A new family of Hadamard matrices of order $4(2q^2+1)$

TL;DR

This paper constructs a new family of Hadamard matrices of order $4(2q^2+1)$ for specific prime powers $q$, using difference families and the Wallis-Whiteman array, expanding known Hadamard matrix orders.

Contribution

The paper introduces a novel construction of Hadamard matrices of order $4(2q^2+1)$ for prime powers $q=12c^2+4c+3$, using difference families in finite groups.

Findings

Constructed difference family with specific parameters in ${ m Z}_2 imes { m F}_{q^2}$.

Derived Hadamard matrices of order $4(2q^2+1)$ for the given $q$.

Expanded the class of known Hadamard matrices with new orders.

Abstract

Let $q$ be a prime power of the form $q=12c^2+4c+3$ with $c$ an arbitrary integer. In this paper we construct a difference family with parameters $(2q^2;q^2,q^2,q^2,q^2-1;2q^2-2)$ in ${\mathbb Z}_2\times ({\mathbb F}_{q^2},+)$. As a consequence, by applying the Wallis-Whiteman array, we obtain Hadamard matrices of order $4(2q^2+1)$ for the aforementioned $q$'s.

Tables

Table 1. Table 1 . Cyclotomic numbers of order 8 8 8 : the ( i , j ) 𝑖 𝑗 (i,j) -entry is ( i , j ) 8 subscript 𝑖 𝑗 8 (i,j)_{8} .

Equations

$$\sum_{i=1}^{\ell}B_{i}B_{i}^{(-1)}=\lambda G+\Big{(}\sum_{i=1}^{\ell}k_{i}-\lambda\Big{)}\cdot 0_{G}.$$

$$3,11,19,43,59,179,211,283,563,619,739,1163,1499,1979,2083,2411,3011,$$

$$3539,4259,4723,7603,8011,8219,10211,11411,12163,14011,14563,14843,$$

$$17483,20011,23059,25579,26699,28619,29803,30203,33923,36083,36523,$$

$$41539,49411,54139,55219,55763,59083,60779,63659,65419,69011,70843,$$

$$75211,80363,81019,82339,83003,88411,93283.$$

$$H_i=$$

$$L_i=$$

$$S_i=$$

$$D_i=$$

$$S_i{S}_j={\mathbb{F}}_{q^2}$$

$$D_i{D}_i^{(-1)}=\frac{q^2-5}{4}{D}_i+\frac{q^2-1}{4}{\overline{D}}_i+\frac{q^2-1}{2}\cdot 0_{{\mathbb{F}}_{q^2}}.$$

$$\sum _{i=0}^3{B}_i{B}_i^{(-1)}=(2q^2-2)({\mathbb{Z}}_2\times {\mathbb{F}}_{q^2})+(2q^2+1)\cdot (0,0_{{\mathbb{F}}_{q^2}}).$$

$$B_0$$

$$B_1$$

$$\sum_{i=0,1}B_{i}B_{i}^{(-1)}=\{0\}\times\Big{(}(q^{2}-2){\mathbb{F}}_{q^{2}}^{\ast}+(2q^{2}-1)\cdot 0_{{\mathbb{F}}_{q^{2}}}\Big{)}+\{1\}\times\Big{(}2D_{0}-2D_{2}+(q^{2}-1){\mathbb{F}}_{q^{2}}\Big{)}.$$

$$\sum _{i=0,1}{B}_i{B}_i^{(-1)}=$$

$$\displaystyle\,\,+\{1\}\times\Big{(}-2D_{0}D_{0}^{(-1)}+2D_{2}D_{2}^{(-1)}+2|D_{0}|{\mathbb{F}}_{q^{2}}\Big{)}.$$

$$\sum _{i=0,2}{D}_i{D}_i^{(-1)}=$$

$$=$$

$$-D_0{D}_0^{(-1)}+D_2{D}_2^{(-1)}=$$

$$=$$

$$E_0{E}_1^{(-1)}+E_1{E}_0^{(-1)}+E_2{E}_3^{(-1)}+E_3{E}_2^{(-1)}=(q-1{)}^2{\mathbb{F}}_{q^2}+2D_0-2D_2.$$

$$B_2$$

$$B_3$$

$$\sum _{i=2,3}{B}_i{B}_i^{(-1)}=\{0\}\times (2q^2\cdot 0_{{\mathbb{F}}_{q^2}}+q^2{\mathbb{F}}_{q^2}^{\ast })+\{1\}\times ((q^2-1){\mathbb{F}}_{q^2}-2D_0+2D_2).$$

$$\sum _{i=2,3}{B}_i{B}_i^{(-1)}=$$

$$+\{1\}\times (-E_0{E}_1^{(-1)}-E_1{E}_0^{(-1)}-E_2{E}_3^{(-1)}-E_3{E}_2^{(-1)}+2(|E_0|+|E_2|){\mathbb{F}}_{q^2}).$$

$$\begin{gathered} \sum _{i=0}^3{E}_i{E}_i^{(-1)}+2(q^2-|E_1|-|E_3|){\mathbb{F}}_{q^2} \\ =2q^2\cdot 0_{{\mathbb{F}}_{q^2}}+q^2{\mathbb{F}}_{q^2}^{\ast }. \end{gathered}$$

$$-E_0{E}_1^{(-1)}-E_1{E}_0^{(-1)}-E_2{E}_3^{(-1)}-E_3{E}_2^{(-1)}+2(|E_0|+|E_2|){\mathbb{F}}_{q^2}=(q^2-1){\mathbb{F}}_{q^2}-2D_0+2D_2.$$

$$H$$

$$M_i$$

$${\mathbf{D}}_i$$

$$J_1=\{y+2+4i(\mathrm{mod}q+1):i\in \{0,1,\dots ,m-1\}\}\text{mbox}and$$

$$J_2=\{y+4i(\mathrm{mod}q+1):i\in \{0,1,\dots ,m-1\}\}.$$

$${\mathbf{B}}_2={\mathbf{B}}_3=\{y-2+4i(\mathrm{mod}q+1):i\in \{0,1,\dots ,m-1\}\}.$$

$$E_0=$$

$$E_2=$$

Full text

Koji Momihara was supported by JSPS under Grant-in-Aid for Young Scientists (B) 17K14236 and Scientific Research (B) 15H03636.

Qing Xiang was supported by an NSF grant DMS-1600850.

1. Introduction

A Hadamard matrix of order $v$ is a $v\times v$ matrix $H$ with entries $\pm 1$ such that $HH^{\top}=vI$, where $I$ is the identity matrix. It can be easily shown that if $H$ is a Hadamard matrix of order $v$, then $v=1,2$, or $4t$ for some positive integer $t$. A long-standing conjecture in combinatorics states that a Hadamard matrix of order $v$ exists for every $v\equiv 0\,({\mathrm{mod\,\,}}{4})$. Despite the work of many researchers, the conjecture is far from being resolved. Currently it is still not known whether the set of orders of Hadamard matrices has positive density. For some sparse infinite subsequences of $\{4t:t=1,2,3,\ldots\}$, it is often possible to construct Hadamard matrices of order $v$ for every $v$ belonging to the subsequences. The most famous examples are the Paley constructions which produce Hadamard matrices of order $q+1$ if $q$ is a prime power congruent to 3 modulo 4, and Hadamard matrices of order $2(q+1)$ if $q$ is a prime power congruent to 1 modulo 4. As further examples, we mention that for prime powers $q\equiv 1\,({\mathrm{mod\,\,}}{4})$ or $q\equiv 3\,({\mathrm{mod\,\,}}{8})$, Xia and Liu [12, 14] construct Hadamard matrices of order $4q^{2}$; for $q\equiv 7\,({\mathrm{mod\,\,}}{8})$, the first author, Ma and Schmidt [4] construct two possibly infinite families of Hadamard matrices of order $4q^{2}$. All these constructions are based on cyclotomy of finite fields. The Paley constructions use the nonzero squares of ${\mathbb{F}}_{q}$. The constructions by Xia and Liu [12, 14], and by Leung, Ma, and Schmidt [4] use the $4^{\rm th}$, $8^{\rm th}$ and $(q+1)^{\rm th}$ cyclotomic classes of ${\mathbb{F}}_{q^{2}}$. The main idea behind the constructions of Xia/Liu and Leung/Ma/Schmidt is to use cyclotomic classes of finite fields to construct a difference family with appropriate parameters in an abelian group $G$.

Throughout this paper, we will use the following notation. Let $(G,+)$ be an additively written finite abelian group and let $G^{\ast}:=G\setminus\{0_{G}\}$. For any subset $D$ in $G$, we define $D^{(-1)}:=\{-x:x\in D\}$, $\overline{D}:=G^{\ast}\setminus D$, and $D^{c}:=G\setminus D$. Furthermore, we will identify $D$ with the group ring element $\sum_{x\in D}x\in{\mathbb{Z}}[G]$ when there is no confusion.

Let $B_{i}$, $i=1,2,\ldots,\ell$, be $k_{i}$-subsets of $G$. The set ${\mathcal{B}}=\{B_{i}:i=1,2,\ldots,\ell\}$ is called a difference family with parameters $(v;k_{1},k_{2},\ldots,k_{\ell};\lambda)$ in $G$ if the list of differences “$x-y,x,y\in B_{i},x\not=y,i=1,2,\ldots,\ell$” represents every nonzero element of $G$ exactly $\lambda$ times; or equivalently

[TABLE]

Each subset $B_{i}$ is called a block of ${\mathcal{B}}$. We now define two special classes of difference families. A difference family in $G$ with four blocks is said to be of type $H$ if $\sum_{i=1}^{4}k_{i}-|G|=\lambda$; and of type $H_{4}^{\ast}$ if $\sum_{i=1}^{4}k_{i}-(|G|+1)=\lambda$.

It is well known that if there is a difference family of type $H$ in $G$, then we obtain a Hadamard matrix of order $4|G|$ by plugging the group invariant $(-1,1)$ matrices obtained from its blocks into the Goethals-Seidel array [1]. In the literature, difference families of type $H$ have been extensively studied [4, 12, 13, 14, 15, 16, 17].

On the other hand, from a difference family of type $H_{4}^{\ast}$ in a finite abelian group $G$, we obtain a Hadamard matrix of order $4(|G|+1)$ by plugging the the group invariant $(-1,1)$ matrices obtained from its blocks into the Wallis-Whiteman array [10, Theorem 4.17]. Indeed difference families of type $H_{4}^{\ast}$ are particularly interesting as the orders of the Hadamard matrices obtained from the difference families are no longer of the form $4|G|$, but of the form $4(|G|+1)$. Very recently, the first and second authors [5] gave two new constructions of difference families of type $H_{4}^{\ast}$ with parameters $(2n;n,n,n,n-1;2n-2)$. Difference families with these parameters were initially considered by Whiteman [11], who obtained one infinite family. Soon afterwards, Spence [8] came up with two new families whose constructions are based on relative difference sets. On the other hand, the existence of difference families with parameters $(2n;n,n,n,n-1;2n-2)$ in dihedral groups was also studied in [2, 3, 7]. Let us summarize all known constructions of difference families of type $H_{4}^{\ast}$ with parameters $(2n;n,n,n,n-1;2n-2)$.

Theorem 1.1**.**

There exists a difference family of type $H_{4}^{\ast}$ with parameters $(2n;n,n,n,$ $n-1;2n-2)$ if $n$ satisfies any of the following conditions:

• (1)

[11, 7]* $n=q$ and $2q-1$ are both prime powers.*

• (2)

[8]* $q=2n+1$ is a prime power for which there exists a nonnegative integer $s$ such that $(q-2^{s+1}-1)/2^{s+1}$ is an odd prime power.*

• (3)

[8]* $n=q$ is a prime power such that $q\equiv 1\,({\mathrm{mod\,\,}}{4})$, and $q-2$ is also a prime power.*

• (4)

[5]* $n=9^{t_{0}}q_{1}^{4t_{1}}q_{2}^{4t_{2}}\cdots q_{s}^{4t_{s}}$, where $p_{i}$, $i=1,2,\ldots,s$, are prime powers and $t_{i}$, $i=0,1,\ldots,s$, are nonnegative integer.*

• (5)

[5]* $n=q^{2}$ with $q$ a prime power such that $q\equiv 1\,({\mathrm{mod\,\,}}{4})$.*

In particular, there exists a Hadamard matrix of order $4(2n+1)$ if $n$ satisfies any of the above conditions.

In this paper, we obtain a new series of difference families of type $H_{4}^{*}$ with parameters $(2q^{2};q^{2},q^{2},$ $q^{2},q^{2}-1;2q^{2}-2)$ where $q$ is a prime power congruent to 3 modulo 8 satisfying some extra condition. The construction uses $8^{\rm th}$ cyclotomic classes of ${\mathbb{F}}_{q^{2}}$ and “half lines” in $\mathsf{AG}(2,q)$. In [4, 12, 14], the main idea is to construct difference families of type $H$ in the group $({\mathbb{F}}_{q^{2}},+)$. Our approach here is analogous to that of [5]; the main difference here is the usage of Paley type partial difference sets. The following are our main results.

Theorem 1.2**.**

Let $q$ be a prime power of the form $q=12c^{2}+4c+3$ with $c$ an arbitrary integer, and let $n=q^{2}$. Then there exists a difference family with parameters $(2n;n,n,n,n-1;2n-2)$ in ${\mathbb{Z}}_{2}\times({\mathbb{F}}_{q^{2}},+)$.

By plugging the group invariant $(1,-1)$ matrices obtained from the blocks of the difference family in Theorem 1.2 into the Wallis-Whiteman array, we immediately obtain the following:

Theorem 1.3**.**

Let $q$ be a prime power of the form $q=12c^{2}+4c+3$ with $c$ an arbitrary integer, and let $n=q^{2}$. Then there exists a Hadamard matrix of order $4(2n+1)$.

We remark that there are $386$ prime powers of the form $q=12c^{2}+4c+3<10^{7}$ while there are $166181$ prime powers $q<10^{7}$ such that $q\equiv 3\,({\mathrm{mod\,\,}}{8})$. The first $58$ prime powers of the form $q=12c^{2}+4c+3<10^{5}$ are listed below:

[TABLE]

2. The construction

We first fix our notation. Let $q$ be a prime power such that $q\equiv 3\,({\mathrm{mod\,\,}}{4})$. Let $\omega$ be a primitive element of ${\mathbb{F}}_{q^{2}}$ and let $0_{{\mathbb{F}}_{q^{2}}}$ denote the zero of ${\mathbb{F}}_{q^{2}}$. For any fixed positive integer $N$ dividing $q^{2}-1$, define $C_{i}^{(N,q^{2})}=\omega^{i}\langle\omega^{N}\rangle$, $i=0,1,\ldots,N-1$, called the $N^{\rm th}$* cyclotomic classes* of ${\mathbb{F}}_{q^{2}}$. Furthermore, define

[TABLE]

Note that each $S_{i}$ is a line through the origin of $\mathsf{AG}(2,q)$; for this reason the $H_{i}$’s are called half lines [18]. In the group ring ${\mathbb{Z}}[({\mathbb{F}}_{q^{2}},+)]$, we have

[TABLE]

Lemma 2.1**.**

For $i=0,1,2,3$, $D_{i}$ is a Paley type partial difference set in $({\mathbb{F}}_{q^{2}},+)$. In particular,

[TABLE]

For a proof of Lemma 2.1, we refer the reader to [6, p. 216]. The strongly regular Cayley graph, ${\rm Cay}({\mathbb{F}}_{q^{2}},D_{0})$, is often called a Peisert graph.

Our objective is to construct difference families with parameters $(2q^{2};q^{2},q^{2}-1,q^{2},q^{2};2q^{2}-2)$ in ${\mathbb{Z}}_{2}\times{\mathbb{F}}_{q^{2}}$. So we need to find four blocks $B_{0},B_{1},B_{2},B_{3}$ with $|B_{i}|=q^{2}$, $i=0,2,3$, and $|B_{1}|=q^{2}-1$, in ${\mathbb{Z}}_{2}\times{\mathbb{F}}_{q^{2}}$ such that

[TABLE]

To construct the first two blocks, we make use of the Paley type partial difference sets $D_{0}$ and $D_{2}$ defined above. Note that $D_{2}=\omega^{2}D_{0}$ and $D_{2}=\overline{D_{0}}$. In ${\mathbb{Z}}_{2}\times{\mathbb{F}}_{q^{2}}$, we set

[TABLE]

Then $|B_{0}|=q^{2}$ and $|B_{1}|=q^{2}-1$.

Proposition 2.2**.**

With $B_{0},B_{1}$ defined as above, we have

[TABLE]

Proof.

It is clear that

[TABLE]

By Lemma 2.1, we have

[TABLE]

and

[TABLE]

It is now straight forward to obtain (2.2) from (2.3), (2.4) and (2.5). ∎

To construct the remaining blocks of the desired difference family, we need difference families of type $H$ in ${\mathbb{F}}_{q^{2}}$ that satisfy certain conditions.

Proposition 2.3**.**

Suppose ${\mathcal{E}}=\{E_{i}:i=0,1,2,3\}$ is a difference family of type $H$ in ${\mathbb{F}}_{q^{2}}$ such that $|E_{0}|=|E_{1}|=|E_{2}|=|E_{3}|=(q^{2}-q)/2$ and

[TABLE]

Let $B_{0},B_{1}$ be defined as above and set

[TABLE]

Then $\{B_{0},B_{1},B_{2},B_{3}\}$ is a difference family with parameters $(2q^{2};q^{2},q^{2}-1,q^{2},q^{2};2q^{2}-2)$ in ${\mathbb{Z}}_{2}\times{\mathbb{F}}_{q^{2}}$.

Proof.

First of all, we have $|B_{2}|=q^{2}+|E_{0}|-|E_{1}|=q^{2}$ and $|B_{3}|=q^{2}+|E_{2}|-|E_{3}|=q^{2}$. In view of (2.2), it suffices to show that

[TABLE]

It is clear that

[TABLE]

Since $\{E_{i}:i=0,1,2,3\}$ is a difference family of type $H$ and $|E_{1}|+|E_{3}|=q^{2}-q$, we have

[TABLE]

On the other hand, by the assumption (2.6) and $|E_{0}|+|E_{2}|=q^{2}-q$, we have

[TABLE]

The proposition now follows from (2.7), (2.8), and (2.9). ∎

To construct difference families of type $H$ in ${\mathbb{F}}_{q^{2}}$ satisfying the conditions in Proposition 2.3, it is then natural to consider those constructed in [4].

Lemma 2.4**.**

([4, Lemma 4 and Corollary 5])*

Let $q\equiv 3\,({\mathrm{mod\,\,}}{4})$ be a prime power and let $e$ be the exact power of $2$ dividing $q+1$. Let $\alpha<e$ be an odd number and set $\beta=\frac{qe-\alpha(q+1)}{2e}$. Let ${\bf A}\subseteq\{0,1,\ldots,2e-1\}$ and ${\bf B}_{0},\ldots,{\bf B}_{e-1}\subseteq\{0,1,\ldots,q\}$ with $|{\bf A}|=\alpha$, $|{\bf B}_{0}|=\cdots=|{\bf B}_{e-1}|=\beta$ such that $b\not\equiv a\,({\mathrm{mod\,\,}}{e})$ for all $a\in{\bf A}$ and $b\in\bigcup_{r=0}^{e-1}{\bf B}_{r}$. Set*

[TABLE]

Then $|{\bf D}_{i}|=\frac{q(q-1)}{2}$ for $i=0,1,\ldots,e-1$, and $\{{\bf D}_{i}:i=0,1,\ldots,e-1\}$ forms a difference family in $({\mathbb{F}}_{q^{2}},+)$ with $\lambda=\frac{eq(q-2)}{4}$.

We now assume that $q$ is a prime power and $q=8m+3$ for some positive integer $m$. In view of Lemma 2.4, we need a set ${\bf A}\subseteq\{0,1,\ldots,7\}$ with $|{\bf A}|=3$, and four subsets ${\bf B}_{i}$, $i=0,1,2,3$, of $\{0,\ldots,q\}$, each of size $m$, satisfying certain conditions.

First, we require $I\cap\{x+4\,({\mathrm{mod\,\,}}{8}):x\in I\}=\emptyset$. Since $|I|=3$, the condition $I\cap\{x+4\,({\mathrm{mod\,\,}}{8}):x\in I\}=\emptyset$ simply means that $I$ contains exactly one odd or exactly one even element, say, $y\in I$. (Note that such an $I$ clearly exists, for example, take $I=\{0,1,3\}$; and in this case $y=0$.) Next, we define two $m$-subsets of $\{0,1,\ldots,q\}$:

[TABLE]

[TABLE]

Now, using the notation in Lemma 2.4, we set $e=4$, $\alpha=3$ and $\beta=m$. Let ${\bf A}=I$, ${\bf B}_{0}={\bf B}_{1}=J_{1}$,

[TABLE]

It is then straight forward to check that the conditions in Lemma 2.4 are all satisfied. Therefore we obtain a difference family $\{{\bf D}_{i}:i=0,1,2,3\}$. However, for our purpose, we need to set $E_{0}={\bf D}_{0},E_{1}={\bf D}_{2},E_{2}={\bf D}_{1}$ and $E_{3}={\bf D}_{3}$. In terms of $I,J_{1},J_{2}$, we have the following:

[TABLE]

By Lemma 2.4, $\{E_{i},i=0,1,2,3\}$ is a difference family of type $H$ in $({\mathbb{F}}_{q^{2}},+)$. Furthermore, $|E_{i}|=\frac{q^{2}-q}{2}$ for $i=0,1,2,3$. It therefore remains to show the following:

Theorem 2.5**.**

The $E_{i}$’s defined in (2.10) satisfy the equation (2.6). In particular, there is a difference family with parameters $(2q^{2};q^{2},q^{2},q^{2},q^{2}-1;2q^{2}-2)$ in ${\mathbb{Z}}_{2}\times({\mathbb{F}}_{q^{2}},+)$.

3. Proof of Theorem  2.5

To prove Theorem  2.5, we need to compute $E_{0}E_{1}^{(-1)}+E_{0}E_{1}^{(-1)}+E_{2}E_{3}^{(-1)}+E_{3}E_{2}^{(-1)}$. As in the case of Lemma 4 in [4], it will make the computations easier if we write each $E_{i}$ in a different form (i.e., as a union of $H_{i}$’s and $L_{j}$’s). Recall that $q=8m+3$ is a prime power. We define

[TABLE]

Here we use the notation $K+1=\{x+1:x\in K\}$. Note that $|I_{1}|=|I_{2}|=3(q+1)/4$. Recall that

[TABLE]

We write

[TABLE]

[TABLE]

Observe that the following conditions are satisifed:

• (1)

Since $I\cap\{x+4\,({\mathrm{mod\,\,}}{8}):x\in I\}=\emptyset$, we have $I_{i}\cap\{h+(q+1)\,({\mathrm{mod\,\,}}{2(q+1)}):h\in I_{i}\}=\emptyset$ for $i=1,2$,

• (2)

$a\not\equiv b\,({\mathrm{mod\,\,}}{q+1})$ for all $a\in I_{i},b\in J_{i}$, $i=1,2$,

• (3)

$|I_{1}|+2|J_{1}|=|I_{2}|+2|J_{2}|=q$,

• (4)

$J_{1}\subseteq I_{2}^{\prime}\cup J_{2}$ and $J_{2}\subseteq I_{1}^{\prime}\cup J_{1}$, where $I_{j}^{\prime}=\{i\,({\mathrm{mod\,\,}}{q+1}):i\in I_{j}\}$ for $j=1,2$.

Lemma 3.1**.**

In the group ring ${\mathbb{Z}}[({\mathbb{F}}_{q^{2}},+)]$, $E_{0}E_{1}^{(-1)}+E_{1}E_{0}^{(-1)}=$

[TABLE]

where $\lambda_{1}=|I_{1}||J_{2}|+|I_{2}||J_{1}|+2|J_{1}||J_{2}|$ and $\lambda_{2}=|I_{1}^{\prime}||J_{2}|+|I_{2}^{\prime}||J_{1}|+2|J_{1}||J_{2}|-|I_{1}^{\prime}\cap J_{2}|-|I_{2}^{\prime}\cap J_{1}|-2|J_{1}\cap J_{2}|$.

Proof.

Note that $L_{i}^{(-1)}=L_{i}$. We first expand the expression $E_{0}E_{1}^{(-1)}+E_{0}E_{1}^{(-1)}$ and obtain the following:

[TABLE]

[TABLE]

Note that $S_{i}=L_{i}+0_{{\mathbb{F}}_{q^{2}}}$. So, we may replace each $L_{i}$ by $S_{i}-0_{{\mathbb{F}}_{q^{2}}}$ in the above sum and we get

[TABLE]

Observe that $H_{i}+H_{q+1+i}=S_{i}-0_{{\mathbb{F}}_{q^{2}}}$ for $i=0,1,\ldots,q$ and (3) holds. Also note that

[TABLE]

[TABLE]

We then have

[TABLE]

On the other hand, $S_{i}S_{j}={\mathbb{F}}_{q^{2}}$ whenever $i\neq j$. Therefore, by the conditions (2) and (4), for distinct $u,v$ in $\{1,2\}$,

[TABLE]

[TABLE]

(3.1) now follows easily from (3.2), (3.3) and (3.4). ∎

Now, replace $I_{i}$ with $I_{i}+1$, and $J_{i}$ with $J_{i}+1$ in the argument above and observe that condition (2), (3) and (4) still hold. We immediately get the following:

Lemma 3.2**.**

In the group ring ${\mathbb{Z}}[({\mathbb{F}}_{q^{2}},+)]$, $E_{2}E_{3}^{(-1)}+E_{3}E_{2}^{(-1)}=$

[TABLE]

where $\lambda_{1}=|I_{1}||J_{2}|+|I_{2}||J_{1}|+2|J_{1}||J_{2}|$ and $\lambda_{2}=|I_{1}^{\prime}||J_{2}|+|I_{2}^{\prime}||J_{1}|+2|J_{1}||J_{2}|-|I_{1}^{\prime}\cap J_{2}|-|I_{2}^{\prime}\cap J_{1}|-2|J_{1}\cap J_{2}|$.

Lemma 3.3**.**

Let $E_{i}$, $i=0,1,2,3$, be defined as in (2.10). Recall that $q=8m+3$. Then, we have

[TABLE]

where $Z=8m(4m+1)\cdot 0_{{\mathbb{F}}_{q^{2}}}+m(28m+5){\mathbb{F}}_{q^{2}}^{\ast}$.

Proof.

Applying Lemmas 3.1 and 3.2, we obtain

[TABLE]

Since $|I_{1}^{\prime}|=|I_{2}^{\prime}|=6m+3$, we have

[TABLE]

and

[TABLE]

Note that $\sum_{j=0}^{3}C_{i+j}^{(4,q^{2})}={\mathbb{F}}_{q^{2}}^{\ast}$, $|I|=3$ and $|J_{1}|=|J_{2}|=m$. Hence,

[TABLE]

Furthermore, it is clear that

[TABLE]

Our lemma now follows from (3.6) with (3.7) and (3.8). ∎

To finish our proof, we need to evaluate $C_{i}^{(8,q^{2})}{C_{j}^{(8,q^{2})}}^{(-1)}$. The coefficient $c_{x}$ of $x\in{\mathbb{F}}_{q^{2}}$ in $C_{i}^{(8,q^{2})}{C_{j}^{(8,q^{2})}}^{(-1)}$ is \big{|}(C_{j}^{(8,q^{2})}+x)\cap C_{i}^{(8,q^{2})}\big{|}. If $x\in C_{h}^{(8,q^{2})}$, it is clear that c_{x}=\big{|}(C_{j-h}^{(8,q^{2})}+1)\cap C_{i-h}^{(8,q^{2})}\big{|}. The numbers (i,j)_{N}=\big{|}(C_{i}^{(N,q^{2})}+1)\cap C_{j}^{(N,q^{2})}\big{|}, $i,j=0,1,\ldots,N-1$, are called $N^{\rm th}$* cyclotomic numbers*. In our case, $q\equiv 3\,({\mathrm{mod\,\,}}{8})$ is a prime power. In view of [9, Lemma 30], we obtain the following:

Proposition 3.4**.**

Let $q\equiv 3\,({\mathrm{mod\,\,}}{8})$ be a prime power. Then the cyclotomic numbers $(i,j)_{8}$, $i,j=0,1,\ldots,7$, in ${\mathbb{F}}_{q^{2}}$ are determined by Table 1 and the relations:

[TABLE]

where $a,b$ are specified by the unique proper representation of $q^{2}=a^{2}+2b^{2}$ with $a\equiv 1\,({\mathrm{mod\,\,}}{4})$. Note that there is no restriction on the sign of $b$.

Theorem 3.5**.**

Suppose $q^{2}=a^{2}+2b^{2}$ is the unique proper representation with $a\equiv 1\,({\mathrm{mod\,\,}}{4})$. Theorem 2.5 holds if either of the following conditions is satisfied.

• (a)

$I=\{0,2,3\}$* and $3q=a+4b+16$.*

• (b)

$I=\{0,2,7\}$* and $3q=a-4b+16$.*

Proof.

By Lemma 3.3, it is sufficient to show the following:

[TABLE]

We give a proof only in the case where $3q=a+4b+16$. The proof for the case where $3q=a-4b+16$ is similar.

Define $D=\bigcup_{i\in I}C_{i}^{(8,q^{2})}$, and let $c_{x}$ denote the coefficient of $x\in{\mathbb{F}}_{q^{2}}$ in $U$. To show that $c_{0}=\frac{q^{2}-1}{2}$, it is sufficient to check number of pairs $(i,j)\in I\times I$ such that $i\equiv j+2\,({\mathrm{mod\,\,}}{8})$ and $i+2\equiv j\,({\mathrm{mod\,\,}}{8})$. Clearly, the solution is $(2,0)$ and $(0,2)$ in each case respectively. Therefore, $c_{0}=2\times 2\times\frac{q^{2}-1}{8}=\frac{q^{2}-1}{2}$. To prove that (3.9) holds, it is enough to see that $c_{1}=c_{\omega}=c_{\omega^{2}}+4=c_{\omega^{3}}+4$ since $c_{w^{i}}=c_{w^{i+4j}}$ for all $i,j$. On the other hand, $c_{x}$ for $x\in{\mathbb{F}}_{q^{2}}^{\ast}$ is given by

[TABLE]

Hence, the system of equations $c_{1}=c_{\omega}=c_{\omega^{2}}+4=c_{\omega^{3}}+4$ is reformulated as

[TABLE]

Noting that $|\omega D\cap(\omega^{3}D+1)|+|\omega^{3}D\cap(\omega D+1)|=|\omega^{-3}D\cap(\omega^{-1}D+1)|+|\omega^{-1}D\cap(\omega^{-3}D+1)|$, the equations above are reduced to

[TABLE]

and

[TABLE]

Let

[TABLE]

Then, (3.10) and (3.11) are rewritten as $N_{1}=N_{2}$ and $N_{3}=N_{4}+4$, respectively. From the definition of $I$ and Table 1 of Proposition 3.4, we have

[TABLE]

It is clear that $N_{1}=N_{2}$. By the evaluations for $n_{1},n_{2},\ldots,n_{8}$ in Proposition 3.4, we have $N_{3}=(18q^{2}+28q-8a-32b-46)/64$ and $N_{4}=(18q^{2}+20q+8a+32b-46)/64$. Hence, $N_{3}=N_{4}+4$ if and only if $3q=a+4b+16$. This shows that (3.9) holds if $3q=a+4b+16$. ∎

It is not difficult to see that the condition $q^{2}=a^{2}+2b^{2}$ with $3q=a\pm 4b+16$ and $a\equiv 1\,({\mathrm{mod\,\,}}{4})$ is equivalent to that $q$ has the form $q=12c^{2}+4c+3$ with $c$ an arbitrary integer; in this case, $a=4c^{2}+12c+1$ and $b=\pm(8c^{2}-2)$. Hence, by Theorem 3.5 and Proposition 2.3, Theorem 1.2 now follows.

To see whether we have constructed an infinite family of Hadamard matrices in Theorem 1.3, a natural question arises: are there infinitely many prime powers $q$ of the form $q=12c^{2}+4c+3$ with $c$ an integer? We believe that there are infinitely many primes of the form $12c^{2}+4c+3$ with $c$ an integer. But this is probably very difficult to prove. On the other hand, we conjecture that there are no proper prime powers $q$ of the form $q=12c^{2}+4c+3$ ($c$ is an integer). That is, we conjecture that there are no solutions to the equation

[TABLE]

where $c$ is an integer, and $p$ is a prime. Some evidence is given in Introduction, namely all $58$ prime powers listed in (1.1) are actually primes.