =0}^{n-1}{R}_{\mathit{a}}\left(\tau \right){\xi }^{-j\tau },j=0,1,\cdots ,n-1.$$

(6)

For the HMs in the form of GS type (2), four circulant matrices therein have the following property.

Lemma 1 

([11]). Let $A,B,C$ and D denote the four circulant matrices of order n whose first rows are four sequences $\mathit{a}={\left\{a_i\right\}}_{i=0}^{n-1},\mathit{b}={\left\{b_i\right\}}_{i=0}^{n-1},\mathit{c}={\left\{c_i\right\}}_{i=0}^{n-1}$ and $\mathit{d}={\left\{d_i\right\}}_{i=0}^{n-1}$, respectively. Then, $A{A}^T+B{B}^T+C{C}^T+D{D}^T=4n{I}_n$ if and only if

$$\parallel {\mathsf{\Phi }}_{\mathit{a}}\left({\xi }^j\right){\parallel }^2+\parallel {\mathsf{\Phi }}_{\mathit{b}}\left({\xi }^j\right){\parallel }^2+\parallel {\mathsf{\Phi }}_{\mathit{c}}\left({\xi }^j\right){\parallel }^2+{\parallel {\mathsf{\Phi }}_{\mathit{d}}\left({\xi }^j\right)\parallel }^2=4n,$$

(7)

where ξ is the n-th root of unity and $j=0,1,\cdots ,n-1$.

Proof. 

It follows immediately from (1) and (6). □

Remark 1. 

By Lemma 1, the relationship is now transformed from the circulant matrices to the associated polynomials.

In the later statements, when the polynomials, sequences, and the coefficients appear in the same place, without special clarifications, we denote by the capital letter, for example, $F_i\left(\xi \right)$ polynomials, the bold letter ${\mathit{f}}_i$ sequences and the lower case letter $f_{ij}$ corresponding coefficients, where i and j rely on different cases.

2.2. GS Sequences

Definition 1 

(GS sequences, [23]). Four $\pm 1$ sequences ${\mathit{q}}_i=(q_{i0},q_{i1},\cdots ,q_{i,n-1}),i=1,2,3,4$ are said to be a quad of GS sequences, if their associated polynomials $Q_i\left(\xi \right)=q_{i0}+q_{i1}\xi +\cdots +q_{i,n-1}{\xi }^{n-1}$ satisfy

$$\sum _{i=1}^4{\parallel Q_i\left(\xi \right)\parallel }^2=4n,$$

(8)

where ξ is the n-th root of unity.

As a special case of the GS sequences, a quad of Williamson sequences will be defined analogously, which requires the symmetry additionally.

Definition 2 

(Williamson sequences). Four $\pm 1$ sequences ${\mathit{w}}_i=(w_{i0},\cdots ,w_{i,n-1})$, $i=1,2,3,4$, are said to be a quad of Williamson sequences, if their associated polynomials $W_i\left(\xi \right)=w_{i0}+w_{i1}\xi +\cdots +w_{i,n-1}{\xi }^{n-1}$ satisfy

$$\sum _{i=1}^4{\parallel W_i\left(\xi \right)\parallel }^2=4n,\overline{W_i\left(\xi \right)}=W_i\left(\xi \right),$$

(9)

where ξ is the n-th root of unity.

2.3. T-Matrices and T-Matrix Sequences

Definition 3 

(T-matrices, [6,30]). Four $(0,1,-1)$ circulant matrices $T_1,T_2,T_3$ and $T_4$ of order m are T-matrices if they satisfy the following conditions:

$$\begin{array}{c}\hfill \begin{array}{ccc}\left(\mathrm{i}\right)\hfill & T_i*T_j^T=0,\hfill & i\ne jandi,j=1,2,3,4,\hfill \\ \left(\mathrm{ii}\right)\hfill & {\displaystyle \sum _{i=1}^4}{T}_i{T}_i^T=m{I}_m.\hfill \end{array}\end{array}$$

where * denotes the Hadamard product.

According to the definition of T-matrices, we define a quad of T-matrix sequences.

Definition 4 

(T-matrix sequences). Let ${\mathit{t}}_i=(t_{i0},t_{i1},\cdots ,t_{i,m-1})$, $i=1,2,3,4$ be the first rows of T-matrices satisfying ${\displaystyle \sum _{i=1}^4}\left|t_{ij}\right|=1,j=0,\cdots ,m-1$. Then, we call ${\mathit{t}}_{\mathit{i}}$ as a quad of T-matrix sequences (TMS).

We give two examples of TMS: for $n=6$,

$$\begin{array}{cc}{\mathit{t}}_1=(1,0,0,0,0,0),\hfill & {\mathit{t}}_2=(0,0,-1,0,-1,0),\hfill \\ {\mathit{t}}_3=(0,0,0,0,0,0),\hfill & {\mathit{t}}_4=(0,1,0,-1,0,1),\hfill \end{array}$$

and for $n=8$,

$$\begin{array}{cc}{\mathit{t}}_1=(1,0,0,0,1,0,0,0),\hfill & {\mathit{t}}_2=(0,-1,0,1,0,1,0,-1),\hfill \\ {\mathit{t}}_3=(0,0,-1,0,0,0,-1,0),\hfill & {\mathit{t}}_4=(0,0,0,0,0,0,0,0).\hfill \end{array}$$

Let ${\mathbb{T}}_i\left(\xi \right)=t_{i0}+t_{i1}\xi +\cdots +t_{i,m-1}{\xi }^{m-1}$ be the associated polynomials of sequences ${\mathit{t}}_i$, $i=1,2,3,4$, with $\xi$ being the m-th root of unity. Then, from Definition 3 and Definition 4, we have the following results.

Lemma 2. 

Let polynomials ${\mathbb{T}}_1\left(\xi \right),{\mathbb{T}}_2\left(\xi \right),{\mathbb{T}}_3\left(\xi \right)$ and ${\mathbb{T}}_4\left(\xi \right)$ be the associated polynomials of TMS ${\mathit{t}}_{\mathit{i}}=(t_{i0},\cdots ,t_{i,m-1}),i=1,2,3,4$, and then the sum ${\displaystyle \sum _{i=1}^4}(\pm {\mathbb{T}}_i\left(\xi \right))$ determines a polynomial whose coefficients are $\pm 1$. Moreover, it holds that

$$\parallel {\mathbb{T}}_1{\left(\xi \right)\parallel }^2+\parallel {\mathbb{T}}_2{\left(\xi \right)\parallel }^2+\parallel {\mathbb{T}}_3{\left(\xi \right)\parallel }^2+{\parallel {\mathbb{T}}_4\left(\xi \right)\parallel }^2=m,$$

(10)

where ξ is the m-th root of unity.

Proof. 

It follows directly from the Parseval relation (5) and Definition 4 of TMS. □

A quad of GS sequences could be obtained directly by utilizing TMS.

Lemma 3 

([31]). Let ${\mathbb{T}}_1\left(\xi \right),{\mathbb{T}}_2\left(\xi \right),{\mathbb{T}}_3\left(\xi \right),{\mathbb{T}}_4\left(\xi \right)$ of order m be the associated polynomials of TMS ${\mathit{t}}_1,{\mathit{t}}_2,{\mathit{t}}_3,{\mathit{t}}_4$. The coefficients of the following four polynomials:

$$\begin{array}{cc}\hfill Q_1\left(\xi \right)& =-{\mathbb{T}}_1\left(\xi \right)+{\mathbb{T}}_2\left(\xi \right)+{\mathbb{T}}_3\left(\xi \right)+{\mathbb{T}}_4\left(\xi \right),\hfill \\ \hfill Q_2\left(\xi \right)& ={\mathbb{T}}_1\left(\xi \right)-{\mathbb{T}}_2\left(\xi \right)+{\mathbb{T}}_3\left(\xi \right)+{\mathbb{T}}_4\left(\xi \right),\hfill \\ \hfill Q_3\left(\xi \right)& ={\mathbb{T}}_1\left(\xi \right)+{\mathbb{T}}_2\left(\xi \right)-{\mathbb{T}}_3\left(\xi \right)+{\mathbb{T}}_4\left(\xi \right),\hfill \\ \hfill Q_4\left(\xi \right)& ={\mathbb{T}}_1\left(\xi \right)+{\mathbb{T}}_2\left(\xi \right)+{\mathbb{T}}_3\left(\xi \right)-{\mathbb{T}}_4\left(\xi \right),\hfill \end{array}$$

i.e., ${\mathit{q}}_1,{\mathit{q}}_2,{\mathit{q}}_3,{\mathit{q}}_4$, are a quad of GS sequences, where ξ is the m-th root of unity.

Now, we present a novel construction of a quad of GS sequences different from the method by using TMS in Lemma 3.

Theorem 1. 

For four sequences ${\mathit{h}}_i=(h_{i0},h_{i1},\cdots ,h_{i,n-1})$ consisting of $\{0,1,-1\}$ with associated polynomials $H_i\left(\xi \right)=h_{i0}+h_{i1}\xi \cdots +h_{i,n-1}{\xi }^{n-1}$, where ξ is the n-th root of unity, $i=1,2,3,4$, if they satisfy the conditions

$$\begin{array}{c}\hfill \begin{array}{cc}\left(\mathrm{i}\right)\hfill & h_{i0}=0,i=1,2,3,4,\hfill \\ \left(\mathrm{ii}\right)\hfill & |h_{1k}|+|h_{2k}|+|h_{3k}|+|h_{4k}|=1,k=1,\cdots ,n-1,\hfill \\ \left(\mathrm{iii}\right)\hfill & {\displaystyle \sum _{i=1}^4}{\parallel H_i\left(\xi \right)+\frac{1}{2}\parallel }^2=n,\hfill \end{array}\end{array}$$

(11)

then there exists a quad of GS sequences whose associated polynomials $Q_1\left(\xi \right),\cdots ,Q_4\left(\xi \right)$ satisfy

$$\begin{array}{cc}\hfill Q_1\left(\xi \right)& =1-H_1\left(\xi \right)+H_2\left(\xi \right)+H_3\left(\xi \right)+H_4\left(\xi \right),\hfill \\ \hfill Q_2\left(\xi \right)& =1+H_1\left(\xi \right)-H_2\left(\xi \right)+H_3\left(\xi \right)+H_4\left(\xi \right),\hfill \\ \hfill Q_3\left(\xi \right)& =1+H_1\left(\xi \right)+H_2\left(\xi \right)-H_3\left(\xi \right)+H_4\left(\xi \right),\hfill \\ \hfill Q_4\left(\xi \right)& =1+H_1\left(\xi \right)+H_2\left(\xi \right)+H_3\left(\xi \right)-H_4\left(\xi \right).\hfill \end{array}$$

(12)

Proof. 

It is easy to verify that the coefficients of $Q_i\left(\xi \right)$ are $\pm 1$ due to conditions (i) and (ii) of (11). Additionally, from (iii) of (11), we have

$$\begin{array}{cc}\hfill & \parallel Q_1{\left(\xi \right)\parallel }^2+\parallel Q_2{\left(\xi \right)\parallel }^2+\parallel Q_3{\left(\xi \right)\parallel }^2+{\parallel Q_4\left(\xi \right)\parallel }^2\hfill \\ \hfill =& 4+4(\parallel H_1\left(\xi \right){\parallel }^2+\parallel H_2\left(\xi \right){\parallel }^2+\parallel H_3\left(\xi \right){\parallel }^2+\parallel H_4\left(\xi \right){\parallel }^2)\hfill \\ \hfill & +2\left(\overline{H_1\left(\xi \right)}+\overline{H_2\left(\xi \right)}+\overline{H_3\left(\xi \right)}+\overline{H_4\left(\xi \right)}\right)+2\left(H_1\left(\xi \right)+H_2\left(\xi \right)+H_3\left(\xi \right)+H_4\left(\xi \right)\right)\hfill \\ \hfill =& 4(\parallel H_1\left(\xi \right)+\frac{1}{2}{\parallel }^2+\parallel H_2\left(\xi \right)+\frac{1}{2}{\parallel }^2+\parallel H_3\left(\xi \right)+\frac{1}{2}{\parallel }^2+\parallel H_4\left(\xi \right)+\frac{1}{2}{\parallel }^2)\hfill \\ \hfill =& 4n,\hfill \end{array}$$

(13)

which completes the proof. □

Two examples of GS sequences as Definition 1 are presented to verify this theorem: for $n=11$,

$$\begin{array}{cc}{\mathit{h}}_1=(0,1,-1,1,0,0,0,0,0,0,0),\hfill & {\mathit{h}}_2=(0,0,0,0,-1,0,0,1,0,-1,0),\hfill \\ {\mathit{h}}_3=(0,0,0,0,0,0,0,0,1,0,1),\hfill & {\mathit{h}}_4=(0,0,0,0,0,-1,-1,0,0,0,0),\hfill \end{array}$$

which lead to the GS sequences of order 11

$$\begin{array}{cc}{\mathit{q}}_1=(1,-1,1,-1,-1,-1,-1,1,1,-1,1),\hfill & {\mathit{q}}_2=(1,1,-1,1,1,-1,-1,-1,1,1,1),\hfill \\ {\mathit{q}}_3=(1,1,-1,1,-1,-1,-1,1,-1,-1,-1),\hfill & {\mathit{q}}_4=(1,1,-1,1,-1,1,1,1,1,-1,1),\hfill \end{array}$$

and for $n=13$,

$$\begin{array}{cc}{\mathit{h}}_1=(0,1,-1,1,1,1,0,0,0,0,0,0,0),\hfill & {\mathit{h}}_2=(0,0,0,0,0,0,-1,1,0,0,0,0,0),\hfill \\ {\mathit{h}}_3=(0,0,0,0,0,0,0,0,-1,0,-1,0,1),\hfill & {\mathit{h}}_4=(0,0,0,0,0,0,0,0,0,-1,0,1,0),\hfill \end{array}$$

with which we obtain the GS sequences of order 13

$$\begin{array}{cc}\hfill {\mathit{q}}_1& =(1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,1,1),\hfill \\ \hfill {\mathit{q}}_2& =(1,1,-1,1,1,1,1,-1,-1,-1,-1,1,1),\hfill \\ \hfill {\mathit{q}}_3& =(1,1,-1,1,1,1,-1,1,1,-1,1,1,-1),\hfill \\ \hfill {\mathit{q}}_4& =(1,1,-1,1,1,1,-1,1,-1,1,-1,-1,1).\hfill \end{array}$$

Remark 2. 

Theorem 1 provides a method to construct GS sequences by using T-matrix type sequences. Note that, for $i=1,2,3,4$, if polynomials $H_i\left(\xi \right)$ in (12) are symmetrical, the coefficients of $Q_i\left(\xi \right)$ are actually a quad of Williamson sequences as Definition 2.

3. Main Results

Even if the conditions of a quad of GS sequences are weaker than Williamson sequences, actually it is still not easy to construct GS sequences directly. In this section, two indirect methods will be proposed, which utilize the properties of symmetry and antisymmetry together with some sequences known beforehand.

Definition 5 

(Symmetry and antisymmetry). Let $F_i\left(\xi \right)$ be a polynomial with coefficients ${\mathit{f}}_i=(f_{i0},\cdots ,f_{i,n-1})$. $F_i\left(\xi \right)$ being symmetrical (or antisymmetrical) if they satisfy

$$\overline{F_i\left(\xi \right)}=F_i\left(\xi \right)(or\overline{F_i\left(\xi \right)}=-F_i\left(\xi \right)),$$

where ξ is the n-th root of unity. That is, the coefficients $(f_{i0},\cdots ,f_{i,n-1})$ satisfy $f_{ij}=f_{i,n-j}$ ($f_{ij}=-f_{i,n-j}$, respectively), $j=1,2,\cdots ,n-1$.

Lemma 4. 

Given polynomials $G_i\left(\xi \right)=g_{i0}+g_{i1}\xi +...+g_{i,n-1}{\xi }^{n-1}$ for $i=1,\cdots ,8$, $G_1\left(\xi \right),G_2\left(\xi \right)$, $G_3\left(\xi \right)$, $G_4\left(\xi \right)$ are symmetrical, and $G_5\left(\xi \right),G_6\left(\xi \right),G_7\left(\xi \right)$, $G_8\left(\xi \right)$ are antisymmetrical, where ξ is the n-th root of unity. Then, there exist four polynomials $F_1\left(\xi \right),F_2\left(\xi \right),F_3\left(\xi \right),F_4\left(\xi \right)$

$$\begin{array}{cc}\hfill F_1\left(\xi \right)& =G_1\left(\xi \right)-G_2\left(\xi \right)-G_3\left(\xi \right)-G_4\left(\xi \right)+G_5\left(\xi \right)+G_6\left(\xi \right)+G_7\left(\xi \right)+G_8\left(\xi \right),\hfill \\ \hfill F_2\left(\xi \right)& =-G_1\left(\xi \right)+G_2\left(\xi \right)-G_3\left(\xi \right)-G_4\left(\xi \right)+G_5\left(\xi \right)-G_6\left(\xi \right)-G_7\left(\xi \right)+G_8\left(\xi \right),\hfill \\ \hfill F_3\left(\xi \right)& =-G_1\left(\xi \right)-G_2\left(\xi \right)+G_3\left(\xi \right)-G_4\left(\xi \right)-G_5\left(\xi \right)+G_6\left(\xi \right)-G_7\left(\xi \right)+G_8\left(\xi \right),\hfill \\ \hfill F_4\left(\xi \right)& =-G_1\left(\xi \right)-G_2\left(\xi \right)-G_3\left(\xi \right)+G_4\left(\xi \right)-G_5\left(\xi \right)-G_6\left(\xi \right)+G_7\left(\xi \right)+G_8\left(\xi \right),\hfill \end{array}$$

(14)

satisfying

$$\begin{array}{cc}\hfill \sum _{i=1}^4{\parallel F_i\left(\xi \right)\parallel }^2& =\sum _{i=1}^4{F}_i\left(\xi \right)\overline{F_i\left(\xi \right)}=4\sum _{i=1}^8{\parallel G_i\left(\xi \right)\parallel }^2.\hfill \end{array}$$

(15)

Proof. 

Duo to the properties of symmetry and antisymmetry, we have $G_i\left(\xi \right)=\overline{G_i\left(\xi \right)}$, $i=1,2,3,4$ and $G_i\left(\xi \right)=-\overline{G_i\left(\xi \right)},i=5,6,7,8$, which leads to the result after some tedious calculation. □

Lemma 4 implies that the construction of polynomials ${\left\{F_i\right\}}_{i=1}^4$ could be changed to find some appropriate polynomials ${\left\{G_i\right\}}_{i=1}^8$. In order to obtain new GS sequences, we now introduce the definition of a k-partition which is a special case of L-matrices ([6], Definition 4.15), i.e., the appropriate polynomials ${\left\{G_i\right\}}_{i=1}^8$ we find.

Definition 6 

(k-partition). If polynomials $G_i\left(\xi \right)=g_{i0}+g_{i1}\xi +\cdots +g_{i,n-1}{\xi }^{n-1}$, $i=1,\cdots ,k$, satisfy

$$\begin{array}{ccc}\left(\mathrm{i}\right)\hfill & g_{ij}\in \{0,1,-1\},\hfill & i,j=0,1,\cdots ,n-1,\hfill \\ \left(\mathrm{ii}\right)\hfill & {\displaystyle \sum _{i=1}^k}\left|g_{ij}\right|=1,\hfill & j=0,1,\cdots ,n-1,\hfill \\ \left(\mathrm{iii}\right)\hfill & {\displaystyle \sum _{i=1}^k}{\parallel G_i\left(\xi \right)\parallel }^2=n,\hfill \end{array}$$

(16)

where ξ is the n-th root of unity, then we call ${\left\{G_i\left(\xi \right)\right\}}_{i=1}^k$ a k-partition of sequences with $\pm 1$, abbreviation as k-partition without confusion.

We show some examples of 8-partition as Definition 6: for $n=10$

$$\begin{array}{cc}{\mathit{g}}_{\mathbf{1}}=(1,0,0,0,-1,0,-1,0,0,0),\hfill & {\mathit{g}}_{\mathbf{2}}=(0,0,-1,0,0,0,0,0,-1,0),\hfill \\ {\mathit{g}}_{\mathbf{3}}=(0,0,0,0,0,-1,0,0,0,0),\hfill & {\mathit{g}}_{\mathbf{4}}=(0,0,0,1,0,0,0,1,0,0),\hfill \\ {\mathit{g}}_{\mathbf{5}}=(0,1,0,0,0,0,0,0,0,-1),\hfill & {\mathit{g}}_{\mathbf{6}}=(0,0,0,0,0,0,0,0,0,0),\hfill \\ {\mathit{g}}_{\mathbf{7}}=(0,0,0,0,0,0,0,0,0,0),\hfill & {\mathit{g}}_{\mathbf{8}}=(0,0,0,0,0,0,0,0,0,0),\hfill \end{array}$$

(17)

and for $n=12$

$$\begin{array}{cc}{\mathit{g}}_{\mathbf{1}}=(1,0,0,0,0,0,-1,0,0,0,0,0),\hfill & {\mathit{g}}_{\mathbf{2}}=(0,0,0,1,0,0,0,0,0,1,0,0),\hfill \\ {\mathit{g}}_{\mathbf{3}}=(0,1,0,0,0,0,0,0,0,0,0,1),\hfill & {\mathit{g}}_{\mathbf{4}}=(0,0,1,0,0,0,0,0,0,0,1,0),\hfill \\ {\mathit{g}}_{\mathbf{5}}=(0,0,0,0,1,0,0,0,-1,0,0,0),\hfill & {\mathit{g}}_{\mathbf{6}}=(0,0,0,0,0,1,0,-1,0,0,0,0),\hfill \\ {\mathit{g}}_{\mathbf{7}}=(0,0,0,0,0,0,0,0,0,0,0,0),\hfill & {\mathit{g}}_{\mathbf{8}}=(0,0,0,0,0,0,0,0,0,0,0,0).\hfill \end{array}$$

(18)

Comparing Definition 4 with Definition 6, it is easy to see that actually T-matrices sequences are the special cases of 4-partitions. As an extension, we will next investigate how to utilize 8-th partitions to obtain more GS sequences with different structures and orders.

Corollary 1. 

Let $\left\{G_i\left(\xi \right)\right\}$ be an 8-partition, where $G_1\left(\xi \right),\cdots ,G_4\left(\xi \right)$ are symmetrical and$G_5\left(\xi \right),...,G_8\left(\xi \right)$ are antisymmetrical. Then, for $F_i\left(\xi \right)$ defined in (14), the coefficient sequences ${\mathit{f}}_i$ of associated polynomials $F_i\left(\xi \right)$, $i=1,2,3,4$, make up a quad of GS sequences.

Proof. 

Since $\left\{G_i\left(\xi \right)\right\}$ is an 8-partition, together with the statement (ii) of (16), the coefficients of $F_i\left(\xi \right)$ consist of $\pm 1$. Combining the fact

$$\sum _{i=1}^4\parallel F_i{\left(\xi \right)\parallel }^2=4\sum _{i=1}^8{\parallel G_i\left(\xi \right)\parallel }^2=4n$$

with the definition of GS sequences (1), we arrive at the result. □

According to two 8-partition of length $n=10$ (17) and $n=12$ (18), we obtain two quad of GS sequences as Definition 1 based on the construction in Lemma 4. For $n=10$, the GS sequences are

$$\begin{array}{cc}{\mathit{f}}_1=(1,1,1,-1,-1,1,-1,-1,1,-1),\hfill & {\mathit{f}}_2=(-1,1,-1,-1,1,1,1,-1,-1,-1),\hfill \\ {\mathit{f}}_3=(-1,-1,1,-1,1,-1,1,-1,1,1),\hfill & {\mathit{f}}_4=(-1,-1,1,1,1,1,1,1,1,1),\hfill \end{array}$$

and, for $n=12$, the GS sequences are

$$\begin{array}{cc}\hfill {\mathit{f}}_1=& (1,-1,-1,-1,1,1,-1,-1,-1,-1,-1,-1),\hfill \\ \hfill {\mathit{f}}_2=& (-1,-1,-1,1,1,-1,1,1,-1,1,-1,-1),\hfill \\ \hfill {\mathit{f}}_3=& (-1,1,-1,-1,-1,1,1,-1,1,-1,-1,1),\hfill \\ \hfill {\mathit{f}}_4=& (-1,-1,1,-1,-1,-1,1,1,1,-1,1,-1).\hfill \end{array}$$

Remark 3. 

Corollary 1 is a direct conclusion of Lemma 4, and indicates that we now could turn to find an 8-partition instead of a quad of GS sequences.

Theorem 2. 

Let two classes of polynomials ${\left\{G_i\left({\xi }^m\right)\right\}}_{i=1}^8$ and ${\left\{E_i\left({\xi }^n\right)\right\}}_{i=1}^8$ be symmetrical for $i=1,2,3,4$ and antisymmetrical for $i=5,6,7,8$. Both $\left\{G_i\left({\xi }^m\right)\right\}$ and $\left\{E_i\left({\xi }^n\right)\right\}$ are 8-partitions, where ξ is the $mn$-th root of unity with $(m,n)=1$. Define eight polynomials ${\left\{L_i\left(\xi \right)\right\}}_{i=1}^8$ by these two classes of polynomials as

$$\begin{array}{cc}\hfill L_1\left(\xi \right)=& G_1\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_2\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_3\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_4\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_5\left({\xi }^n\right)+G_6\left({\xi }^m\right)E_6\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_7\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_8\left({\xi }^n\right),\hfill \\ \hfill L_2\left(\xi \right)=& G_1\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_3\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_6\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_5\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_8\left({\xi }^n\right)-G_8\left({\xi }^m\right)E_7\left({\xi }^n\right),\hfill \\ \hfill L_3\left(\xi \right)=& G_1\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_2\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_7\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_8\left({\xi }^n\right)-G_7\left({\xi }^m\right)E_5\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_6\left({\xi }^n\right),\hfill \\ \hfill L_4\left(\xi \right)=& G_1\left({\xi }^m\right)E_4\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_1\left({\xi }^n\right)\hfill \\ \hfill & -G_5\left({\xi }^m\right)E_8\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_7\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_6\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_5\left({\xi }^n\right),\hfill \\ \hfill L_5\left(\xi \right)=& G_1\left({\xi }^m\right)E_5\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_6\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_7\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_8\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_1\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_7\left({\xi }^m\right)E_3\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_4\left({\xi }^n\right),\hfill \\ \hfill L_6\left(\xi \right)=& G_1\left({\xi }^m\right)E_6\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_5\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_8\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_7\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_2\left({\xi }^n\right)+G_6\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_4\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_3\left({\xi }^n\right),\hfill \\ \hfill L_7\left(\xi \right)=& G_1\left({\xi }^m\right)E_7\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_8\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_5\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_6\left({\xi }^n\right)\hfill \\ \hfill & +G_5\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_4\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_1\left({\xi }^n\right)-G_8\left({\xi }^m\right)E_2\left({\xi }^n\right),\hfill \\ \hfill L_8\left(\xi \right)=& G_1\left({\xi }^m\right)E_8\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_7\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_6\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_5\left({\xi }^n\right)\hfill \\ \hfill & -G_5\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_6\left({\xi }^m\right)E_3\left({\xi }^n\right)+G_7\left({\xi }^m\right)E_2\left({\xi }^n\right)+G_8\left({\xi }^m\right)E_1\left({\xi }^n\right).\hfill \end{array}$$

(19)

Then $L_i\left(\xi \right)$ are symmetrical for $i=1,2,3,4$ and antisymmetrical for $i=5,6,7,8$, and the coefficients of the following polynomials:

$$\begin{array}{cc}\hfill Q_1\left(\xi \right)& =L_1\left(\xi \right)-L_2\left(\xi \right)-L_3\left(\xi \right)-L_4\left(\xi \right)+L_5\left(\xi \right)+L_6\left(\xi \right)+L_7\left(\xi \right)+L_8\left(\xi \right),\hfill \\ \hfill Q_2\left(\xi \right)& =-L_1\left(\xi \right)+L_2\left(\xi \right)-L_3\left(\xi \right)-L_4\left(\xi \right)+L_5\left(\xi \right)-L_6\left(\xi \right)-L_7\left(\xi \right)+L_8\left(\xi \right),\hfill \\ \hfill Q_3\left(\xi \right)& =-L_1\left(\xi \right)-L_2\left(\xi \right)+L_3\left(\xi \right)-L_4\left(\xi \right)-L_5\left(\xi \right)+L_6\left(\xi \right)-L_7\left(\xi \right)+L_8\left(\xi \right),\hfill \\ \hfill Q_4\left(\xi \right)& =-L_1\left(\xi \right)-L_2\left(\xi \right)-L_3\left(\xi \right)+L_4\left(\xi \right)-L_5\left(\xi \right)-L_6\left(\xi \right)+L_7\left(\xi \right)+L_8\left(\xi \right),\hfill \end{array}$$

(20)

make up a quad of GS sequences.

Proof. 

Due to

$$\begin{array}{c}\hfill \begin{array}{ccc}\overline{G_i\left({\xi }^m\right)}=G_i\left({\xi }^m\right),\hfill & \overline{E_i\left({\xi }^n\right)}=E_i\left({\xi }^n\right),\hfill & i=1,2,3,4,\hfill \\ \overline{G_i\left({\xi }^m\right)}=-G_i\left({\xi }^m\right),\hfill & \overline{E_i\left({\xi }^n\right)}=-E_i\left({\xi }^n\right),\hfill & i=5,6,7,8,\hfill \end{array}\end{array}$$

correspondingly, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\overline{L_i\left(\xi \right)}=L_i\left(\xi \right),\hfill & i=1,2,3,4,\hfill \\ \overline{L_i\left(\xi \right)}=-L_i\left(\xi \right),\hfill & i=5,6,7,8.\hfill \end{array}\end{array}$$

Since $\left\{E_i\left({\xi }^n\right)\right\}$ and $\left\{G_i\left({\xi }^m\right)\right\}$ are 8-partitions, it holds that

$$\begin{array}{c}\hfill \sum _{i=1}^8\parallel E_i\left({\xi }^n\right){\parallel }^2=m\mathrm{and}\sum _{i=1}^8{\parallel G_i\left({\xi }^m\right)\parallel }^2=n,\end{array}$$

which leads to the fact that $L_i\left(\xi \right)$ are an 8-partition satisfying

$$\sum _{i=1}^8\parallel L_i{\left(\xi \right)\parallel }^2=\sum _{i=1}^8\parallel G_i\left({\xi }^m\right){\parallel }^2\sum _{i=1}^8{\parallel E_i\left({\xi }^n\right)\parallel }^2=mn,$$

(21)

and the coefficients of $Q_i\left(\xi \right)$ are $\pm 1$. Then, we obtain

$$\begin{array}{cc}\hfill \sum _{i=4}^4{\parallel Q_i\left(\xi \right)\parallel }^2=& 4\sum _{i=1}^8\parallel L_i{\left(\xi \right)\parallel }^2=4\sum _{i=1}^8\parallel G_i\left({\xi }^m\right){\parallel }^2\sum _{i=1}^8{\parallel E_i{\xi }^n\parallel }^2=4mn.\hfill \end{array}$$

(22)

Here, the tedious calculation in identities (21) and (22) is completed with the help of MATLAB. Using Corollary 1, we obtain that the coefficients ${\mathit{q}}_i$ of corresponding polynomials $Q_i\left(\xi \right)$, $i=1,2,3,4$, are a quad of GS sequences. □

Theorem 2 provides another way to obtain the sequences ${\left\{G_i\right\}}_{i=1}^8$ in Lemma 4, i.e., the sequences ${\left\{L_i\right\}}_{i=1}^8$ in Theorem 2.

Remark 4. 

For now, it is not easy to find an 8-partition with two large n, as stated in Theorem 2, with the requirements of symmetry and antisymmetry. Hence, it could be reduced to find a 4-partition and a class of symmetrical sequences, which will lead to two known conclusions [32]. It could be considered as special cases of our results shown in Lemma 5 and Corollary 2.

Lemma 5. 

Given a class of symmetrical polynomials ${\left\{G_i\left({\xi }^m\right)\right\}}_{i=1}^4$ and a 4-partition $\left\{E_i\left({\xi }^n\right)\right\}$, and defining ${\tilde{L}}_i\left(\xi \right)$ as

$$\begin{array}{cc}\hfill {\tilde{L}}_1\left(\xi \right)=& G_1\left({\xi }^m\right)\overline{E_1\left({\xi }^n\right)}+G_2\left({\xi }^m\right)E_2\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_3\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_4\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_2\left(\xi \right)=& G_1\left({\xi }^m\right)\overline{E_2\left({\xi }^n\right)}-G_2\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_3\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_3\left(\xi \right)=& G_1\left({\xi }^m\right)\overline{E_3\left({\xi }^n\right)}-G_2\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_2\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_4\left(\xi \right)=& G_1\left({\xi }^m\right)\overline{E_4\left({\xi }^n\right)}+G_2\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_1\left({\xi }^n\right),\hfill \end{array}$$

(23)

then we have

$$\sum _{i=1}^4\parallel {\tilde{L}}_i{\left(\xi \right)\parallel }^2=\sum _{i=1}^4\parallel G_i\left({\xi }^m\right){\parallel }^2\sum _{i=1}^4{\parallel E_i\left({\xi }^n\right)\parallel }^2,$$

(24)

where ξ is the $mn$-th root of unity and $(m,n)=1$.

Proof. 

The symmetry of ${\left\{G_i\left({\xi }^m\right)\right\}}_{i=1}^4$ yields

$$\begin{array}{cc}\hfill \parallel {\tilde{L}}_1{\left(\xi \right)\parallel }^2& ={\tilde{L}}_1\left(\xi \right)\overline{{\tilde{L}}_1\left(\xi \right)}\hfill \\ \hfill & =G_1\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}\overline{E_1\left({\xi }^n\right)}{E}_1\left({\xi }^n\right)+G_1\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}\overline{E_1\left({\xi }^n\right)}\overline{E_2\left({\xi }^n\right)}\hfill \\ \hfill & +G_1\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}\overline{E_1\left({\xi }^n\right)}\overline{E_3\left({\xi }^n\right)}+G_1\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}\overline{E_1\left({\xi }^n\right)}\overline{E_4\left({\xi }^n\right)}\hfill \\ \hfill & +G_2\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_2\left({\xi }^n\right)E_1\left({\xi }^n\right)+G_2\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_2\left({\xi }^n\right)\overline{E_2\left({\xi }^n\right)}\hfill \\ \hfill & +G_2\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_2\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}+G_2\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_2\left({\xi }^n\right)\overline{E_4\left({\xi }^n\right)}\hfill \\ \hfill & +G_3\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)E_1\left({\xi }^n\right)+G_3\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_2\left({\xi }^n\right)}\hfill \\ \hfill & +G_3\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}+G_3\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_4\left({\xi }^n\right)}\hfill \\ \hfill & +G_4\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)E_1\left({\xi }^n\right)+G_4\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)\overline{E_2\left({\xi }^n\right)}\hfill \\ \hfill & +G_4\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}+G_4\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)\overline{E_4\left({\xi }^n\right)},\hfill \end{array}$$

$$\begin{array}{cc}\hfill \parallel {\tilde{L}}_2{\left(\xi \right)\parallel }^2& ={\tilde{L}}_2\left(\xi \right)\overline{{\tilde{L}}_2\left(\xi \right)}\hfill \\ \hfill & =G_1\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}\overline{E_2\left({\xi }^n\right)}{E}_2\left({\xi }^n\right)-G_1\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}\overline{E_1\left({\xi }^n\right)}\overline{E_2\left({\xi }^n\right)}\hfill \\ \hfill & +G_1\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}\overline{E_2\left({\xi }^n\right)}\overline{E_4\left({\xi }^n\right)}-G_1\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}\overline{E_2\left({\xi }^n\right)}\overline{E_3\left({\xi }^n\right)}\hfill \\ \hfill & -G_2\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_2\left({\xi }^n\right)E_1\left({\xi }^n\right)+G_2\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_1\left({\xi }^n\right)\overline{E_1\left({\xi }^n\right)}\hfill \\ \hfill & -G_2\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_1\left({\xi }^n\right)\overline{E_4\left({\xi }^n\right)}+G_2\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_1\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}\hfill \\ \hfill & +G_3\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)E_2\left({\xi }^n\right)-G_3\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)\overline{E_1\left({\xi }^n\right)}\hfill \\ \hfill & +G_3\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}-G_3\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_4\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)}\hfill \\ \hfill & -G_4\left({\xi }^m\right)\overline{G_1\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)E_2\left({\xi }^n\right)+G_4\left({\xi }^m\right)\overline{G_2\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_1\left({\xi }^n\right)}\hfill \\ \hfill & -G_4\left({\xi }^m\right)\overline{G_3\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_4\left({\xi }^n\right)}+G_4\left({\xi }^m\right)\overline{G_4\left({\xi }^m\right)}{E}_3\left({\xi }^n\right)\overline{E_3\left({\xi }^n\right)},\hfill \end{array}$$

and also for $\parallel {\tilde{L}}_i{\left(\xi \right)\parallel }^2,i=3,4$. Here, note that the 2nd, 5th, 12th, and 15th terms in $\parallel {\tilde{L}}_1{\left(\xi \right)\parallel }^2$ will vanish when calculating $\parallel {\tilde{L}}_1{\left(\xi \right)\parallel }^2+{\parallel {\tilde{L}}_2\left(\xi \right)\parallel }^2$, and finally we have

$$\begin{array}{cc}\hfill & \sum _{i=1}^4{\parallel {\tilde{L}}_i\left(\xi \right)\parallel }^2\hfill \\ \hfill & =\parallel G_1\left({\xi }^m\right){\parallel }^2\parallel E_1\left({\xi }^n\right){\parallel }^2+\parallel G_2\left({\xi }^m\right){\parallel }^2\parallel E_2\left({\xi }^n\right){\parallel }^2+\parallel G_3\left({\xi }^m\right){\parallel }^2\parallel E_3\left({\xi }^n\right){\parallel }^2+\parallel G_4\left({\xi }^m\right){\parallel }^2{\parallel E_4\left({\xi }^n\right)\parallel }^2\hfill \\ \hfill & +\parallel G_1\left({\xi }^m\right){\parallel }^2\parallel E_2\left({\xi }^n\right){\parallel }^2+\parallel G_2\left({\xi }^m\right){\parallel }^2\parallel E_1\left({\xi }^n\right){\parallel }^2+\parallel G_3\left({\xi }^m\right){\parallel }^2\parallel E_4\left({\xi }^n\right){\parallel }^2+\parallel G_4\left({\xi }^m\right){\parallel }^2{\parallel E_3\left({\xi }^n\right)\parallel }^2\hfill \\ \hfill & +\parallel G_1\left({\xi }^m\right){\parallel }^2\parallel E_3\left({\xi }^n\right){\parallel }^2+\parallel G_2\left({\xi }^m\right){\parallel }^2\parallel E_4\left({\xi }^n\right){\parallel }^2+\parallel G_3\left({\xi }^m\right){\parallel }^2\parallel E_1\left({\xi }^n\right){\parallel }^2+\parallel G_4\left({\xi }^m\right){\parallel }^2{\parallel E_2\left({\xi }^n\right)\parallel }^2\hfill \\ \hfill & +\parallel G_1\left({\xi }^m\right){\parallel }^2\parallel E_4\left({\xi }^n\right){\parallel }^2+\parallel G_2\left({\xi }^m\right){\parallel }^2\parallel E_3\left({\xi }^n\right){\parallel }^2+\parallel G_3\left({\xi }^m\right){\parallel }^2\parallel E_2\left({\xi }^n\right){\parallel }^2+\parallel G_4\left({\xi }^m\right){\parallel }^2{\parallel E_1\left({\xi }^n\right)\parallel }^2\hfill \\ \hfill & =\sum _{i=1}^4\parallel G_i\left({\xi }^m\right){\parallel }^2\sum _{i=1}^4{\parallel E_i\left({\xi }^n\right)\parallel }^2.\hfill \end{array}$$

This completes the proof. □

Furthermore, if the coefficient sequences ${\mathit{g}}_i$ of associated polynomials $G_i\left({\xi }^m\right)$ are a quad of Williamson sequences, we present the main result that a quad of GS sequences has been obtained, as shown in the following corollary:

Corollary 2. 

Let $G_1\left({\xi }^m\right),\cdots ,G_4\left({\xi }^m\right)$ be four associated polynomials of a quad of Williamson sequences and $E_1\left({\xi }^n\right),\cdots ,E_4\left({\xi }^n\right)$ be a 4-partition. Then, the coefficient sequences ${\tilde{\mathit{l}}}_i$ of polynomials $\{{\tilde{L}}_i\}$ defined as

$$\begin{array}{cc}\hfill {\tilde{L}}_1\left(\xi \right)=& G_1\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_2\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_3\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_4\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_2\left(\xi \right)=& G_1\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_3\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_3\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_3\left(\xi \right)=& G_1\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_2\left({\xi }^m\right)E_4\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_1\left({\xi }^n\right)+G_4\left({\xi }^m\right)E_2\left({\xi }^n\right),\hfill \\ \hfill {\tilde{L}}_4\left(\xi \right)=& G_1\left({\xi }^m\right)E_4\left({\xi }^n\right)+G_2\left({\xi }^m\right)E_3\left({\xi }^n\right)-G_3\left({\xi }^m\right)E_2\left({\xi }^n\right)-G_4\left({\xi }^m\right)E_1\left({\xi }^n\right),\hfill \end{array}$$

(25)

are a quad of GS sequences, where ξ is the $mn$-th root of unity.

Proof. 

The coefficients of $G_1\left({\xi }^m\right),G_2\left({\xi }^m\right),G_3\left({\xi }^m\right),G_4\left({\xi }^m\right)$ are a quad of Williamson sequences and thus their coefficients belong to $\{1,-1\}$. For $E_1\left({\xi }^n\right),\cdots ,E_4\left({\xi }^n\right)$ being a 4-partition and from the condition (ii) of Definition 6, the coefficients of $\widetilde{L_i}\left(\xi \right)$ are made up of $\pm 1$, too. Meanwhile, four polynomials $G_1\left({\xi }^m\right),G_2\left({\xi }^m\right),G_3\left({\xi }^m\right),G_4\left({\xi }^m\right)$ satisfy

$$\parallel G_1\left({\xi }^m\right){\parallel }^2+\parallel G_2\left({\xi }^m\right){\parallel }^2+\parallel G_3\left({\xi }^m\right){\parallel }^2+{\parallel G_4\left({\xi }^m\right)\parallel }^2=4n.$$

(26)

Similar to Lemma 5, we obtain the desired result

$$\begin{array}{cc}\hfill \sum _{i=1}^4{\parallel {\tilde{L}}_i\left(\xi \right)\parallel }^2& =\sum _{i=1}^4\parallel G_i\left({\xi }^m\right){\parallel }^2\sum _{i=1}^4{\parallel E_i\left({\xi }^n\right)\parallel }^2=4mn.\hfill \end{array}$$

(27)

This completes the proof. □

Next, we present an example to verify Corollary 2. For $m=7,n=6$, ${\left\{{\mathit{g}}_i\right\}}_{i=1}^4$ is a quad of Williamson sequences of order n, and ${\left\{{\mathit{e}}_i\right\}}_{i=1}^4$ is a 4-partition,

$$\begin{array}{cc}{\mathit{g}}_1=(1,0,0,0,0,0),\hfill & {\mathit{g}}_2=(0,0,0,1,0,0),\hfill \\ {\mathit{g}}_3=(0,-1,0,0,0,-1),\hfill & {\mathit{g}}_4=(0,0,-1,0,1,0),\hfill \end{array}$$

and

$$\begin{array}{cc}{\mathit{e}}_1=(1,1,-1,-1,-1,-1,1),\hfill & {\mathit{e}}_2=(1,-1,1,1,1,1,-1),\hfill \\ {\mathit{e}}_3=(1,1,-1,1,1,-1,1),\hfill & {\mathit{e}}_4=(1,1,-1,1,1,-1,1).\hfill \end{array}$$

Thus, as the construction of (25), the GS sequences are

$$\begin{array}{cc}\hfill {\tilde{\mathit{l}}}_1& =(1,-1,1,1,1,1,1,-1,-1,1,1,-1,-1,-1,-1,-1,-1,-1,-1,1,-1,\hfill \\ \hfill & 1,1,1,-1,-1,1,-1,1,-1,-1,-1,-1,1,1,-1,1,1,-1,1,-1,-1),\hfill \\ \hfill {\tilde{\mathit{l}}}_2& =(1,-1,-1,1,-1,1,-1,-1,1,1,-1,-1,1,-1,1,-1,1,-1,1,1,1,\hfill \\ \hfill & -1,-1,1,1,-1,-1,-1,-1,-1,1,-1,1,1,-1,-1,-1,1,1,1,1,-1),\hfill \\ \hfill {\tilde{\mathit{l}}}_3& =(1,1,-1,-1,1,-1,1,1,1,1,1,-1,-1,1,-1,-1,1,-1,1,-1,1,\hfill \\ \hfill & -1,-1,-1,1,-1,-1,-1,1,1,-1,-1,-1,1,-1,1,1,-1,-1,-1,1,1),\hfill \\ \hfill {\tilde{\mathit{l}}}_4& =(1,-1,-1,1,1,1,1,1,1,-1,1,1,-1,-1,1,1,1,1,1,1,1,\hfill \\ \hfill & 1,-1,1,1,1,-1,1,-1,-1,-1,1,-1,-1,-1,1,1,1,-1,1,1,-1).\hfill \end{array}$$

Remark 5. 

From Corollary 2, for now, the construction eventually becomes to find a quad of Williamson sequences and a 4-partition. It is meaningful since, in practice, some Williamson sequences have been constructed, for example, in [11], and a class of TMS itself is a 4-partition.

4. Conclusions

Due to the important application of Hadamard matrices, it is meaningful to find some novel methods to construct them. An alternative way is making use of GS sequences based on the combination of 4-partition from the existing works. It is noted, however, that, for some large n, there do not exist for 4-partitions but exist for 8-partitions, so that we are motivated to design GS sequences by utilizing 8-partitions as a natural generalization of 4-partition.

To this end, we first construct a quad of GS sequences based on the T-matrices directly. Next, we introduce the definitions of k-partition and utilize the properties of symmetry and antisymmetry. As a result, the construction process turns to finding some k-partitions. More specifically, this paper shows that a class of 8-partition with some (anti)symmetry properties could give a quad of GS sequences. As the special case $k=4$, one can use the existing results, a quad of Williamson sequences and a quad of TMS, to construct Hadamard matrices immediately with more orders, compared with only using a quad of Williamson sequences directly. In the future, the constructions of 8-partition will be investigated which could be used to construct GS sequences with more orders by using the methods proposed in this paper. Meanwhile, it has the potential to enhance k following the methods developed in this paper, to find more orders n for k-partitions and further GS sequences.