# Frames over finite fields:
Equiangular lines in orthogonal geometry

Gary R. W. Greaves  School of Physical and Mathematical Sciences, Nanyang
Technological University, Singapore 637371<EMAIL_ADDRESS>,  Joseph W.
Iverson  Department of Mathematics, Iowa State University, Ames, IA USA 50011
<EMAIL_ADDRESS>,  John Jasper  Department of Mathematics and Statistics,
South Dakota State University, Brookings, SD USA 57007
<EMAIL_ADDRESS>and  Dustin G. Mixon  Department of Mathematics, The
Ohio State University, Columbus, OH USA 43210<EMAIL_ADDRESS>
###### Abstract.

We investigate equiangular lines in finite orthogonal geometries, focusing
specifically on equiangular tight frames (ETFs). In parallel with the known
correspondence between real ETFs and strongly regular graphs (SRGs) that
satisfy certain parameter constraints, we prove that ETFs in finite orthogonal
geometries are closely aligned with a modular generalization of SRGs. The
constraints in our finite field setting are weaker, and all but 18 known SRG
parameters on $v\leq 1300$ vertices satisfy at least one of them. Applying our
results to triangular graphs, we deduce that Gerzon’s bound is attained in
finite orthogonal geometries of infinitely many dimensions. We also
demonstrate connections with real ETFs, and derive necessary conditions for
ETFs in finite orthogonal geometries. As an application, we show that Gerzon’s
bound cannot be attained in a finite orthogonal geometry of dimension 5.

###### Key words and phrases:

equiangular lines, equiangular tight frames, finite fields, strongly regular
graphs

###### 2010 Mathematics Subject Classification:

Primary: 51E99. Secondary: 05E30, 42C15, 52C35.

##  1\. Introduction

Over the real number field, a sequence $\mathscr{L}=\\{\ell_{j}\\}_{j\in[n]}$
of lines through the origin of $\mathbb{R}^{d}$ is called equiangular if there
are constants $a,b\geq 0$ and nonzero column vectors $\varphi_{j}\in\ell_{j}$
such that $\varphi_{j}^{\top}\varphi_{j}=a$ and
$(\varphi_{i}^{\top}\varphi_{j})^{2}=b$ for every $i\neq j$ in
$[n]:=\\{1,\dotsc,n\\}$. As a special instance, the representatives
$\\{\varphi_{j}\\}_{j\in[n]}$ are said to form an equiangular tight frame
(ETF) of size $d\times n$ if $\mathscr{L}$ is equiangular and the matrix
$\Phi=\left[\begin{array}[]{ccc}\varphi_{1}&\cdots&\varphi_{n}\end{array}\right]\in\mathbb{R}^{d\times
n}$ satisfies $\Phi\Phi^{\top}=cI$ for some $c$. ETFs are known to create
optimal packings in projective space $\mathbb{R}\mathrm{P}^{d-1}$, and for
this reason and others they find applications in areas such as compressed
sensing [3], wireless communication [24], digital fingerprinting [19], and
quantum information theory [21].

In this paper, we investigate the related phenomenon of ETFs over finite
fields (Definition 2.8). Frames and equiangular lines over finite fields are
introduced in the companion paper [14], which develops the basic theory and
continues on to study ETFs in unitary geometries (akin to complex ETFs). Here
we focus on ETFs in orthogonal geometries, which are the finite field analogs
of real ETFs.

In the real setting, ETFs are deeply connected with the theory of strongly
regular graphs (SRGs). Using a construction of Seidel, an SRG with parameters
$(v,k,\lambda,\mu)$ creates a real ETF with $n=v+1$ vectors whenever $k=2\mu$,
and, up to a change of line representatives, every ETF of $n>d+1$ vectors in
$\mathbb{R}^{d}$ arises this way [22, 26, 29]. In another construction, an SRG
with parameters $(v,k,\lambda,\mu)$ creates a real ETF with $n=v$ vectors
whenever $v=4k-2\lambda-2\mu$, and this correspondence uniquely characterizes
real ETFs with _centroidal symmetry_ [26, 11]. From this perspective, real
ETFs may be viewed as a subclass of SRGs, where about 27% of known SRG
parameters on $v\leq 1300$ vertices satisfy one of the constraints $k=2\mu$ or
$v=4k-2\lambda-2\mu$ [6]. This relationship has proven to be fruitful for the
theories of both SRGs and ETFs, and results from each area have been used to
establish both existence and nonexistence in the other [6, 12, 18, 25, 10, 1,
2].

A similar phenomenon occurs for ETFs in finite orthogonal geometries, but in
this setting we encounter the more general notion of a modular strongly
regular graph (Definition 4.1). As with SRGs, modular SRGs are described by
parameters $(v,k,\lambda,\mu)$, and an ETF can be obtained if and only if
constraints like those above are satisfied (Theorem 4.3 and Theorem 4.5).
However, in the finite field setting these parameter constraints need only be
satisfied modulo the field characteristic. Thanks to this weaker condition, a
much larger fraction of SRGs can be identified with ETFs over finite fields,
and we observe that 98% of known SRG parameters on $v\leq 1300$ vertices occur
in this way (Remark 5.2).

These constructions produce remarkable examples of ETFs over finite fields. In
the real setting, Gerzon’s bound states that $\mathbb{R}^{d}$ admits a system
of $n$ equiangular lines only if $n\leq\binom{d+1}{2}$. Saturation is
empirically rare, and examples of $n=\binom{d+1}{2}$ equiangular lines in
$\mathbb{R}^{d}$ are known only for $d\in\\{2,3,7,23\\}$. It is an open
problem whether the bound can be attained in any other dimension, and it is
known that this can happen only for $d\geq 119$ such that $d+2$ is an odd
square [17, 18, 4]. Gerzon’s bound also applies in our finite field setting
(Proposition 2.9). However, we now find a completely different behavior: when
applied to triangular graphs, our results show that Gerzon’s bound is
saturated in some $d$-dimensional finite orthogonal geometry whenever $|d-7|$
is not a power of 2 (Theorem 5.4).

This is all the more surprising since we show that ETFs in finite orthogonal
geometries are closely aligned with real ETFs. If a $d\times n$ real ETF
exists, then $d\times n$ ETFs exist in finite orthogonal geometries of all but
finitely many characteristics (Proposition 3.2). Conversely, if a $d\times n$
ETF exists in a finite orthogonal geometry of characteristic $p>2n-5$, then a
real $d\times n$ ETF also exists (Proposition 3.3). In this sense, ETFs in
finite orthogonal geometries may be seen as approximations of real ETFs, and
there is only a narrow window in which existence may differ between the two
settings. Nevertheless, marked deviations occur within that window, and our
results imply the existence of a large quantity of ETFs having sizes that are
not known to exist over either the real or complex numbers (Example 5.1).

The paper is laid out as follows. The next section provides background on
SRGs, orthogonal geometry, and frames and equiangular lines over finite
fields. In Section 3, we examine interactions between real ETFs and those in
finite orthogonal geometries, and provide additional necessary conditions for
the existence of the latter. Section 4 develops the crucial relationship
between ETFs in finite orthogonal geometries and modular SRGs. In Section 5 we
give consequences of that theory and find many new ETF sizes over finite
fields, including those that attain Gerzon’s bound in infinitely many
dimensions. Finally, in Section 6 we give a computer-assisted proof that
Gerzon’s bound is never attained in a 5-dimensional finite orthogonal geometry
of odd characteristic.

##  2\. Preliminaries

###  2.1. Review of strongly regular graphs

The reader may consult [9] for background on strongly regular graphs. In this
paper, all graphs are assumed to be simple and undirected. If $\Gamma$ is a
graph on the $v$-element vertex set $V$, then its $\\{0,1\\}$-adjacency matrix
is defined as the $V\times V$ matrix $A$ with $A_{ij}=1$ if vertices $i$ and
$j$ are adjacent, and $A_{ij}=0$ otherwise, whereas its Seidel adjacency
matrix is the $V\times V$ matrix $\Sigma$ with entries

$\Sigma_{ij}=\begin{cases}-1,&\text{if $i$ and $j$ are adjacent};\\\
0,&\text{if }i=j;\\\ 1,&\text{otherwise.}\end{cases}$

We denote $J_{v}$ or simply $J$ for the $v\times v$ matrix of all ones, and
$\mathbf{1}_{v}$ or $\mathbf{1}$ for the column vector of all ones. Then the
two adjacency matrices are related by

(2.1) $\Sigma=J-2A-I.$

By definition, a nontrivial strongly regular graph (SRG) $\Gamma$ is neither
complete nor edgeless, and there are parameters $v,k,\lambda,\mu$ such that
its $\\{0,1\\}$-adjacency matrix $A$ has size $v\times v$ and satisfies

(2.2) $AJ=kJ,\qquad A^{2}=\mu J+(\lambda-\mu)A+(k-\mu)I.$

We abbreviate this by saying $\Gamma$ is a $(v,k,\lambda,\mu)$-SRG. It has
exactly two eigenvalues $r>s$ with eigenvectors outside the span of
$\mathbf{1}$, namely

$r=\frac{1}{2}\left[\lambda-\mu+\sqrt{(\lambda-\mu)^{2}+4(k-\mu)}\right],\quad
s=\frac{1}{2}\left[\lambda-\mu-\sqrt{(\lambda-\mu)^{2}+4(k-\mu)}\right].$

Each satisfies the equation

(2.3) $x^{2}=(\lambda-\mu)x+(k-\mu)\qquad\text{for }x\in\\{r,s\\},$

as well as the relations

(2.4) $r+s=\lambda-\mu,\qquad rs=\mu-k,\qquad(k-r)(k-s)=\mu v.$

The orthogonal projections

$E_{r}=\tfrac{1}{r-s}(A-sI-\tfrac{k-s}{v}J),\qquad
E_{s}=\tfrac{1}{s-r}(A-rI-\tfrac{k-r}{v}J)$

together with $\tfrac{1}{v}J$ form a basis of mutually orthogonal idempotents
for the algebra spanned by $A,I,J$. Here $I=\tfrac{1}{v}J+E_{r}+E_{s}$ and
$A=k(\tfrac{1}{v}J)+rE_{r}+sE_{s}$. We denote
$f=\operatorname{rank}E_{r}=(r-s)^{-1}(s-k-vs)$ and
$g=\operatorname{rank}E_{s}=(s-r)^{-1}(r-k-vr)$. If $f\neq g$ then $r$ and $s$
are integers, and if $f=g$ then $(v,k,\lambda,\mu)=(4\mu+1,2\mu,\mu-1,\mu)$.
The complementary graph $\overline{\Gamma}$ is also an SRG, with parameters
$v^{\prime}=v$, $k^{\prime}=v-k-1$, $\lambda^{\prime}=v-2k+\mu-2$,
$\mu^{\prime}=v-2k+\lambda$ and $r^{\prime}=-s-1$, $s^{\prime}=-r-1$,
$f^{\prime}=g$, $g^{\prime}=f$.

We denote the eigenvalues of the Seidel adjacency matrix $\Sigma=J-2A-I$ of
$\Gamma$ by

(2.5) $\theta_{k}=v-2k-1,\qquad\theta_{r}=-2r-1,\qquad\theta_{s}=-2s-1.$

Here $\Sigma=\theta_{k}(\tfrac{1}{v}J)+\theta_{r}E_{r}+\theta_{s}E_{s}$, and
in particular $\Sigma\mathbf{1}=\theta_{k}\mathbf{1}$. From (2.4) we obtain

(2.6)
$\tfrac{1}{v}(\theta_{k}-\theta_{r})(\theta_{k}-\theta_{s})=v-4k+2\lambda+2\mu,$

and the spectral decomposition of $\Sigma$ implies

(2.7)
$(\Sigma-\theta_{r}I)(\Sigma-\theta_{s}I)=(\theta_{k}-\theta_{r})(\theta_{k}-\theta_{s})(\tfrac{1}{v}J)=(v-4k+2\lambda+2\mu)J.$

###  2.2. Review of orthogonal geometry

The reader may consult [15] for background on orthogonal geometry.

######  Definition 2.1.

Given an odd prime power $q$, a quadratic space over $\mathbb{F}_{q}$ is a
finite-dimensional vector space $W$ equipped with a form
$\langle\cdot,\cdot\rangle\colon W\times W\to\mathbb{F}_{q}$ that satisfies
the following:

  * (O1)

for every $u\in W$, the induced mapping $\langle u,\cdot\rangle\colon
W\to\mathbb{F}_{q}$ is linear,

  * (O2)

$\langle u,w\rangle=\langle w,u\rangle$ for every $u,w\in W$,

  * (O3)

if $u\in W$ satisfies $\langle u,w\rangle=0$ for every $w\in W$, then $u=0$.

In other words, $\langle\cdot,\cdot\rangle$ is a nondegenerate symmetric
bilinear form on $W$. We also say that $\langle\cdot,\cdot\rangle$ induces an
orthogonal geometry on $W$.

In Definition 2.1, if $\\{e_{j}\\}_{j\in[d]}$ is a basis for $W$, then the
form $\langle\cdot,\cdot\rangle$ above is completely determined by its Gram
matrix $M=\begin{bmatrix}\langle e_{i},e_{j}\rangle\end{bmatrix}_{i,j\in[d]}$.
It is symmetric and invertible, and conversely any symmetric invertible
$d\times d$ matrix over $\mathbb{F}_{q}$ is the Gram matrix of an orthogonal
geometry on $W$. The determinant of the Gram matrix $M$ lies in the
multiplicative group $\mathbb{F}_{q}^{\times}$ of nonzero elements in
$\mathbb{F}_{q}$, and if we denote

$\mathbb{F}_{q}^{\times
2}=\\{\alpha^{2}:\alpha\in\mathbb{F}_{q}^{\times}\\}\leq\mathbb{F}_{q}^{\times}$

for the subgroup of nonzero quadratic residues, then the quadratic space $W$
is said to have discriminant

$\operatorname{discr}(W)=(\det M)\,\mathbb{F}_{q}^{\times
2}\in\mathbb{F}_{q}^{\times}/\mathbb{F}_{q}^{\times 2}.$

The latter does not depend on the choice of basis for $W$.

######  Definition 2.2.

For an odd prime power $q$, the real model on $\mathbb{F}_{q}^{d}$ is given by
the nondegenerate symmetric bilinear form $(x,y)=x^{\top}y$.

Up to isometric isomorphism, a given vector space $W$ has exactly two types of
orthogonal geometry: one for which the discriminant is trivial (square-type),
and another for which it is not. For example, the real model on the vector
space $W=\mathbb{F}_{q}^{d}$ has Gram matrix $M=I$ with $\det M=1$, so it is
an orthogonal geometry with square-type discriminant. As an example of the
other type of orthogonal geometry on $\mathbb{F}_{q}^{d}$, let
$\zeta\in\mathbb{F}_{q}^{\times}$ be a primitive element and create
$\langle\cdot,\cdot\rangle$ from the Gram matrix
$M=\operatorname{diag}(1,\dotsc,1,\zeta)$. Then $\det
M=\zeta\notin\mathbb{F}_{q}^{\times 2}$, so that the geometry induced by
$\langle\cdot,\cdot\rangle$ on $\mathbb{F}_{q}^{d}$ has nontrivial
discriminant.

If $W$ is a quadratic space, then a subspace $U\leq W$ is called totally
isotropic if $\langle u,w\rangle=0$ for every $u,w\in U$. In that case $\dim
U\leq\tfrac{1}{2}\dim W$.

###  2.3. Review of frames and equiangular lines

The basic theory of frames and equiangular lines over finite fields is
developed in the companion paper [14]. We summarize the relevant portions
here.

Throughout the paper, we abuse notation by identifying a sequence
$\Phi=\\{\varphi_{j}\\}_{j\in[n]}$ of vectors in a quadratic space $W$ with
its synthesis operator $\Phi\colon\mathbb{F}_{q}^{n}\to W$ given by

$\Phi\\{x_{i}\\}_{i\in[n]}=\sum_{i\in[n]}x_{i}\varphi_{i}.$

Its analysis operator $\Phi^{\dagger}\colon W\to\mathbb{F}_{q}^{n}$ is given
by
$\Phi^{\dagger}w=\begin{bmatrix}\langle\varphi_{i},w\rangle\end{bmatrix}_{i\in[n]}$,
and its Gram matrix is the matrix for $\Phi^{\dagger}\Phi$ in the standard
basis, namely
$G=\begin{bmatrix}\langle\varphi_{i},\varphi_{j}\rangle\end{bmatrix}\in\mathbb{F}_{q}^{n\times
n}$.

######  Definition 2.3.

Let $W$ be a $d$-dimensional quadratic space over $\mathbb{F}_{q}$, where $q$
is odd. A sequence $\Phi=\\{\varphi_{j}\\}_{j\in[n]}$ in $W$ is called a frame
of size $d\times n$ if it spans $W$. A tight frame is a frame $\Phi$ for which
there exists a frame constant $c\in\mathbb{F}_{q}$ such that
$\Phi\Phi^{\dagger}=cI$. We also call $\Phi$ a $c$-tight frame in that case.
If $\Phi=\\{\varphi_{j}\\}_{j\in[n]}$ is a $c$-tight frame and there exists
$a\in\mathbb{F}_{q}$ such that $\langle\varphi_{j},\varphi_{j}\rangle=a$ for
every $j\in[n]$, then we call $\Phi$ an equal norm tight frame, or an
$(a,c)$-NTF.

Unlike the real setting, it may happen that the frame constant equals zero. We
emphasize that the vectors of a $0$-tight frame span the ambient space by
definition. If $\Phi$ is an $(a,c)$-NTF of size $d\times n$, then the equal
traces of $\Phi^{\dagger}\Phi$ and $\Phi\Phi^{\dagger}$ give

(2.8) $na=dc.$

######  Proposition 2.4 ([14]).

If $\Phi$ is a $d\times n$ frame, then it is a $0$-tight frame if and only if
$\operatorname{im}\Phi^{\dagger}$ is a totally isotropic subspace of
$\mathbb{F}_{q}^{n}$ in the real model. In particular, a $0$-tight frame of
size $d\times n$ exists only if $n\geq 2d$.

As in the real setting, it is possible to infer the existence of a frame from
a corresponding Gram matrix. In addition, the Gram matrix uniquely determines
the discriminant of the ambient orthogonal geometry, which may be detected
through the following notion.

######  Definition 2.5.

Given an $n\times n$ matrix
$A=\left[\begin{array}[]{ccc}a_{1}&\cdots&a_{n}\end{array}\right]=\begin{bmatrix}A_{ij}\end{bmatrix}_{i,j\in[n]}$,
choose any indexing set $J\subseteq[n]$ such that $\\{a_{j}\\}_{j\in J}$ forms
a basis for $\operatorname{im}A$. Then the principal submatrix
$A_{b}:=\begin{bmatrix}A_{ij}\end{bmatrix}_{i,j\in J}$ is called a basic
submatrix of $A$.

In the following characterization, notice the absence of any condition akin to
positive semidefiniteness.

######  Proposition 2.6 ([14]).

Let $G\in\mathbb{F}_{q}^{n\times n}$ ($q$ odd) be a square matrix, and let
$G_{b}$ be a basic submatrix (Definition 2.5). Given a quadratic space $W$,
$G$ is the Gram matrix of a frame for $W$ if and only if all of the following
are satisfied:

  * (i)

$G^{\top}=G$,

  * (ii)

$\operatorname{rank}G=\dim W$,

  * (iii)

$(\det G_{b})\mathbb{F}_{q}^{\times 2}=\operatorname{discr}(W)$.

In particular, $G$ is symmetric if and only if it is the Gram matrix of a
frame for an orthogonal geometry with discriminant $(\det
G_{b})\mathbb{F}_{q}^{\times 2}$ on $\mathbb{F}_{q}^{\operatorname{rank}G}$.

######  Proposition 2.7 ([14]).

Suppose $G\in\mathbb{F}_{q}^{n\times n}$ is the Gram matrix of a frame $\Phi$.
Then, $G^{2}=cG$ if and only if $\Phi$ is a $c$-tight frame.

######  Definition 2.8.

Let $\Phi=\\{\varphi_{j}\\}_{j\in[n]}$ be a sequence of vectors in a quadratic
space $W$ over $\mathbb{F}_{q}$. For $a,b\in\mathbb{F}_{q}$, $\Phi$ is called
an $(a,b)$-equiangular system if:

  * (i)

$\langle\varphi_{j},\varphi_{j}\rangle=a$ for every $j\in[n]$,

  * (ii)

$\langle\varphi_{i},\varphi_{j}\rangle^{2}=b$ for every $i\neq j$ in $[n]$.

If this occurs and $\varphi_{j}\neq 0$ for every $j$, then we call
$\mathscr{L}=\\{\operatorname{span}\varphi_{j}\\}_{j\in[n]}$ a sequence of
equiangular lines. An $(a,b)$-equiangular system that is also a $c$-tight
frame is known as an equiangular tight frame, or $(a,b,c)$-ETF.

We have the following version of Gerzon’s bound [17].

######  Proposition 2.9 (Gerzon’s bound, [14]).

Suppose $q$ is an odd prime power and $\Phi$ is an $(a,b)$-equiangular system
of $n$ vectors in a $d$-dimensional quadratic space over $\mathbb{F}_{q}$,
such that $a^{2}\neq b$. Then $n\leq\binom{d+1}{2}$, and equality holds only
if $\Phi$ is an ETF.

The parameters of an $(a,b,c)$-ETF are related in a manner similar to that
given by the Welch bound in the real and complex settings [30].

######  Proposition 2.10 ([14]).

If $\Phi$ is an $(a,b,c)$-ETF of size $d\times n$, then

$a(c-a)=(n-1)b.$

Finally, the sizes of ETFs with nonzero frame constant come in Naimark
complementary pairs. We emphasize that the two geometries mentioned below may
have different discriminants.

######  Proposition 2.11 (Naimark complements, [14]).

If $c\neq 0$ and there is an $(a,b,c)$-ETF in an orthogonal geometry on
$\mathbb{F}_{q}^{d}$ ($q$ odd), then there is a $(c-a,b,c)$-ETF in an
orthogonal geometry on $\mathbb{F}_{q}^{n-d}$.

###  2.4. Initial observations

Before beginning in earnest, we make a few basic observations about ETFs in
orthogonal geometries. First of all, the existence of an ETF may be detected
from its Gram matrix, where the following is immediate from Proposition 2.6
and Proposition 2.7.

######  Proposition 2.12.

Suppose $G$ is a symmetric $n\times n$ matrix with entries in
$\mathbb{F}_{q}$, where $q$ is odd. Given $a,b,c\in\mathbb{F}_{q}$, $G$ is the
Gram matrix of an $(a,b,c)$-ETF in an orthogonal geometry on
$\mathbb{F}_{q}^{d}$ if and only if all of the following hold:

  * (ii)

$G_{ii}=a$ for every $i\in[n]$,

  * (ii)

$(G_{ij})^{2}=b$ for every $i\neq j$ in $[n]$,

  * (iii)

$G^{2}=cG$,

  * (iv)

$\operatorname{rank}G=d$.

######  Remark 2.13.

Proposition 2.12 fails to mention the type of orthogonal geometry on
$\mathbb{F}_{q}^{d}$, even though it is uniquely determined by $G$ and can be
recovered from the determinant of a basic submatrix. This is easy to compute
in practice but may be difficult to predict in general. For this reason, we
shall often be deliberately ambiguous as we have done in Proposition 2.12,
referring only to “an” orthogonal geometry without specifying its
discriminant.

In any case, the discriminant can always be made unambiguous by passing to a
quadratic extension field: any symmetric $G\in\mathbb{F}_{q}^{n\times n}$ is
the Gram matrix of a frame for $\mathbb{F}_{q^{2}}^{\operatorname{rank}G}$ in
the real model, since the determinant of a basic submatrix lies in
$\mathbb{F}_{q}^{\times}\leq\mathbb{F}_{q^{2}}^{\times 2}$. In particular, if
an $(a,b,c)$-ETF exists in an orthogonal geometry on $\mathbb{F}_{q}^{d}$,
then an $(a,b,c)$-ETF exists in the real model on $\mathbb{F}_{q^{2}}^{d}$.

######  Remark 2.14.

We will see later that, as far as existence of $d\times n$ ETFs is concerned,
one may assume the field order $q=p^{l}$ is either prime (when $n\neq 2d$) or
prime squared (when $n=2d$). As in the previous remark, one may further assume
the geometry is the real model by taking $q=p^{2}$ (when $n\neq 2d$) or
$q=p^{4}$ (when $n=2d$). See Theorem 3.4.

######  Remark 2.15.

The parameters of an $(a,b,c)$-ETF may be altered by rescaling its Gram
matrix. If $G\in\mathbb{F}_{q}^{n\times n}$ is the Gram matrix of an
$(a,b,c)$-ETF in an orthogonal geometry and
$\alpha\in\mathbb{F}_{q}^{\times}$, then $\alpha G$ is the Gram matrix of an
$(\alpha a,\alpha^{2}b,\alpha c)$-ETF in an orthogonal geometry of the same
dimension. Notice that the discriminant changes if the dimension is odd and
$\alpha\notin\mathbb{F}_{q}^{\times 2}$. If $n>d$ then $b\neq 0$, since the
vectors of a $(0,0)$-equiangular system span a totally isotropic subspace (and
therefore cannot form a frame), while those of an $(a,0)$-equiangular system
with $a\neq 0$ are linearly independent (by the usual argument). As $b$ is a
quadratic residue, we can normalize to ensure $b=1$ whenever $n>d$. In that
case the ambiguity associated with $\alpha=-1$ remains. (We usually do not
normalize $a$ or $c$ since each is often equal to $0$.) After such
normalization, the Gram matrix takes the form $G=S+aI$, where $S$ has zeros on
the diagonal and $\pm 1$ off the diagonal. Here, $S$ is known as the signature
matrix of the rescaled ETF. Up to a change in sign, it is uniquely determined
by the unscaled ETF.

We end our preliminary discussion with some trivial ETF constructions.

######  Example 2.16.

For any $n\geq 2$ and any odd prime power $q=p^{l}$, the matrices
$I,J\in\mathbb{F}_{q}^{n\times n}$ satisfy $I^{2}=I$ and $J^{2}=nJ$. The
former produces a $(1,0,1)$-ETF in an orthogonal geometry of dimension $d=n$
over $\mathbb{F}_{q}$. The latter produces a $(1,1,n)$-ETF of size $1\times
n$. When $p$ fails to divide $n$, we can take a Naimark complement to obtain
an $(n-1,1,n)$-ETF of size $(n-1)\times n$.

##  3\. Field conditions

In this section, we provide conditions under which the signature matrix of an
ETF in one field produces another ETF in a different field. Using this
technique, we demonstrate a close correspondence between real ETFs and ETFs in
finite orthogonal geometries. In the finite field setting, we show that in
most cases the parameters of an $(a,b,c)$-ETF belong to the base field
$\mathbb{F}_{p}$, and so its Gram matrix factors to produce another ETF of the
same size in an orthogonal geometry over the base field. Considering relations
between the parameters, we deduce necessary conditions on $d,n,p$ such that a
$d\times n$ ETF exists in a finite orthogonal geometry of characteristic $p$.

We use the following notation when moving between fields. Let
$R\subset\mathbb{C}$ be a unital ring of algebraic integers (that is, zeroes
of monic polynomials with integer coefficients), and let $\pi\colon R\to F$ be
a unital ring homomorphism into a finite field $F$ of characteristic $p$. (For
instance, $R=\mathbb{Z}$ and $F=\mathbb{F}_{p}$.) Given a matrix $M$ with
entries in $R$, we denote $\overline{M}=\pi(M)$ for its entrywise image over
$F$ and $\operatorname{rank}_{F}\overline{M}$ for its rank over $F$. In case
$M$ has integer entries, we also write
$\operatorname{rank}_{p}\overline{M}:=\operatorname{rank}_{\mathbb{F}_{p}}\overline{M}=\operatorname{rank}_{F}\overline{M}$.
We routinely abuse notation by identifying integers $k\in\mathbb{Z}$ with
their images $\overline{k}=\pi(k)\in\mathbb{F}_{p}\leq F$. The symbols
$\equiv$ and $\equiv_{p}$ indicate equality in $F$ and $\mathbb{F}_{p}$,
respectively, after appropriate applications of $\pi\colon R\to F$. These
abuses should not cause any confusion in practice, and the exact meaning
should be clear from context.

The following proposition will prove indispensible. It summarizes aspects of
Propositions 13.3.1, 13.3.2, and 13.3.4 of [7]. In [7] it is assumed that $R$
is the entire ring of algebraic integers in $\mathbb{C}$. However, a careful
reading shows that the same proofs apply in our more general setting.

######  Proposition 3.1 ([7]).

Let $M$ be an $n\times n$ matrix all of whose entries and eigenvalues lie in
$R$.

  * (a)

We have $\operatorname{rank}_{F}\overline{M}\leq\operatorname{rank}M$, with
equality if $M$ is diagonalizable and every nonzero eigenvalue $\theta$
satisfies $\overline{\theta}\neq 0$.

  * (b)

Suppose $M$ is nonzero, singular, and symmetric with integral entries and
constant row sum $\theta_{0}$, and that the $\theta_{0}$-eigenspace is one-
dimensional. Denote the minimal polynomial of $M$ by $(x-\theta_{0})h(x)$, and
define $\epsilon=1$ if $\overline{h(M)}=0$ and $\epsilon=0$ otherwise. If
every other nonzero eigenvalue $\theta\neq\theta_{0}$ of $M$ satisfies
$\overline{\theta}\neq 0$, then
$\operatorname{rank}_{F}\overline{M}=\operatorname{rank}M-\epsilon$.

###  3.1. Interactions between real and finite fields

######  Proposition 3.2.

Suppose $S\in\mathbb{Z}^{n\times n}$ is the signature matrix of a real
$d\times n$ ETF with $n>d+1$. Let $q=p^{l}$ be any odd prime power, but if
$n=2d$ assume there exists $\delta\in\mathbb{F}_{q}$ with $\delta^{2}\equiv
n-1$. Then there exist $a,c\in\mathbb{F}_{q}$ such that
$\overline{S}+aI\in\mathbb{F}_{q}^{n\times n}$ is the Gram matrix of an
$(a,1,c)$-ETF of $n$ vectors in an orthogonal geometry on
$\mathbb{F}_{q}^{d^{\prime}}$, where $d^{\prime}\leq d$ and

$a\equiv\begin{cases}\sqrt{\frac{d(n-1)}{n-d}},&\text{if }n\neq 2d;\\\
\delta,&\text{if }n=2d\end{cases}\qquad\text{and}\qquad
c\equiv\begin{cases}\sqrt{\frac{n^{2}(n-1)}{d(n-d)}},&\text{if }n\neq 2d;\\\
2\delta,&\text{if }n=2d.\end{cases}$

(The square roots are integers in the indicated cases.) Furthermore,
$d^{\prime}=d$ if $c\not\equiv 0$.

###### Proof.

Let $G=S+\hat{a}I\in\mathbb{R}^{n\times n}$ be the Gram matrix of a $d\times
n$ ETF with frame constant $\hat{c}\in\mathbb{R}$. Since $G\neq 0$ satisfies
$G^{2}=\hat{c}G$, $S$ has eigenvalues $-\hat{a}$ and $\hat{c}-\hat{a}$, where

$\hat{a}=\sqrt{\frac{d(n-1)}{n-d}},\qquad\hat{c}=\tfrac{d}{n}\hat{a}=\sqrt{\frac{n^{2}(n-1)}{d(n-d)}}.$

(See [25, Section 4].)

First suppose $n\neq 2d$. Corollary 13 of [25] states that $S$ has integer
eigenvalues, so $\hat{a},\hat{c}\in\mathbb{Z}$. Therefore $G$ has integer
entries and integer spectrum $\sigma(G)=\\{0^{n-d},\hat{c}^{d}\\}$, where
superscripts denote geometric multiplicities. Its image
$\overline{G}\in\mathbb{F}_{q}^{n\times n}$ satisfies
$\overline{G}^{2}=\hat{c}\overline{G}$, and, by Proposition 2.12, it is the
Gram matrix of an $(\hat{a},1,\hat{c})$-ETF in an orthogonal geometry on
$\mathbb{F}_{q}^{d^{\prime}}$, where
$d^{\prime}=\operatorname{rank}_{p}\overline{G}$. We have $d^{\prime}\leq d$
by Proposition 3.1(a), and equality holds if $\hat{c}\not\equiv_{p}0$.

Now suppose $n=2d$. We have $\hat{a}=\sqrt{n-1}$ and $\hat{c}=2\sqrt{n-1}$, so
the entries and eigenvalues of $G$ lie in the ring $R=\mathbb{Z}[\sqrt{n-1}]$.
Let $\pi\colon R\to\mathbb{F}_{q}$ be the unique unital ring homomorphism for
which $\pi(\sqrt{n-1})=\delta$. In this case we consider
$\overline{G}=\pi(G)\in\mathbb{F}_{q}^{n\times n}$. As above, $\overline{G}$
is the Gram matrix of a $(\delta,1,2\delta)$-ETF in an orthogonal geometry on
$\mathbb{F}_{q}^{d^{\prime}}$, and $d^{\prime}\leq d$ with equality if
$\delta\neq 0$. ∎

We have a partial converse to Proposition 3.2.

######  Proposition 3.3.

Suppose there is an ETF of $n$ vectors in an orthogonal geometry on
$\mathbb{F}_{q}^{d}$. If $q=p^{l}$ is odd and $p>2n-5$, then there is an ETF
of $n$ vectors in $\mathbb{R}^{d}$.

We do not know if $p>2n-5$ is a sharp bound.

###### Proof.

By applying Proposition 2.11 if necessary, we may assume that $n\geq 2d$.
(Note that if the frame constant is zero then we already have $n\geq 2d$ by
Proposition 2.4.) We may also assume that $n\geq d+2\geq 4$, since otherwise a
real ETF of the desired size is already known to exist. Then it is easy to
show

$\max\\{d,n-d,n-1\\}\leq 2n-5<p,$

so that $d\not\equiv_{p}0$ and $n-d\not\equiv_{p}0$.

Let $G=S+aI\in\mathbb{F}_{q}^{n\times n}$ be the Gram matrix of an
$(a,1,c)$-ETF, where the signature matrix $S\in\mathbb{F}_{q}^{n\times n}$ has
zeros on the diagonal and $\pm 1$ off the diagonal. Here
$G^{2}=cG=\tfrac{n}{d}aG$ by (2.8), or equivalently,

$S^{2}=\tfrac{n-2d}{d}aS+\tfrac{n-d}{d}a^{2}I.$

On the other hand, $S^{2}$ has entries in $\mathbb{F}_{p}$, and its diagonal
is constantly $n-1$. Therefore $\beta:=\tfrac{n-2d}{d}a$ lies in
$\mathbb{F}_{p}$, and $\tfrac{n-d}{d}a^{2}=n-1$. Consequently,

(3.1) $\beta^{2}=\frac{(n-2d)^{2}(n-1)}{d(n-d)}.$

We are going to lift $S$ to a matrix over $\mathbb{Z}$ and recover a real ETF.
Let $\pi\colon\mathbb{Z}\to\mathbb{F}_{q}$ be the unique unital ring
homomorphism. Define $\hat{\beta}$ to be the unique integer with
$-\tfrac{1}{2}(p-1)\leq\hat{\beta}\leq\tfrac{1}{2}(p-1)$ and
$\pi(\hat{\beta})=\beta$, and define $\hat{S}\in\mathbb{Z}^{n\times n}$ to be
the unique matrix with $\hat{S}_{ij}\in\\{0,\pm 1\\}$ and
$\pi(\hat{S}_{ij})=S_{ij}$ for every $i,j$. Since $\pi$ extends to a
homomorphism of matrix rings, we have
$\pi\left[(\hat{S}^{2})_{ij}-\hat{\beta}\hat{S}_{ij}\right]=0$ whenever $i\neq
j$. On the other hand, when $i\neq j$,

$0\leq\left|(\hat{S}^{2})_{ij}-\hat{\beta}\hat{S}_{ij}\right|\leq\left|(\hat{S}^{2})_{ij}\right|+\left|\hat{\beta}\hat{S}_{ij}\right|\leq(n-2)+|\hat{\beta}|<p$

since $n<\tfrac{1}{2}(p+5)$ and $|\hat{\beta}|\leq\tfrac{1}{2}(p-1)$.
Therefore $(\hat{S}^{2})_{ij}-\hat{\beta}\hat{S}_{ij}=0$ whenever $i\neq j$,
and

$\hat{S}^{2}=\hat{\beta}\hat{S}+(n-1)I.$

Hence $\hat{S}$ has just two eigenvalues, and one is negative since the trace
is zero. Writing $-\hat{a}$ for the negative eigenvalue, we see that
$\hat{G}:=\hat{S}+\hat{a}I$ is the Gram matrix of an ETF in
$\mathbb{R}^{d^{\prime}}$ for some $d^{\prime}$. Furthermore, proceeding as
above we find

(3.2)
$\hat{\beta}^{2}=\frac{(n-2d^{\prime})^{2}(n-1)}{d^{\prime}(n-d^{\prime})}.$

Arguing by cases, we now show $d^{\prime}\in\\{d,n-d\\}$. First suppose
$n\equiv_{p}2d$. Then (3.1) gives $\beta=0$, so $\hat{\beta}=0$ and
$n=2d^{\prime}$ by (3.2). Since $n\geq 2d$ and $d\geq 2$ we have $0\leq
n-2d\leq n-4<p$. It follows that $n=2d$, hence $d^{\prime}=d$.

Now suppose $n\not\equiv_{p}2d$. Rewriting (3.2), we find that $d^{\prime}$ is
a root of

$h(x)=(4n-4+\hat{\beta}^{2})x^{2}-n(\hat{\beta}^{2}+4n-4)x+n^{2}(n-1)\in\mathbb{Z}[x].$

Likewise, (3.1) says that $d$ is a root of

$\overline{h}(x)=(4n-4+\beta^{2})x^{2}-n(\beta^{2}+4n-4)x+n^{2}(n-1)\in\mathbb{F}_{p}[x].$

By inspection of (3.1), we see that the other root of $\overline{h}$ is $n-d$.
We have $d\not\equiv_{p}n-d$ by case assumption, so these are the only roots
of $\overline{h}$ in $\mathbb{F}_{p}$. Applying $\pi$ to $d^{\prime}$ and the
coefficients of $h$, we deduce that either $d^{\prime}\equiv_{p}d$ or
$d^{\prime}\equiv_{p}n-d$. In the first case, observe

$0\leq|d-d^{\prime}|=\max\\{d-d^{\prime},d^{\prime}-d\\}\leq n-2<p,$

so that $d^{\prime}=d$. In the second case, we have $n-(d+d^{\prime})\leq
n-2<p$ and $(d+d^{\prime})-n\leq d<p$, so $0\leq|n-d-d^{\prime}|<p$ and
$d^{\prime}=n-d$. Now taking a Naimark complement produces a real $d\times n$
ETF. ∎

###  3.2. Additional constraints

The following theorem and its corollary may be seen as finite field versions
of [25, Theorem 15]. Our proof uses a similar strategy.

######  Theorem 3.4.

Suppose there is an $(a,1,c)$-ETF of $n$ vectors in an orthogonal geometry on
$\mathbb{F}_{q}^{d}$, where $q=p^{l}$ is an odd prime power. Then
$a,c\in\mathbb{F}_{p^{j}}$ for some $j\in\\{1,2\\}$, with $j=1$ if $n\neq 2d$
or $c=0$. Consequently, there is an ETF of $n$ vectors in an orthogonal
geometry on $\mathbb{F}_{p^{j}}^{d}$.

We emphasize that the hypothesis $n\neq 2d$ is not to be confused with the
stronger condition $n\not\equiv_{p}2d$.

###### Proof.

Let $G\in\mathbb{F}_{q}^{n\times n}$ be the Gram matrix of the given ETF, and
define $S=G-aI$. Notice that $S$ has entries in $\mathbb{F}_{p}$, and so

(3.3) $\chi_{S}(x):=\det(xI-S)\in\mathbb{F}_{p}[x].$

On the other hand, $G$ has minimal polynomial $m_{G}(x)=x(x-c)$ with only $0$
and $c$ as its roots. If $c\neq 0$, then the Jordan normal form $D$ of $G$ is
diagonal with $d=\operatorname{rank}G$ copies of $c$ and $n-d$ copies of $0$
on the diagonal, and if $c=0$ then $D$ is upper triangular with all zeros on
the diagonal. In either case, we find that
$\chi_{G}(x):=\det(xI-G)=(x-c)^{d}x^{n-d}$, and so

(3.4) $\chi_{S}(x)=\chi_{G}(x+a)=(x+a-c)^{d}(x+a)^{n-d}.$

By considering the Jordan normal form of $G$ again, we see that $d\neq n$. (If
$c=0$ then $d\leq\tfrac{1}{2}n<n$. If $c\neq 0$ and $d=n$, then $D=cI=G$,
despite being the Gram matrix of an $(a,1,c)$-ETF.) Therefore
$\chi_{S}(-a)=0$.

Let $h\in\mathbb{F}_{p}[x]$ be the minimal polynomial of $-a$ over
$\mathbb{F}_{p}$. By definition, this is the unique monic polynomial in
$\mathbb{F}_{p}[x]$ that satisfies $h(-a)=0$ and divides every other
polynomial $k\in\mathbb{F}_{p}[x]$ with $k(-a)=0$. Equivalently, it is the
unique irreducible monic polynomial $h(x)\in\mathbb{F}_{p}[x]$ satisfying
$h(-a)=0$. Using (3.3) and (3.4), we see that $h(x)$ divides $\chi_{S}(x)$ in
$\mathbb{F}_{p}[x]$, and that it has no roots in an algebraic closure of
$\mathbb{F}_{p}$ other than $-a$ and, perhaps, $c-a$. Since finite fields are
perfect, $h(x)$ has no repeated roots, and it must be one of two
possibilities:

  * (I)

$h(x)=x+a$, or

  * (II)

$h(x)=(x+a)(x+a-c)$.

In both cases, $-a$ belongs to a quadratic extension of $\mathbb{F}_{p}$, and
so $a\in\mathbb{F}_{p^{2}}$. Considering that $G\neq 0$ has entries in
$\mathbb{F}_{p^{2}}$ and satisfies $G^{2}=cG$, we deduce that
$c\in\mathbb{F}_{p^{2}}$ as well. Moreover, if $c=0$ then option (II) cannot
occur since $h(x)$ has no repeated roots, and in this case we conclude that
$a\in\mathbb{F}_{p}$.

Now suppose $n\neq 2d$ and $c\neq 0$. Assume for the sake of contradiction
that option (II) occurs. Then $h(x)$ is also the minimal polynomial of $c-a$
over $\mathbb{F}_{p}$. In the factorization

$\chi_{S}(x)=\prod_{i=1}^{k}f_{i}(x)$

of $\chi_{S}(x)$ as a product of monic irreducible polynomials
$f_{i}(x)\in\mathbb{F}_{p}[x]$, every factor $f_{i}(x)$ has either $-a$ or
$c-a$ as a root by (3.4). By uniqueness of the minimal polynomial, every
factor $f_{i}(x)$ equals $h(x)$, so that

$\chi_{S}(x)=h(x)^{k}=(x+a)^{k}(x+a-c)^{k}.$

This contradicts (3.4) since $a\neq a-c$ and $d\neq n-d$. Therefore option (I)
holds, and $a,c\in\mathbb{F}_{p}$.

In every case, the “consequentally” statement follows from Proposition 2.12. ∎

######  Corollary 3.5.

Suppose there is an ETF of $n\notin\\{d,2d\\}$ vectors in an orthogonal
geometry on $\mathbb{F}_{q}^{d}$, where $q=p^{l}$ is an odd prime power. If
$d\not\equiv_{p}0$ and $n\not\equiv_{p}1$, then $n\not\equiv_{p}d$ and
$\overline{d(n-1)}(\overline{n-d})^{-1}\in\mathbb{F}_{p}^{\times 2}$.

###### Proof.

As explained in Remark 2.15, there is an $(a,1,c)$-ETF of $n$ vectors in an
orthogonal geometry on $\mathbb{F}_{q}^{d}$, where $a\in\mathbb{F}_{p}$ by
Theorem 3.4. Proposition 2.10 states that
$1=(\overline{n-d})[\overline{d(n-1)}]^{-1}a^{2}$, so $\overline{n-d}\neq 0$
and

$\overline{d(n-1)}(\overline{n-d})^{-1}=a^{2}\in\mathbb{F}_{p}^{\times
2}.\qed$

##  4\. Relationships with modular SRGs

As in the real setting, ETFs in finite orthogonal geometries are closely
associated with graphs given by the sign patterns in their signature matrices.
In this section we study this correspondence in detail. Specifically, let $G$
be the Gram matrix of an $(a,b,c)$-ETF $\Phi$ of $n>d$ vectors in a
$d$-dimensional orthogonal geometry over $\mathbb{F}_{q}$, where $q$ is odd.
After rescaling as in Remark 2.15, we may assume $b=1$. Then the signature
matrix $S:=G-aI$ of $\Phi$ is symmetric with zeros on the diagonal and $\pm 1$
off the diagonal, and so it is the Seidel adjacency matrix of a graph $\Gamma$
on $n$ vertices. A second graph related to $\Phi$ is obtained by multiplying
each of the frame vectors by $\pm 1$ to produce another $(a,1,c)$-ETF
$\Phi^{\prime}$ with Gram matrix

$G^{\prime}=\left[\begin{array}[]{cc}a&\mathbf{1}_{v}^{\top}\\\
\mathbf{1}_{v}&\Sigma+aI_{v}\end{array}\right].$

Here $\Phi^{\prime}$ is said to have normalized signature matrix, and $\Sigma$
is the Seidel adjacency matrix of a graph $\Gamma^{\prime}$ on $n-1$ vertices.

In this section we characterize graphs $\Gamma^{\prime}$ for which there
exists $a$ such that $G^{\prime}$ above is the Gram matrix of an ETF (Theorem
4.3). We also characterize graphs $\Gamma$ for which there exists $a$ such
that $G$ above is the Gram matrix of an ETF and the all-ones vector is an
eigenvector of $G$ (Theorem 4.5). In both cases, we are led to a modular
generalization of strongly regular graphs. SRGs form a special subclass, and
for this subclass we identify the ETF size and parameters directly from the
SRG parameters (Theorem 4.4 and Theorem 4.6).

We frequently encounter matrices $S\in\\{0,1,-1\\}^{n\times n}$, and we
sometimes abuse notation by using the same symbol for $S\in\mathbb{Z}^{n\times
n}$ as well as its image $\overline{S}\in\mathbb{F}_{q}^{n\times n}$. This
should not cause any confusion in practice.

###  4.1. ETFs with normalized signature matrix

######  Definition 4.1.

Given an odd prime $p$, a graph $\Gamma$ on vertex set $V$ of size $v$ is
called a $p$-modular strongly regular graph with parameters $v,k,\lambda,\mu$
(briefly, a $(v,k,\lambda,\mu)\text{-SRG}_{p}$) if the following hold:

  * (i)

every vertex $i\in V$ has valency $k_{i}\equiv_{p}k$,

  * (ii)

whenever $i,j\in V$ are adjacent, they have exactly
$\lambda_{ij}\equiv_{p}\lambda$ neighbors in common,

  * (iii)

whenever $i,j\in V$ are distinct and non-adjacent, they have exactly
${\mu_{ij}\equiv_{p}\mu}$ neighbors in common.

Equivalently, $\Gamma$ is a $(v,k,\lambda,\mu)\text{-SRG}_{p}$ if and only if
its $\\{0,1\\}$-adjacency matrix $A$ satisfies $A^{2}\equiv_{p}\mu
J+(\lambda-\mu)A+(k-\mu)I$. Applying the operators on both sides of the
equivalence to the vector $\mathbf{1}$, we see that the parameters of a
$(v,k,\lambda,\mu)\text{-SRG}_{p}$ necessarily satisfy

(4.1) $k(k-\lambda-1)\equiv_{p}\mu(v-k-1).$

The following characterization of modular SRGs can be proved easily using
(2.1) and the inversion of a $4\times 4$ matrix. We omit details.

######  Lemma 4.2.

Let $\Sigma$ be the Seidel adjacency matrix of a graph $\Gamma$ on $v$
vertices. The following are true for any choice of odd prime $p$:

  * (a)

If $\Gamma$ is a $(v,k,\lambda,\mu)\text{-SRG}_{p}$, then
$\Sigma\mathbf{1}\equiv_{p}(v-2k-1)\mathbf{1}$ and

$\Sigma^{2}\equiv_{p}(v-4k+2\lambda+2\mu)J+2(\mu-\lambda-1)\Sigma+(4k-2\lambda-2\mu-1)I.$

  * (b)

Suppose there exist $\alpha,\beta,\gamma,\theta$ in an extension field
$\mathbb{F}_{q}\geq\mathbb{F}_{p}$ such that
$\Sigma\mathbf{1}\equiv\theta\mathbf{1}$ and $\Sigma^{2}\equiv\alpha
J+\beta\Sigma+\gamma I$ in $\mathbb{F}_{q}$. Then
$\alpha,\beta,\gamma,\theta\in\mathbb{F}_{p}$, and $\Gamma$ is a
$(v,k,\lambda,\mu)\text{-SRG}_{p}$ for

(4.2) $\left\\{\begin{array}[]{rcl}v&\equiv&\alpha+\gamma+1\\\\[4.0pt]
k&\equiv&\tfrac{1}{2}(\alpha+\gamma-\theta)\\\\[4.0pt]
\lambda&\equiv&\tfrac{1}{4}(2\alpha-\beta+\gamma-2\theta-3)\\\\[4.0pt]
\mu&\equiv&\tfrac{1}{4}(2\alpha+\beta+\gamma-2\theta+1),\end{array}\right.$

where the inverses are taken in $\mathbb{F}_{p}$.

The theorem below may be considered as a $p$-modular version of Seidel’s
classical theorem relating strongly regular graphs with real ETFs [22, 26,
29].

######  Theorem 4.3.

Let $\Sigma$ be the Seidel adjacency matrix of a graph $\Gamma$ on $v=n-1$
vertices, and define

$S=\left[\begin{array}[]{cc}0&\mathbf{1}_{v}^{\top}\\\
\mathbf{1}_{v}&\Sigma\end{array}\right].$

Given an odd prime power $q=p^{l}$, there exist $a,c\in\mathbb{F}_{q}$ such
that ${S+aI}\in\mathbb{F}_{q}^{n\times n}$ is the Gram matrix of an
$(a,1,c)$-ETF if and only if $\Gamma$ is a $(v,k,\lambda,\mu)\text{-SRG}_{p}$
and all of the following hold in $\mathbb{F}_{q}$:

  * (i)

$k\equiv_{p}2\mu$,

  * (ii)

$v\equiv_{p}3k-2\lambda-1$,

  * (iii)

there exists $\delta\in\mathbb{F}_{q}$ such that
$\delta^{2}\equiv(\lambda-\mu)^{2}+4(k-\lambda)$.

In that case, $S+aI$ is the Gram matrix of an $(a,1,c)$-ETF if and only if
$a=\lambda-\mu+\epsilon\delta+1$ and $c=2\epsilon\delta$ for some
$\epsilon\in\\{\pm 1\\}$.

If $\mu\not\equiv_{p}0$, then condition (i) above implies (ii) for any
$(v,k,\lambda,\mu)\text{-SRG}_{p}$, as seen from (4.1). Similarly, (ii)
implies (i) if $k-\lambda-1\not\equiv_{p}0$. However, (i) and (ii) are
independent in general.

###### Proof.

First we make a reduction. For any choice of $a,c\in\mathbb{F}_{q}$, the
matrix

$G:=S+aI=\left[\begin{array}[]{cc}a&\mathbf{1}_{v}^{\top}\\\
\mathbf{1}_{v}&\Sigma+aI_{v}\end{array}\right]\in\mathbb{F}_{q}^{n\times n}$

satisfies $G^{2}=cG$ if and only if

$\left[\begin{array}[]{cc}a^{2}+v&\mathbf{1}_{v}^{\top}(\Sigma+2aI_{v})\\\
(\Sigma+2aI_{v})\mathbf{1}_{v}&\Sigma^{2}+J_{v}+2a\Sigma+a^{2}I_{v}\end{array}\right]=\left[\begin{array}[]{cc}ac&c\mathbf{1}_{v}^{\top}\\\
c\mathbf{1}_{v}&c(\Sigma+aI_{v})\end{array}\right].$

This occurs if and only if:

  * (I)

$\Sigma\mathbf{1}_{v}\equiv(c-2a)\mathbf{1}_{v}$,

  * (II)

$\Sigma^{2}\equiv-J_{v}+(c-2a)\Sigma+(ac-a^{2})I_{v}$,

  * (III)

$v\equiv ac-a^{2}$.

In any case $\Sigma^{2}$ has constant diagonal $v-1$, so that (II) is
equivalent to the combination of (II) and (III). By Proposition 2.12, $G$ is
the Gram matrix of an $(a,1,c)$-ETF if and only if (I) and (II) hold.

In the forward direction, suppose there exist $a,c\in\mathbb{F}_{q}$ such that
$G$ is the Gram matrix of an $(a,1,c)$-ETF. We prove (i)–(iii) and deduce the
values of $a$ and $c$. Since (I) and (II) hold,
$\Sigma\mathbf{1}_{v}\equiv\theta\mathbf{1}_{v}$ and $\Sigma^{2}\equiv\alpha
J_{v}+\beta\Sigma+\gamma I_{v}$ for

(4.3) $\alpha=-1,\quad\beta=\theta=c-2a,\quad\gamma=ac-a^{2}.$

By Lemma 4.2(b), $\Gamma$ is a $(v,k,\lambda,\mu)\text{-SRG}_{p}$ with
parameters as in (4.2). In particular,

$2(k-2\mu)\equiv(\theta-\beta)-(\alpha+1)\qquad\text{and}\qquad
2(v-3k+2\lambda+1)\equiv(\theta-\beta)+(\alpha+1).$

Since $\theta=\beta$ and $\alpha=-1$, we have

(4.4) $k\equiv_{p}2\mu\qquad\text{and}\qquad
v\equiv_{p}3k-2\lambda-1\equiv_{p}6\mu-2\lambda-1.$

This gives (i) and (ii). For (iii), observe from (4.3) and (4.2) that
$c=\beta+2a$ and

$v\equiv\gamma=ac-a^{2}=a^{2}+\beta a.$

Completing the square to solve for $a$, we find that

(4.5) $4(a+\tfrac{\beta}{2})^{2}\equiv 4v+\beta^{2}=:\Delta.$

Another application of (4.2) gives $\beta\equiv 2(\mu-\lambda-1)$, so by
(4.4), the right-hand side of (4.5) is

$\Delta\equiv 4(6\mu-2\lambda-1)+4(\mu-\lambda-1)^{2}\equiv
4[(\lambda-\mu)^{2}+4(k-\mu)].$

Therefore $a+\tfrac{\beta}{2}\in\mathbb{F}_{q}$ satisfies
$(\alpha+\tfrac{\beta}{2})^{2}\equiv\tfrac{1}{4}\Delta\equiv(\lambda-\mu)^{2}+4(k-\mu)$.
This gives (iii). Additionally, for either choice of $\delta\in\mathbb{F}_{q}$
such that $\delta^{2}\equiv(\lambda-\mu)^{2}+4(k-\mu)$ we have
$a=-\tfrac{\beta}{2}+\epsilon\delta\equiv\lambda-\mu+\epsilon\delta+1$ for
some $\epsilon\in\\{\pm 1\\}$, while $c=2a+\beta=2\epsilon\delta$.

In the reverse direction, suppose $\Gamma$ is a
$(v,k,\lambda,\mu)\text{-SRG}_{p}$ that satisfies (i)–(iii). Choose
$\delta\in\mathbb{F}_{q}$ such that
$\delta^{2}\equiv(\lambda-\mu)^{2}+4(k-\mu)$. Fix $\epsilon\in\\{\pm 1\\}$,
and define $a=\lambda-\mu+\epsilon\delta+1$ and $c=2\epsilon\delta$. In order
to prove that $G$ is the Gram matrix of an $(a,1,c)$-ETF, we show (I) and
(II). By Lemma 4.2(a), $\Sigma\mathbf{1}_{v}\equiv\theta\mathbf{1}_{v}$ and
$\Sigma^{2}\equiv\alpha J_{v}+\beta\Sigma+\gamma I_{v}$, where

$\displaystyle\alpha$ $\displaystyle\equiv v-4k+2\lambda+2\mu\equiv-1,$
$\displaystyle\beta$ $\displaystyle\equiv 2(\mu-\lambda-1),$
$\displaystyle\gamma$ $\displaystyle\equiv 4k-2\lambda-2\mu-1\equiv v,$
$\displaystyle\theta$ $\displaystyle\equiv v-2k-1\equiv 2(\mu-\lambda-1).$

We have $c-2a\equiv 2(\mu-\lambda-1)\equiv\beta\equiv\theta$, which gives (I).
For (II), observe as above that $4(a+\tfrac{\beta}{2})^{2}=4\delta^{2}\equiv
4v+\beta^{2}$, hence $a^{2}+\beta\equiv v$. Consequently,

$ac-a^{2}=a^{2}+\beta a\equiv v\equiv\gamma.$

Comparing coefficients, we find that (II) holds. ∎

######  Theorem 4.4.

Let $\Sigma$ be the Seidel adjacency matrix of a nontrivial SRG with $f\neq
g$. Define

$G=\left[\begin{array}[]{cc}2r+1&\mathbf{1}_{v}^{\top}\\\
\mathbf{1}_{v}&\Sigma+(2r+1)I\end{array}\right]\in\mathbb{Z}^{n\times n},$

where $n=v+1$. Suppose $p$ is an odd prime that divides both $k-2\mu$ and
$v-3k+2\lambda+1$. Then $\overline{G}\in\mathbb{F}_{p}^{n\times n}$ is the
Gram matrix of a $(2r+1,1,2r-2s)$-ETF in an orthogonal geometry on
$\mathbb{F}_{p}^{d}$, where

$d=\operatorname{rank}_{p}\overline{\Sigma+(2r+1)I}.$

Furthermore, $d=g+1$ if $r\not\equiv_{p}s$, whereas $d\leq\min\\{f+1,g+1\\}$
if $r\equiv_{p}s$.

We omit the case $f=g$ for simplicity, since in that case the matrix $G$ above
is already the Gram matrix of a real ETF with $n=2d$, and Proposition 3.2
applies.

###### Proof.

Recall our notation (2.5) for the eigenvalues of $\Sigma$, where
$2r+1=-\theta_{r}$. Theorem 4.3 implies that $\overline{G}$ is the Gram matrix
of an $(a,b,c)$-ETF with the given parameters, and the dimension is
$d=\operatorname{rank}_{p}\overline{G}$ by Proposition 2.12. It remains to
prove that
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$,
and to deduce the “furthermore” statements.

To begin, we argue in cases to show
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$.
First suppose $\theta_{k}\not\equiv_{p}\theta_{r}$. Then
$x:=\overline{(\theta_{k}-\theta_{r})}^{-1}\overline{\mathbf{1}_{v}}\in\mathbb{F}_{p}^{v}$
satisfies $\overline{(\Sigma-\theta_{r}I)}x=\overline{\mathbf{1}_{v}}$. Taking
transposes, we see that $\overline{\mathbf{1}_{v}^{\top}}$ lies in the row
space of $\overline{\Sigma-\theta_{r}I}$, and so

$\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}=\operatorname{rank}_{p}\overline{\left[\begin{array}[]{c}\mathbf{1}_{v}^{\top}\\\
\Sigma-\theta_{r}I\end{array}\right]}.$

If we can show that $\overline{\mathbf{1}_{v}^{\top}x}\equiv-\theta_{r}$, then
we can continue to find

$\operatorname{rank}_{p}\overline{\left[\begin{array}[]{c}\mathbf{1}_{v}^{\top}\\\
\Sigma-\theta_{r}I\end{array}\right]}=\operatorname{rank}_{p}\overline{\left[\begin{array}[]{cc}-\theta_{r}&\mathbf{1}_{v}^{\top}\\\
\mathbf{1}_{v}&\Sigma-\theta_{r}I\end{array}\right]},$

where the additional column on the right already lies in the column space on
the left, as the image of $x$. It will follow that
$\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}=\operatorname{rank}_{p}\overline{G}$
as desired. In fact, the definition of $x$ gives
$\overline{\mathbf{1}_{v}^{\top}x}=\overline{(\theta_{k}-\theta_{r})}^{-1}\overline{v}$,
so we need only show $v\equiv_{p}-\theta_{r}(\theta_{k}-\theta_{r})$. In other
words, to prove
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$
it suffices in this case to show

(4.6) $v\equiv_{p}(2r+1)(v-2k+2r).$

By applying the relation $r^{2}=(\lambda-\mu)r+(k-\mu)$ from (2.3), we find
that

$\displaystyle(2r+1)(v-2k+2r)$ $\displaystyle=4r^{2}+(2v-4k+2)r+v-2k$
$\displaystyle=(2v-4k+4\lambda-4\mu+2)r+v+2k-4\mu.$

Substituting $v\equiv_{p}3k-2\lambda-1$, we see that the coefficient on $r$
above is congruent to $2k-4\mu$. Since $k\equiv_{p}2\mu$ we deduce (4.6).
Therefore,
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$
when $\theta_{k}\not\equiv_{p}\theta_{r}$.

Next we prove that
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$
in the case $\theta_{k}\equiv_{p}\theta_{r}$. Eventually we will find a
suitable choice for $x$ to use the argument above. In the meantime, observe
that, by (2.6) and our hypotheses,

$\tfrac{1}{v}(\theta_{k}-\theta_{r})(v-2k+2s)=v-4k+2\lambda+2\mu\equiv_{p}-1.$

Rewriting the left-hand side, we find

$-1\equiv_{p}\theta_{k}-\theta_{r}-\frac{2(\theta_{k}-\theta_{r})(k-s)}{v},$

the fraction being an integer. By assumption
$\theta_{k}-\theta_{r}\equiv_{p}0$, thus

(4.7) $\frac{2(\theta_{k}-\theta_{r})(k-s)}{v}\in\mathbb{Z}\setminus
p\mathbb{Z},$

and in particular $\theta_{k}-\theta_{r}\neq 0$.

Now we can produce $x$. Working over $\mathbb{R}$ for the moment, recall that
$\operatorname{ker}(\Sigma-\theta_{r}I)$ contains the column space of
$(r-s)E_{r}=A-sI-(k-s)v^{-1}J$. For an integer $m$ yet to be determined,
define vectors
$y=\left[\begin{array}[]{cccc}m&0&\dotso&0\end{array}\right]^{\top}$ and
$x=\tfrac{1}{\theta_{k}-\theta_{r}}\mathbf{1}_{v}+(r-s)E_{r}y$ in
$\mathbb{R}^{v}$. Simplifying, we find

(4.8)
$x=(A-sI)y+\frac{v-(\theta_{k}-\theta_{r})(k-s)m}{v(\theta_{k}-\theta_{r})}\mathbf{1}_{v}.$

Our goal is to choose $m$ in such a way that all factors of $p$ are cleared
from the denominator of (4.8). In that case, we can interpret $x$ as a vector
over $\mathbb{F}_{p}$, and the same relations that helped above will again
hold. Let $p^{\ell_{1}}$, $p^{\ell_{2}}$, and $p^{\ell_{3}}$ be the largest
powers of $p$ that divide $v$, $\theta_{k}-\theta_{r}$, and $k-s$,
respectively. Then $p^{\ell_{1}+\ell_{2}}$ is the largest power of $p$ that
occurs in the denominator of (4.8), and (4.7) implies
$\ell_{1}={\ell_{2}+\ell_{3}}$. Hence,

$\operatorname{gcd}\Bigl{(}p^{\ell_{1}+\ell_{2}},(\theta_{k}-\theta_{r})(k-s)\Bigr{)}=\operatorname{gcd}\bigl{(}p^{\ell_{1}+\ell_{2}},p^{\ell_{1}}\bigr{)}=p^{\ell_{1}}$

divides $v$. By Bezout’s identity, there is an integer $m$ for which
$v-(\theta_{k}-\theta_{r})(k-s)m$ is a multiple of $p^{\ell_{1}+\ell_{2}}$.
With $m$ so defined, the denominator of (4.8) clears as desired. We have
$(\Sigma-\theta_{r}I)x=\mathbf{1}_{v}$ and
$\mathbf{1}_{v}^{\top}x=\tfrac{v}{\theta_{k}-\theta_{r}}$, since $(r-s)E_{r}y$
lies in the kernel of $\Sigma-\theta_{r}I$ and the orthogonal complement of
$\mathbf{1}_{v}$. Additionally, all factors of $p$ clear from the denominator
of $\tfrac{v}{\theta_{k}-\theta_{r}}$ by (4.7). Overall, we may interpret $x$
as a vector in $\mathbb{F}_{p}^{v}$ for which the desired relations hold, and
the same argument as above shows that
$\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$.

It remains to prove our “furthermore” statements about
$d=\operatorname{rank}_{p}\overline{\Sigma-\theta_{r}I}$. We are going to use
Proposition 3.1. Define $M={\Sigma+(2r+1)I}\in\mathbb{R}^{v\times v}$, so that
$d=\operatorname{rank}_{p}\overline{M}$. Here $M$ has eigenvalues
$\theta_{k}-\theta_{r}$, $0$, $\theta_{s}-\theta_{r}$. By (2.7) it satisfies

(4.9) $M\bigl{(}M+(\theta_{r}-\theta_{s})I\bigr{)}=(v-4k+2\lambda+2\mu)J.$

Applying our hypotheses $v\equiv_{p}3k-2\lambda-1$ and $2\mu\equiv_{p}k$, we
find

(4.10)
$\overline{M\Bigl{(}M+(\theta_{r}-\theta_{s})I\Bigr{)}}=-\overline{J}\neq 0.$

Since $MJ=(\theta_{k}-\theta_{r})J$, we deduce from (4.9) and (4.10) that $M$
has minimal polynomial

$x(x+\theta_{r}-\theta_{s})(x+\theta_{r}-\theta_{k})=:(x+\theta_{r}-\theta_{k})h(x)$

and spectrum
$\sigma(M)=\\{(\theta_{k}-\theta_{r})^{1},0^{f},(\theta_{s}-\theta_{r})^{g}\\}$,
with multiplicities denoted by superscripts and all three eigenvalues
distinct.

We argue in cases. If $r\not\equiv_{p}s$ and
$\theta_{k}-\theta_{r}\not\equiv_{p}0$, then
$\theta_{s}-\theta_{r}\not\equiv_{p}0$ and Proposition 3.1(a) immediately
gives $d=g+1$. If $r\not\equiv_{p}s$ and $\theta_{k}-\theta_{r}\equiv_{p}0$,
then Proposition 3.1(b) applies to produce $d=g+1-\epsilon$, where
$\epsilon=0$ since $\overline{h(M)}\neq 0$ by (4.10). Hence $d=g+1$ in this
case as well. Finally, suppose $r\equiv_{p}s$. Then
$\theta_{r}\equiv_{p}\theta_{s}$, and
$\overline{M}=\overline{\Sigma-\theta_{r}I}=\overline{\Sigma-\theta_{s}I}$. By
Proposition 3.1(a),
$\operatorname{rank}_{p}\overline{M}\leq\operatorname{rank}(\Sigma-\theta_{r}I)=g+1$
and also
$\operatorname{rank}_{p}\overline{M}\leq\operatorname{rank}(\Sigma-\theta_{s}I)=f+1$.
This completes the proof. ∎

###  4.2. ETFs with centroidal symmetry

So far we have related graphs on $v$ vertices with ETFs having $n=v+1$
vectors. Now we consider ETFs with $n=v$ vectors. Specifically, we
characterize ETFs having centroidal symmetry, in the sense that the all-ones
vector is an eigenvector of the Gram matrix [11].

######  Theorem 4.5.

Let $\Sigma$ be the Seidel adjacency matrix of a graph $\Gamma$ on $v$
vertices. The following are equivalent for any choice of odd prime power
$q=p^{l}$:

  * (i)

there exist $a,c,\theta\in\mathbb{F}_{q}$ such that $G=\Sigma+aI$ is the Gram
matrix of an $(a,1,c)$-ETF, and $G\mathbf{1}\equiv\theta\mathbf{1}$,

  * (ii)

$\Gamma$ is a $(v,k,\lambda,\mu)\text{-SRG}_{p}$ with
$v\equiv_{p}4k-2\lambda-2\mu$, and there exists $\delta\in\mathbb{F}_{q}$ such
that $\delta^{2}\equiv(\lambda-\mu)^{2}+4(k-\mu)$.

In that case, $\Sigma+aI$ is the Gram matrix of an $(a,1,c)$-ETF if and only
if $a\equiv\lambda-\mu+\epsilon\delta+1$ and $c\equiv 2\epsilon\delta$ for
some $\epsilon\in\\{\pm 1\\}$.

###### Proof.

Suppose (i) holds. Then $\Sigma\mathbf{1}=(\theta-a)\mathbf{1}$, and the
equation $G^{2}=cG$ produces $\Sigma^{2}=(c-2a)\Sigma+a(c-a)I$. From Lemma 4.2
and the linear independence of $J,\Sigma,I$, we deduce that $\Gamma$ is a
$(v,k,\lambda,\mu)\text{-SRG}_{p}$ with $v-4k+2\lambda+2\mu\equiv 0$,
$2(\mu-\lambda-1)\equiv c-2a$, and $4k-2\lambda-2\mu-1\equiv a(c-a)$. In
particular, $c=2(a+\mu-\lambda-1)$ and

$\displaystyle 4k-2\lambda-2\mu-1$ $\displaystyle\equiv a(a+2\mu-2\lambda-2)$
$\displaystyle\equiv[(a+\mu-\lambda-1)-(\mu-\lambda-1)][(a+\mu-\lambda-1)+(\mu-\lambda-1)]$
$\displaystyle\equiv(a+\mu-\lambda-1)^{2}-(\lambda-\mu+1)^{2}.$

Rearranging, we find

$(a+\mu-\lambda-1)^{2}\equiv(\lambda-\mu)^{2}+4(k-\mu).$

Hence there exists $\delta\in\mathbb{F}_{q}$ such that
$\delta^{2}=(\lambda-\mu)^{2}+4(k-\mu)$, and for either choice of such
$\delta$ there exists $\epsilon\in\\{\pm 1\\}$ such that
$a+\mu-\lambda-1=\epsilon\delta$ and $c=2\epsilon\delta$.

Now assume (ii). Pick $\epsilon\in\\{\pm 1\\}$, and define
$a=\lambda-\mu+\epsilon\delta+1$ and $c=2\epsilon\delta$. Then $c-2a\equiv
2(\mu-\lambda-1)$ and

$a(c-a)\equiv[\epsilon\delta+\lambda-\mu-1][\epsilon\delta-(\lambda-\mu-1)]\equiv\delta^{2}-(\lambda-\mu+1)^{2}\equiv
4k-2\lambda-2\mu-1.$

From these identities, the assumption $v-4k+2\lambda+2\mu\equiv_{p}0$, and
Lemma 4.2(a), we conclude that $\Sigma^{2}\equiv(c-2a)\Sigma+a(c-a)I$. For
$G=\Sigma+aI$, this says $G^{2}=cG$. Finally, Lemma 4.2(a) supplies
$\theta\in\mathbb{F}_{q}$ such that $\Sigma\mathbf{1}=(\theta-a)\mathbf{1}$,
that is, $G\mathbf{1}=\theta\mathbf{1}$. ∎

######  Theorem 4.6.

Let $\Sigma$ be the Seidel adjacency matrix of a nontrivial SRG with $f\neq
g$, and define $G=\Sigma+(2r+1)I$. Suppose $p$ is an odd prime that divides
$v-4k+2\lambda+2\mu$. Then $\overline{G}$ is the Gram matrix of a
$(2r+1,1,2r-2s)$-ETF with $n=v$ vectors in an orthogonal geometry on
$\mathbb{F}_{p}^{d}$, where $d=\operatorname{rank}_{p}\overline{G}$.
Furthermore:

  * (a)

If $r\not\equiv_{p}s$ and $v-2k+2r\not\equiv_{p}0$, then $d=g+1$.

  * (b)

If $r\not\equiv_{p}s$ and $v-2k+2r\equiv_{p}0$, then $d=g$.

  * (c)

If $r\equiv_{p}s$, then $d\leq\min\\{f+1,g+1\\}$.

  * (d)

Let $p^{m}$ be the largest power of $p$ that divides $v$. If $p^{m+1}$ divides
$v-2k+2r$, then $d\leq g$.

Our proof of (d) uses a strategy suggested in [7, §13.7]. The assumption
$f\neq g$ is given only to make it obvious that $G$ has integer entries. In
fact, if $f=g$ then $v-4k+2\lambda+2\mu=-1$, and there is no choice of $p$ to
satisfy the hypotheses above.

###### Proof.

Recall our notation (2.5) for the eigenvalues of $\Sigma$. By (2.7) and our
assumption $v-4k+2\lambda+2\mu\equiv_{p}0$, we see that $G=\Sigma-\theta_{r}I$
satisfies

(4.11)
$\overline{G\Bigl{(}G+(\theta_{r}-\theta_{s})I\Bigr{)}}=\overline{(v-4k+2\lambda+2\mu)J}=0.$

Therefore $\overline{G}^{2}=(\theta_{s}-\theta_{r})\overline{G}$, and it
follows from Proposition 2.12 that $\overline{G}$ is the Gram matrix of an ETF
with the given parameters in an orthogonal geometry on $\mathbb{F}_{p}^{d}$,
where $d=\operatorname{rank}_{p}\overline{G}$.

It remains to prove (a)–(d). Observe that $G=\Sigma-\theta_{r}I$ has spectrum

$\sigma(G)={\\{(\theta_{k}-\theta_{r})^{1},0^{f},(\theta_{s}-\theta_{r})^{g}\\}}$

with multiplicities denoted by superscripts, where it is possible that some
eigenvalues coincide. We have $\operatorname{rank}G=g+1$ if
$\theta_{k}\neq\theta_{r}$, and $\operatorname{rank}G=g$ otherwise.

Proposition 3.1(a) immediately implies (a), since
$\theta_{k}-\theta_{r}=v-2k+2r$ and $\theta_{s}-\theta_{r}=2(s-r)$. If
$\theta_{k}=\theta_{r}$, then (b) follows as well. On the other hand, if
$\theta_{k}\neq\theta_{r}$ then $G$ has minimal polynomial

$x(x-\theta_{k}+\theta_{r})(x-\theta_{s}+\theta_{r})=:(x-\theta_{k}+\theta_{r})\,h(x),$

and $\overline{h(G)}=0$ by (4.11). In this case, (b) follows from Proposition
3.1(b). For (c), we observe that
$\operatorname{rank}_{p}\overline{G}\leq\operatorname{rank}G\leq g+1$, and if
$r\equiv_{p}s$ then $\theta_{r}\equiv_{p}\theta_{s}$ and likewise

$\overline{G}=\overline{\Sigma-\theta_{s}I}\leq\operatorname{rank}(\Sigma-\theta_{s}I)\leq
f+1.$

Hence, $d\leq\min\\{f+1,g+1\\}$ if $r\equiv_{p}s$.

Finally, we prove (d). Assume $p^{m+1}$ divides $v-2k+2r$. Then $p^{m}$
divides $k-r$. Put $t=\tfrac{k-r}{p^{m}}$ and $z=\tfrac{v}{p^{m}}$, so that
$z-2t\equiv_{p}0$, and define $H=z(s-r)E_{s}$. We will show $\overline{H}$ is
a nonzero scalar multiple of $\overline{G}$, and it will follow that

$d=\operatorname{rank}_{p}\overline{G}=\operatorname{rank}_{p}\overline{H}\leq\operatorname{rank}H=\operatorname{rank}E_{s}=g.$

Using the definition of $E_{s}$ and the fact that $\Sigma=J-2A-I$, we can
rewrite

$H=zA-zrI-\tfrac{z(k-r)}{v}J=zA-zrI-tJ=(z-2t)A-t\Sigma-(zr+t)I.$

Since $z-2t\equiv_{p}0$ we have
$\overline{H}=-t\overline{\Sigma}-(zr+t)\overline{I}$. Next, the relation
$z\equiv_{p}2t$ implies $zr+t\equiv_{p}t(2r+1)$. It follows that
$\overline{H}=-t\overline{G}$. Finally, $t=\frac{k-r}{p^{m}}\not\equiv_{p}0$
since $p^{m+1}$ divides $v-2k+2r$ but not $v$. This completes the proof. ∎

######  Remark 4.7.

In one view of Theorem 4.6, the real positive-semidefinite matrix
$G=\Sigma+(2r+1)I$ represents a real equiangular system that is not
necessarily tight, but that becomes tight when projected into
$\mathbb{F}_{p}^{v\times v}$. In the case where $r\not\equiv_{p}s$, we could
alternatively view $\overline{G}$ as arising from a real _two-distance tight
frame_ [5] that is projected into a finite field in such a way that the two
distances become opposites. Indeed, both of $E_{s}$ and $I-E_{r}$ are Gram
matrices of two-distance tight frames, and each has rational entries. If $p$
divides $m:=\frac{v-2k+2r}{\operatorname{gcd}(v,k-r)}$, then there is an
integer $c$ such that $cE_{s}\in\mathbb{Z}^{v\times v}$ and
$\overline{G}=\overline{cE_{s}}$. If $p$ does not divide $m$, then we can
similarly obtain $\overline{G}$ from projecting a multiple of $I-E_{r}$. We
omit the details. However, in the case where $r\equiv_{p}s$, there are
examples where $\overline{G}$ cannot be obtained by projecting any multiple of
$E_{s}$, $E_{r}$, ${I-E_{s}}$, or ${I-E_{r}}$, since any integral multiple of
any of these matrices is congruent to $J$ mod $p$. For instance, this occurs
when $(v,k,\lambda,\mu)=(81,20,1,6)$ and $p=3$.

In fact, Theorem 4.6 completely captures the phenomenon of projecting real
two-distance tight frames into finite fields to make them equiangular. Indeed,
let $H\in\mathbb{R}^{n\times n}$ be the Gram matrix of a non-equiangular two-
distance $c$-tight frame. Suppose $H$ has integer entries and there is an odd
prime power $p$ for which the projection
$\overline{H}\in\mathbb{F}_{p}^{n\times n}$ is nonzero and represents an
equiangular system. By Theorem 1.2 in [5], there is a nontrivial SRG for which
$H$ is one of $cE_{s}$, $cE_{r}$, $c(I-E_{r})$, or $c(I-E_{s})$. We may assume
that $H=cE_{s}$ by replacing the graph with its complement or $H$ with $cI-H$,
as necessary. Then one can show that $f\neq g$, that $p$ divides
$v-4k+2\lambda+2\mu$, and that $\overline{H}$ is a scalar multiple of
$\overline{\Sigma+(2r+1)I}$. We omit the details.

##  5\. Examples and applications for SRGs

We now give consequences of the preceding theory. Most known SRGs imply the
existence of ETFs over finite fields, and we collect small ETF sizes that
occur in this way. Using the infinite family of triangular graphs, we deduce
that Gerzon’s bound is attained in orthogonal geometries of infinitely many
dimensions. This stands in contrast with the real setting, where it is an open
problem if Gerzon’s bound is saturated in $\mathbb{R}^{d}$ for some
$d\notin\\{2,3,7,23\\}$. Many unknown SRGs would also imply the existence of
ETFs, and this provides necessary conditions for their existence. We collect
this data as well.

###  5.1. Existence

######  Example 5.1.

Table 1 lists some ETFs in finite orthogonal geometries that arise from our
constructions. We created the table by iterating through SRG parameters in
Brouwer’s table [6], as implemented in [28]. For each parameter set, we
proceeded as follows. If $k\neq 2\mu$ then we applied Theorem 4.4 for all
primes that satisfy its hypotheses. Where $r\equiv_{p}s$, we created the Gram
matrix for a single SRG with the given parameters, and computed its rank to
find $d$. If $k=2\mu$ then the same construction from Theorem 4.4 creates a
real ETF, and Proposition 3.2 applies. Our table omits ETFs that occur in this
way, except in those cases where the resulting frame constant is $c=0$ (since
then Proposition 3.2 cannot predict the dimension). In those cases, we again
constructed the Gram matrix from a single SRG and recorded its rank. For each
parameter set, we then followed a similar procedure using Theorem 4.6, where
the condition $k=2\mu$ was replaced by $v=4k-2\lambda-2\mu$.

Table 1 describes our results with $d<40$. We do not claim to record all
consequences of Theorem 4.4 and Theorem 4.6 from known SRGs in this range. In
particular, we did not attempt to compute dimensions for nonisomorphic SRGs
with equal parameters in cases where the parameters may not predict $d$. (In
fact, it is known that $p$-rank can be an effective way to distinguish
nonisomorphic SRGs [8].)

$p$ | $d$ | $n$ | $a$ | $c$
---|---|---|---|---
3 | 4 | 10 | 0 | 0
3 | 7 | 28 | 0 | 0
3 | 9 | 25 | 0 | 1
3 | 9 | 37 | 0 | 1
3 | 10 | 37 | 0 | 0
3 | 10 | 55 | 0 | 0
3 | 12 | 36 | 1 | 0
3 | 12 | 49 | 0 | 1
3 | 12 | 67 | 0 | 1
3 | 13 | 91 | 0 | 0
3 | 14 | 36 | 1 | 0
3 | 14 | 45 | 1 | 0
3 | 15 | 64 | 0 | 1
3 | 15 | 106 | 0 | 1
3 | 16 | 82 | 0 | 0
3 | 16 | 136 | 0 | 0
3 | 18 | 81 | 1 | 0
3 | 18 | 100 | 0 | 1
3 | 18 | 154 | 0 | 1
3 | 19 | 49 | 1 | 1
3 | 19 | 65 | 1 | 2
3 | 19 | 81 | 1 | 0
3 | 19 | 105 | 1 | 0
3 | 19 | 190 | 0 | 0
3 | 20 | 46 | 0 | 0
3 | 20 | 57 | 1 | 0
3 | 21 | 81 | 1 | 0
3 | 21 | 121 | 0 | 1
3 | 21 | 126 | 1 | 0
3 | 21 | 162 | 1 | 0
3 | 21 | 211 | 0 | 1
3 | 22 | 50 | 1 | 2
3 | 22 | 65 | 1 | 2
3 | 22 | 77 | 1 | 2
3 | 22 | 100 | 1 | 1
3 | 22 | 145 | 0 | 0
3 | 22 | 243 | 1 | 0
3 | 22 | 253 | 0 | 0
3 | 22 | 253 | 1 | 1
3 | 22 | 276 | 1 | 0
3 | 24 | 117 | 1 | 0
3 | 24 | 169 | 0 | 1
3 | 24 | 277 | 0 | 1
3 | 25 | 101 | 1 | 2
3 | 25 | 325 | 0 | 0
3 | 27 | 196 | 0 | 1
3 | 27 | 352 | 0 | 1
3 | 28 | 100 | 1 | 1
3 | 28 | 226 | 0 | 0
3 | 28 | 351 | 1 | 0
3 | 28 | 378 | 1 | 0
3 | 28 | 406 | 0 | 0
3 | 30 | 144 | 1 | 0
3 | 30 | 256 | 0 | 1
3 | 30 | 436 | 0 | 1
3 | 31 | 122 | 1 | 2
3 | 31 | 155 | 1 | 2
3 | 31 | 496 | 0 | 0
3 | 32 | 144 | 1 | 0
3 | 32 | 177 | 1 | 0
3 | 33 | 289 | 0 | 1
3 | 33 | 529 | 0 | 1
3 | 34 | 85 | 1 | 1
3 | 34 | 325 | 0 | 0
3 | 34 | 595 | 0 | 0
3 | 35 | 120 | 1 | 0
3 | 35 | 729 | 1 | 0
3 | 35 | 1080 | 1 | 0
3 | 35 | 1107 | 1 | 0
3 | 36 | 361 | 0 | 1

$p$ | $d$ | $n$ | $a$ | $c$
---|---|---|---|---
3 | 36 | 631 | 0 | 1
3 | 37 | 101 | 1 | 2
3 | 37 | 112 | 1 | 1
3 | 37 | 169 | 1 | 1
3 | 37 | 703 | 0 | 0
3 | 38 | 144 | 1 | 0
3 | 39 | 225 | 1 | 0
3 | 39 | 400 | 0 | 1
3 | 39 | 742 | 0 | 1
5 | 3 | 6 | 0 | 0
5 | 9 | 26 | 0 | 0
5 | 10 | 45 | 2 | 4
5 | 11 | 56 | 2 | 2
5 | 12 | 26 | 0 | 0
5 | 12 | 78 | 2 | 3
5 | 13 | 49 | 2 | 1
5 | 15 | 65 | 2 | 4
5 | 15 | 105 | 2 | 4
5 | 16 | 81 | 2 | 2
5 | 16 | 121 | 2 | 2
5 | 17 | 153 | 2 | 3
5 | 19 | 126 | 0 | 0
5 | 20 | 56 | 0 | 2
5 | 20 | 81 | 0 | 2
5 | 20 | 190 | 2 | 4
5 | 21 | 51 | 0 | 0
5 | 21 | 176 | 0 | 0
5 | 21 | 211 | 2 | 2
5 | 22 | 253 | 2 | 3
5 | 23 | 46 | 0 | 0
5 | 23 | 101 | 0 | 0
5 | 23 | 144 | 2 | 1
5 | 23 | 276 | 0 | 0
5 | 24 | 101 | 0 | 0
5 | 25 | 81 | 0 | 2
5 | 25 | 101 | 0 | 2
5 | 25 | 170 | 2 | 4
5 | 25 | 300 | 2 | 4
5 | 26 | 196 | 2 | 2
5 | 26 | 326 | 2 | 2
5 | 27 | 126 | 0 | 0
5 | 27 | 378 | 2 | 3
5 | 30 | 121 | 0 | 2
5 | 30 | 435 | 2 | 4
5 | 31 | 100 | 1 | 0
5 | 31 | 466 | 2 | 2
5 | 32 | 176 | 0 | 0
5 | 32 | 528 | 2 | 3
5 | 33 | 100 | 1 | 0
5 | 33 | 289 | 2 | 1
5 | 35 | 126 | 0 | 0
5 | 35 | 325 | 2 | 4
5 | 35 | 595 | 2 | 4
5 | 36 | 111 | 2 | 2
5 | 36 | 361 | 2 | 2
5 | 36 | 631 | 2 | 2
5 | 37 | 703 | 2 | 3
5 | 39 | 226 | 0 | 0
7 | 12 | 66 | 3 | 6
7 | 13 | 79 | 3 | 1
7 | 14 | 105 | 3 | 5
7 | 16 | 50 | 0 | 0
7 | 17 | 81 | 3 | 4
7 | 19 | 101 | 3 | 6
7 | 19 | 171 | 3 | 6
7 | 20 | 121 | 3 | 1
7 | 20 | 191 | 3 | 1
7 | 21 | 77 | 2 | 5
7 | 21 | 231 | 3 | 5
7 | 24 | 50 | 0 | 0

$p$ | $d$ | $n$ | $a$ | $c$
---|---|---|---|---
7 | 24 | 100 | 2 | 6
7 | 26 | 325 | 3 | 6
7 | 27 | 117 | 2 | 4
7 | 27 | 352 | 3 | 1
7 | 28 | 406 | 3 | 5
7 | 31 | 121 | 2 | 6
7 | 31 | 256 | 3 | 4
7 | 33 | 290 | 3 | 6
7 | 33 | 528 | 3 | 6
7 | 34 | 145 | 2 | 4
7 | 34 | 324 | 3 | 1
7 | 34 | 562 | 3 | 1
7 | 35 | 85 | 0 | 2
7 | 35 | 630 | 3 | 5
7 | 36 | 169 | 2 | 2
7 | 37 | 344 | 0 | 0
7 | 38 | 247 | 2 | 6
11 | 16 | 120 | 3 | 6
11 | 17 | 137 | 3 | 8
11 | 18 | 171 | 3 | 1
11 | 25 | 169 | 3 | 4
11 | 27 | 197 | 3 | 6
11 | 27 | 351 | 3 | 6
11 | 28 | 225 | 3 | 8
11 | 28 | 379 | 3 | 8
11 | 29 | 435 | 3 | 1
11 | 36 | 122 | 0 | 0
11 | 37 | 223 | 5 | 1
11 | 38 | 703 | 3 | 6
11 | 39 | 97 | 4 | 6
11 | 39 | 742 | 3 | 8
13 | 7 | 14 | 0 | 0
13 | 18 | 153 | 3 | 6
13 | 19 | 172 | 3 | 8
13 | 20 | 210 | 3 | 12
13 | 29 | 225 | 3 | 4
13 | 30 | 155 | 5 | 2
13 | 31 | 257 | 3 | 6
13 | 31 | 465 | 3 | 6
13 | 32 | 289 | 3 | 8
13 | 32 | 497 | 3 | 8
13 | 33 | 561 | 3 | 12
13 | 39 | 247 | 5 | 10
17 | 9 | 18 | 0 | 0
17 | 22 | 231 | 3 | 6
17 | 23 | 254 | 3 | 8
17 | 24 | 300 | 3 | 12
17 | 36 | 127 | 5 | 3
17 | 37 | 361 | 3 | 4
17 | 39 | 401 | 3 | 6
17 | 39 | 741 | 3 | 6
19 | 24 | 276 | 3 | 6
19 | 25 | 301 | 3 | 8
19 | 26 | 351 | 3 | 12
19 | 36 | 222 | 5 | 15
23 | 28 | 378 | 3 | 6
23 | 29 | 407 | 3 | 8
23 | 30 | 465 | 3 | 12
29 | 15 | 30 | 0 | 0
29 | 34 | 561 | 3 | 6
29 | 35 | 596 | 3 | 8
29 | 36 | 666 | 3 | 12
31 | 36 | 630 | 3 | 6
31 | 37 | 667 | 3 | 8
31 | 38 | 741 | 3 | 12
37 | 19 | 38 | 0 | 0
41 | 21 | 42 | 0 | 0
53 | 27 | 54 | 0 | 0
61 | 31 | 62 | 0 | 0
73 | 37 | 74 | 0 | 0

Table 1.  For each row above, an $(a,1,c)$-ETF of $n$ vectors exists in an
orthogonal geometry on $\mathbb{F}_{p}^{d}$. We omit data that can be deduced
from the existence of a known real ETF using Proposition 3.2. Example 5.1
describes our methodology.

######  Remark 5.2.

Theorem 4.3 and Theorem 4.5 describe the construction of $p$-modular SRGs from
ETFs in orthogonal geometries. In many cases, the resulting graphs are SRGs,
and empirically this accounts for a large fraction of known SRGs. By Theorem
4.6, a $(v,k,\lambda,\mu)$-SRG arises from the construction of Theorem 4.5
unless $v-4k+2\lambda+2\mu$ is a power of $2$. By Theorem 4.4, it arises from
the construction of Theorem 4.3 whenever there is an odd prime that divides
both $k-2\mu$ and $v-3k+2\lambda+1$. (Apply Proposition 3.2 if $f=g$.) Using
Brouwer’s table [6], we find that $1142$ out of $1160$ known SRG parameters
with $v\leq 1300$ can be constructed from an ETF using Theorem 4.3 or Theorem
4.5. The exceptions are shown in Table 2. In addition, $2824$ out of $2910$
feasible parameters with $v\leq 1300$ for which existence is unresolved would
arise from an ETF in this way, if the SRG exists.

######  Example 5.3.

The Steiner construction of [22, Theorem 12.1] and [13, Theorem 1] works
equally well over finite fields to create real model ETFs. It produces more
ETF sizes in the finite field setting, since modular Hadamard matrices
outnumber their real counterparts. In particular, if $p$ is an odd prime and
$m\equiv_{p}4$, there is a $(3,1,8)$-ETF of size $\frac{m(m-1)}{2}\times
m^{2}$ in the real model over $\mathbb{F}_{p}$, and it can be chosen to have
centroidal symmetry. In the Steiner construction, take the system with vertex
set $[m]$ and all $2$-subsets as blocks. A modular Hadamard matrix of the
appropriate size is given by

$H=\left[\begin{array}[]{cc}1&\mathbf{1}_{m-1}^{\top}\\\
\mathbf{1}_{m-1}&2I_{m-1}-J_{m-1}\end{array}\right]\in\mathbb{F}_{p}^{m\times
m}.$

Combine these ingredients as in [13, Theorem 1] to obtain the ETF described
above.

Notably, the existence of a real ETF with size $\frac{m(m-1)}{2}\times m^{2}$
is currently unresolved for $m\in\\{22,34\\}$, but the Steiner construction
provides examples in the real model over $\mathbb{F}_{3}$ [12, 6].
Furthermore, when $m$ is odd, Theorem $A$ in [25] implies there is no real ETF
with size $\frac{m(m-1)}{2}\times m^{2}$, but the Steiner construction gives
an example in the real model over some finite field as long as $|m-4|$ is not
a power of $2$.

$v$ | $k$ | $\lambda$ | $\mu$
---|---|---|---
21 | 10 | 3 | 6
40 | 12 | 2 | 4
57 | 24 | 11 | 9

$v$ | $k$ | $\lambda$ | $\mu$
---|---|---|---
70 | 27 | 12 | 9
112 | 30 | 2 | 10
120 | 42 | 8 | 18

$v$ | $k$ | $\lambda$ | $\mu$
---|---|---|---
220 | 84 | 38 | 28
280 | 117 | 44 | 52
512 | 196 | 60 | 84

Table 2.  Except for the nine parameter sets above and their complements,
every known SRG with $v\leq 1300$ vertices can be constructed from an ETF over
a finite field. See Remark 5.2.

Next we interpret Theorem 4.6 for a particular infinite family of SRGs. Given
a positive integer $d$, we write $\overline{T}(d+1)$ for the complement of the
triangular graph whose vertices are the $2$-sets of $\\{1,\dotsc,d+1\\}$, with
vertices adjacent in $\overline{T}(d+1)$ when they have trivial intersection.
It is an SRG with parameters $v=\binom{d+1}{2}$, $k=\tfrac{1}{2}(d-2)(d-1)$,
$\lambda=\tfrac{1}{2}(d-4)(d-3)$, $\mu=\tfrac{1}{2}(d-3)(d-2)$ and $r=1$,
$s=2-d$, $f=\tfrac{1}{2}(d^{2}-1)$, $g=d$ (see [9] for the complementary
parameters).

######  Theorem 5.4.

For every integer $d>1$ and every odd prime $p$ that divides $d-7$, there
exists a $(3,1,12)$-ETF of $n=\binom{d+1}{2}$ vectors in an orthogonal
geometry on $\mathbb{F}_{p}^{d}$.

###### Proof.

Take $\Gamma=\overline{T}(d+1)$ in Theorem 4.6. From the above, we obtain

$v-4k+2\lambda+2\mu=\tfrac{1}{2}(d-7)(d-4),$

with $v-2k+2r=-\tfrac{1}{2}d(d-7)\equiv_{p}0$ and $r-s=d-1\equiv_{p}6$. Since
$p$ is odd and divides $d-7$, Theorem 4.6 produces an ETF with the given
parameters in an orthogonal geometry on $\mathbb{F}_{p}^{d^{\prime}}$. If
$p\neq 3$ then $r\not\equiv_{p}s$, and Theorem 4.6(b) gives $d^{\prime}=g=d$.
If $p=3$ then $d\equiv_{p}1$, and the largest power of $p$ that divides
$v\equiv_{p}1$ is $p^{m}=1$. Here $p^{m+1}=p$ divides $v-2k+2r$, so we can
apply Theorem 4.6(d) to obtain $d^{\prime}\leq g=d$. The inequality cannot be
strict, or else the Gram matrix would produce an ETF that violates Gerzon’s
bound. Therefore $d^{\prime}=d$. ∎

###  5.2. Necessary conditions

######  Example 5.5.

Our results provide necessary conditions for the unresolved existence of SRGs
whose parameters satisfy the hypotheses of Theorem 4.4 or Theorem 4.6.
Specifically, Table 3 lists the parameters of ETFs whose existence would be
implied by such SRGs on $v\leq 300$ vertices. (We omit the SRG parameter $\mu$
since it is uniquely determined by $v,k,\lambda$.) Each row is a consequence
of Theorem 4.4 (if $n=v+1$) or Theorem 4.6 (if $n=v$).

Among the data we see that if Conway’s 99-graph with parameters $(99,14,1,2)$
exists, then there is a $(2,1,4)$-ETF of $n=100$ vectors in an orthogonal
geometry on $\mathbb{F}_{5}^{45}$. The authors do not know if such an ETF
exists. It is well known that there is an ETF of $100$ vectors in
$\mathbb{R}^{45}$, but if we attempt to project it into $\mathbb{F}_{5}$ with
Proposition 3.2, then we obtain a $(4,1,0)$-ETF in dimension $d^{\prime}\leq
45$. These parameters do not match the ones predicted for Conway’s graph, and
in any case we cannot be sure from Proposition 3.2 that $d^{\prime}=45$.

$v$ | $k$ | $\lambda$ | $p$ | $d$ | $n$ | $a$ | $c$
---|---|---|---|---|---|---|---
69 | 48 | 32 | 13 | 24 | 69 | 5 | 3
85 | 54 | 33 | 7 | 35 | 85 | 0 | 4
85 | 70 | 57 | 5 | 35 | 86 | 0 | 4
85 | 70 | 57 | 13 | 35 | 85 | 5 | 1
88 | 27 | 6 | 5 | 32 | 88 | 2 | 3
99 | 14 | 1 | 5 | 45 | 100 | 2 | 4
100 | 33 | 8 | 3 | 34 | 101 | 1 | 2
111 | 30 | 5 | 19 | 36 | 111 | 7 | 1
112 | 36 | 10 | 3 | 48 | 112 | 0 | 2
115 | 18 | 1 | 3 | 46 | 115 | 1 | 1
115 | 18 | 1 | 17 | 45 | 115 | 7 | 16
120 | 34 | 8 | 7 | 52 | 121 | 2 | 6
120 | 84 | 58 | 3 | 57 | 121 | 0 | 2
121 | 36 | 7 | 3 | 37 | 121 | 1 | 1
121 | 36 | 7 | 5 | 36 | 121 | 2 | 2
121 | 48 | 17 | 3 | 48 | 121 | 0 | 1
133 | 32 | 6 | 3 | 57 | 133 | 0 | 2
133 | 32 | 6 | 11 | 56 | 133 | 9 | 9
133 | 88 | 57 | 3 | 57 | 133 | 0 | 1
133 | 88 | 57 | 5 | 56 | 133 | 4 | 2
133 | 108 | 87 | 5 | 57 | 133 | 2 | 3
133 | 108 | 87 | 11 | 56 | 133 | 7 | 7
136 | 105 | 80 | 3 | 52 | 137 | 1 | 2
136 | 105 | 80 | 11 | 52 | 136 | 7 | 9
162 | 21 | 0 | 5 | 57 | 163 | 2 | 3
162 | 21 | 0 | 7 | 56 | 162 | 0 | 4
162 | 92 | 46 | 7 | 24 | 163 | 5 | 1
162 | 112 | 76 | 13 | 64 | 162 | 9 | 11
162 | 138 | 117 | 7 | 70 | 162 | 0 | 4
162 | 138 | 117 | 17 | 70 | 163 | 7 | 1
169 | 42 | 5 | 3 | 43 | 170 | 1 | 2
169 | 42 | 5 | 5 | 43 | 169 | 2 | 1
169 | 42 | 5 | 7 | 42 | 169 | 0 | 5
169 | 56 | 15 | 3 | 57 | 169 | 0 | 2
169 | 56 | 15 | 5 | 56 | 169 | 4 | 1
169 | 70 | 27 | 3 | 70 | 169 | 2 | 2
175 | 108 | 63 | 13 | 43 | 175 | 7 | 4
176 | 25 | 0 | 3 | 55 | 176 | 1 | 2
176 | 25 | 0 | 7 | 56 | 176 | 0 | 6
176 | 25 | 0 | 17 | 56 | 177 | 7 | 3
176 | 70 | 24 | 5 | 56 | 177 | 4 | 3
183 | 52 | 11 | 5 | 61 | 184 | 4 | 1
183 | 52 | 11 | 29 | 60 | 183 | 9 | 26
189 | 128 | 82 | 11 | 29 | 189 | 5 | 3
189 | 140 | 103 | 5 | 91 | 189 | 1 | 4
190 | 144 | 108 | 5 | 76 | 191 | 4 | 4
196 | 39 | 2 | 3 | 49 | 197 | 1 | 2
196 | 39 | 2 | 7 | 49 | 197 | 0 | 5
196 | 39 | 2 | 31 | 48 | 196 | 7 | 26
196 | 45 | 4 | 3 | 46 | 196 | 1 | 1
196 | 75 | 26 | 3 | 76 | 197 | 2 | 1
196 | 114 | 59 | 5 | 25 | 196 | 0 | 2
204 | 28 | 2 | 13 | 84 | 204 | 9 | 7
204 | 140 | 94 | 3 | 69 | 205 | 0 | 1
205 | 68 | 15 | 3 | 40 | 205 | 1 | 1
205 | 68 | 15 | 5 | 40 | 205 | 2 | 4
208 | 45 | 8 | 5 | 91 | 209 | 1 | 4
209 | 156 | 115 | 5 | 76 | 209 | 4 | 1
209 | 156 | 115 | 7 | 77 | 210 | 2 | 5
209 | 156 | 115 | 11 | 77 | 209 | 9 | 4
210 | 33 | 0 | 5 | 55 | 210 | 2 | 4
210 | 33 | 0 | 7 | 56 | 211 | 0 | 3
210 | 76 | 26 | 5 | 96 | 211 | 3 | 3
210 | 132 | 82 | 3 | 100 | 211 | 1 | 1
216 | 129 | 72 | 13 | 44 | 217 | 7 | 10
216 | 172 | 136 | 5 | 86 | 216 | 4 | 4
217 | 128 | 72 | 3 | 63 | 217 | 0 | 2
220 | 72 | 22 | 3 | 100 | 220 | 1 | 1
225 | 48 | 3 | 7 | 49 | 225 | 0 | 2

$v$ | $k$ | $\lambda$ | $p$ | $d$ | $n$ | $a$ | $c$
---|---|---|---|---|---|---|---
225 | 64 | 13 | 7 | 64 | 225 | 2 | 2
225 | 128 | 64 | 5 | 25 | 226 | 0 | 3
231 | 160 | 110 | 5 | 111 | 231 | 3 | 3
232 | 33 | 2 | 3 | 87 | 232 | 0 | 1
232 | 33 | 2 | 19 | 88 | 232 | 9 | 3
232 | 33 | 2 | 23 | 88 | 233 | 9 | 22
232 | 63 | 14 | 3 | 88 | 233 | 2 | 1
232 | 63 | 14 | 11 | 88 | 232 | 0 | 6
232 | 150 | 95 | 11 | 88 | 232 | 0 | 8
232 | 154 | 96 | 3 | 28 | 232 | 2 | 2
232 | 154 | 96 | 37 | 29 | 233 | 5 | 7
235 | 52 | 9 | 3 | 94 | 235 | 2 | 2
235 | 52 | 9 | 7 | 95 | 236 | 4 | 5
235 | 52 | 9 | 23 | 95 | 235 | 11 | 3
235 | 192 | 156 | 11 | 95 | 235 | 9 | 2
236 | 180 | 135 | 37 | 60 | 236 | 7 | 30
238 | 75 | 20 | 7 | 84 | 238 | 4 | 2
243 | 66 | 9 | 13 | 44 | 243 | 7 | 10
245 | 52 | 3 | 3 | 49 | 245 | 1 | 2
245 | 52 | 3 | 23 | 50 | 245 | 7 | 9
245 | 180 | 131 | 3 | 100 | 245 | 2 | 1
245 | 180 | 131 | 19 | 101 | 245 | 11 | 9
246 | 85 | 20 | 7 | 42 | 246 | 0 | 5
249 | 88 | 27 | 17 | 84 | 249 | 11 | 15
250 | 81 | 24 | 7 | 106 | 250 | 6 | 2
250 | 153 | 88 | 3 | 46 | 250 | 1 | 1
259 | 42 | 5 | 5 | 111 | 259 | 1 | 4
259 | 42 | 5 | 23 | 112 | 259 | 11 | 1
261 | 64 | 14 | 5 | 116 | 261 | 3 | 3
261 | 64 | 14 | 13 | 117 | 261 | 0 | 2
261 | 176 | 112 | 5 | 30 | 261 | 0 | 3
261 | 180 | 123 | 13 | 117 | 261 | 0 | 4
261 | 208 | 165 | 5 | 116 | 261 | 1 | 1
261 | 208 | 165 | 19 | 117 | 261 | 11 | 7
265 | 96 | 32 | 3 | 106 | 266 | 1 | 2
265 | 96 | 32 | 17 | 105 | 265 | 13 | 15
266 | 45 | 0 | 13 | 56 | 266 | 7 | 4
273 | 80 | 19 | 3 | 91 | 274 | 2 | 2
273 | 80 | 19 | 41 | 90 | 273 | 11 | 32
273 | 200 | 145 | 7 | 105 | 273 | 4 | 2
276 | 75 | 10 | 3 | 46 | 277 | 1 | 1
276 | 75 | 10 | 11 | 45 | 276 | 7 | 7
276 | 75 | 18 | 11 | 116 | 277 | 2 | 8
276 | 165 | 92 | 17 | 46 | 277 | 7 | 10
280 | 62 | 12 | 3 | 124 | 280 | 1 | 1
280 | 62 | 12 | 17 | 125 | 281 | 13 | 11
280 | 216 | 166 | 3 | 136 | 280 | 1 | 1
280 | 216 | 166 | 5 | 136 | 281 | 3 | 3
285 | 64 | 8 | 7 | 76 | 285 | 2 | 4
285 | 64 | 8 | 11 | 75 | 285 | 9 | 10
286 | 95 | 24 | 3 | 66 | 286 | 0 | 2
286 | 95 | 24 | 5 | 66 | 287 | 4 | 3
286 | 160 | 84 | 3 | 66 | 286 | 0 | 1
288 | 41 | 4 | 29 | 124 | 289 | 11 | 24
288 | 182 | 106 | 3 | 28 | 289 | 2 | 2
288 | 245 | 208 | 5 | 141 | 289 | 1 | 4
289 | 72 | 11 | 3 | 72 | 289 | 0 | 1
289 | 72 | 11 | 7 | 73 | 289 | 2 | 6
289 | 90 | 23 | 3 | 91 | 290 | 2 | 1
289 | 90 | 23 | 5 | 91 | 289 | 1 | 4
289 | 90 | 23 | 7 | 90 | 289 | 4 | 6
289 | 108 | 37 | 3 | 109 | 289 | 1 | 1
289 | 108 | 37 | 5 | 108 | 289 | 3 | 4
289 | 126 | 53 | 3 | 126 | 289 | 0 | 1
297 | 104 | 31 | 7 | 89 | 297 | 4 | 1
300 | 184 | 108 | 7 | 70 | 301 | 2 | 5
300 | 230 | 175 | 13 | 116 | 301 | 11 | 4
300 | 273 | 248 | 11 | 118 | 301 | 7 | 9
300 | 273 | 248 | 13 | 118 | 300 | 7 | 7

Table 3.  Each row contains parameters of an SRG whose existence is
unresolved. If such an SRG exists, then there is an $(a,1,c)$-ETF of $n$
vectors in an orthogonal geometry on $\mathbb{F}_{p}^{d}$. See Example 5.5 for
details.

######  Example 5.6.

The missing Moore graph refers to a possible SRG with parameters
$(3250,57,0,1)$ [16]. At the time of writing its existence is unresolved. If
such an SRG exists, then it produces a $(4,1,8)$-ETF of $n=3251$ vectors in an
orthogonal geometry on $\mathbb{F}_{11}^{1521}$ via Theorem 4.4.

######  Example 5.7.

It is known that there is no $(841,200,87,35)$-SRG [20]. Theorem 4.6 gives
another proof of this fact, since the complement of any such SRG would produce
a $(0,1,1)$-ETF of $n=841$ vectors in an orthogonal geometry on
$\mathbb{F}_{5}^{40}$, and this violates Gerzon’s bound.

######  Example 5.8.

Table 4 summarizes feasible SRG parameters with $v\leq 1300$ that imply
equality in Gerzon’s bound via Theorem 4.6. For the first and last rows,
Theorem 4.6 gives only an upper bound on $d$, but as in the proof of Theorem
5.4 the inequality cannot be strict.

The authors do not know how to construct an ETF with any of the parameters
given in Table 4. In particular, every instance of Gerzon equality that we
currently know satisfies the identity $a^{2}=(d+2)b$, but this equation does
not occur anywhere in Table 4 (where $b=1$). Proposition 2.10 implies that
$a^{2}=(d+2)b$ whenever Gerzon’s bound is saturated, $d\not\equiv_{p}0$, and
$d\not\equiv_{p}1$. The authors do not know if these conditions can be
removed.

$v$ | $k$ | $\lambda$ | $\mu$ | $p$ | $d$ | $a$ | $c$
---|---|---|---|---|---|---|---
351 | 210 | 113 | 144 | 5 | 26 | 0 | 0
703 | 520 | 372 | 420 | 3 | 37 | 2 | 2
1081 | 486 | 177 | 252 | 5 | 46 | 2 | 2
1275 | 728 | 379 | 464 | 7 | 50 | 0 | 0
Table 4.  If an SRG with the given parameters exists, then there is an
$(a,1,c)$-ETF consisting of $n=\binom{d+1}{2}$ vectors in an orthogonal
geometry on $\mathbb{F}_{p}^{d}$. See Example 5.8.

######  Remark 5.9.

Since ETFs over finite fields give necessary conditions for the existence of
SRGs, it would be helpful to have more machinery to prove nonexistence of
equiangular systems with given parameters. Gerzon’s bound on the size of an
equiangular system gives one such tool, but it is a blunt instrument since it
takes little account of the parameters $(a,b)$ of an equiangular system (other
than to require $a^{2}\neq b$). In comparison with the real case, it is
natural to ask if there may be “relative bounds” on the number of vectors in
an $(a,b)$-equiangular system that take greater advantage of the known
parameters. We leave this as an open problem.

######  Problem 5.10 ([14]).

Given an orthogonal geometry over $\mathbb{F}_{q}$ and parameters
$a,b\in\mathbb{F}_{q}$, find an upper bound on the size of an
$(a,b)$-equiangular system that improves upon Gerzon’s bound wherever
possible.

##  6\. Failure of Gerzon equality in dimension 5

By leveraging a finite search space, we now show that Gerzon’s bound cannot be
attained in any finite orthogonal geometry of dimension 5 over a field of odd
characteristic. This is the smallest dimension for which Theorem 5.4 fails to
provide an instance of Gerzon equality.

######  Theorem 6.1 (Computer-assisted result).

For any choice of odd prime power $q$, there does not exist an ETF of $n=15$
vectors in any orthogonal geometry on $\mathbb{F}_{q}^{5}$. In particular, for
any choice of $a,b\in\mathbb{F}_{q}$ with $b\neq a^{2}$, there is no
$(a,b)$-equiangular system of $n=15$ vectors in any orthogonal geometry on
$\mathbb{F}_{q}^{5}$.

Our computer-assisted proof implements the following lemma, which can also be
generalized for unitary geometries. Similar methods have been used to prove
nonexistence of SRGs [1, 2]. Recall that a clique in a graph refers to a
complete subgraph. A clique is maximal if it is not properly contained in any
other clique. The clique number of a graph is the size of its largest clique.

######  Lemma 6.2.

Suppose there is an $(a,1)$-equiangular system $\Phi$ of $n$ distinct vectors
in an orthogonal geometry on $\mathbb{F}_{q}^{d}$, such that
$\operatorname{rank}\Phi>1$. Choose any two linearly independent vectors
$\psi_{1},\psi_{2}\in\mathbb{F}_{q}^{d}$ satisfying
$\langle\psi_{1},\psi_{1}\rangle=\langle\psi_{2},\psi_{2}\rangle=a$ and
$\langle\psi_{1},\psi_{2}\rangle=1$. Define

$V=\\{\psi\in\mathbb{F}_{q}^{d}:\langle\psi,\psi\rangle=a,\quad\langle\psi,\psi_{1}\rangle=1,\quad\langle\psi,\psi_{2}\rangle^{2}=1\\},$

and let $\Gamma$ be the compatibility graph on vertex set $V$ with distinct
vertices $\varphi,\psi$ adjacent whenever
${\langle\varphi,\psi\rangle^{2}=1}$. Then $\Gamma$ contains a clique
$K\subseteq V$ of size $n-2$, and $\Psi=\\{\psi_{1},\psi_{2}\\}\cup K$ is an
$(a,1)$-equiangular system of $n$ vectors. Furthermore, if $\Phi$ is an
$(a,1,c)$-ETF, then $K$ can be chosen so that $\Psi$ is an ETF with the same
parameters.

###### Proof.

Denote $\Phi=\\{\varphi_{j}\\}_{j\in[n]}$, where $\varphi_{1}$ and
$\varphi_{2}$ are linearly independent. For any choice of signs
$\epsilon_{j}\in\\{\pm 1\\}$, the system
$\Phi^{\prime}=\\{\epsilon_{j}\varphi_{j}\\}_{j\in[n]}$ is again an
$(a,1)$-equiangular system, and if $\Phi$ is an $(a,1,c)$-ETF then so is
$\Phi^{\prime}$. By choosing the signs appropriately, we may therefore assume
that $\langle\varphi_{1},\varphi_{j}\rangle=1$ for every $j\geq 2$. The linear
transformation
$T\colon\operatorname{span}\\{\varphi_{1},\varphi_{2}\\}\to\operatorname{span}\\{\psi_{1},\psi_{2}\\}$
given by $T(\varphi_{j})=\psi_{j}$ is easily seen to be an isometric
isomorphism. By Witt’s Extension Theorem (Theorem 5.2 of [15]) it extends to
an isometric isomorphism $U\colon\mathbb{F}_{q}^{d}\to\mathbb{F}_{q}^{d}$. For
every $j\geq 3$, $\psi_{j}:=U(\varphi_{j})$ belongs to $V$, and
$K:=\\{\psi_{j}:j\geq 3\\}$ is a clique in $\Gamma$. The proof is complete
since $\Psi=\\{\psi_{1},\psi_{2}\\}\cup K=\\{\psi_{j}\\}_{j\in[n]}$ is the
image of $\Phi$ under an isometric isomorphism. ∎

Our code for the following computer-assisted proof is provided as an ancillary
file with the arXiv version of this paper.

###### Computer-assisted proof of Theorem 6.1.

Suppose for the sake of contradiction that there is an $(a,b,c)$-ETF of $n=15$
vectors in an orthogonal geometry on $\mathbb{F}_{q}^{5}$. As explained in
Remark 2.15, we may assume that $b=1$, and, by Theorem 3.4, we may take $q=p$
to be prime with $a,c\in\mathbb{F}_{p}$. Furthermore, Proposition 3.3 implies
that $p\leq 2n-5=25$, since there is no ETF of $n=15$ vectors in
$\mathbb{R}^{5}$ by [25, Corollary 16]. If $p\neq 5$ then we may divide (2.8)
to obtain $c=3a$, and then Proposition 2.10 states $2a^{2}=14$, or $a^{2}=7$.
Since $7$ is a square, we conclude that $p\in\\{3,5,7,19\\}$.

When $p\in\\{3,7,19\\}$, there are at most two possibilities for
$a\in\mathbb{F}_{p}$ and two possible orthogonal geometries on
$\mathbb{F}_{p}^{5}$. For each possibility, we used GAP [27] to implement a
choice of $\psi_{1},\psi_{2}\in\mathbb{F}_{p}^{5}$ as in Lemma 6.2. Using the
GRAPE package for GAP [23], we then implemented the compatibility graph
$\Gamma$ of Lemma 6.2 and computed its clique number. The results are
summarized in Table 5. In each case, we found the clique number was less than
$n-2=13$. By Lemma 6.2, we conclude that there is no $(a,1)$-equiangular
system of $n=15$ vectors in an orthogonal geometry on $\mathbb{F}_{p}^{5}$ for
$p\in\\{3,7,19\\}$.

$p$ | $s$ | $a$ | $|V|$ | $\omega(\Gamma)$
---|---|---|---|---
3 | 0 | 1 | 24 | 9
3 | 0 | 2 | 21 | 5
3 | 1 | 1 | 12 | 5
3 | 1 | 2 | 33 | 9
5 | 0 | 0 | 40 | 4
5 | 0 | 1 | 70 | 25
5 | 0 | 2 | 55 | 6

$p$ | $s$ | $a$ | $|V|$ | $\omega(\Gamma)$
---|---|---|---|---
5 | 0 | 3 | 55 | 6
5 | 0 | 4 | 145 | 25
5 | 1 | 0 | 60 | 6
5 | 1 | 1 | 30 | 5
5 | 1 | 2 | 45 | 8
5 | 1 | 3 | 45 | 8
5 | 1 | 4 | 105 | 5

$p$ | $s$ | $a$ | $|V|$ | $\omega(\Gamma)$
---|---|---|---|---
7 | 0 | 0 | 98 | 4
7 | 1 | 0 | 98 | 4
19 | 0 | 8 | 722 | 3
19 | 0 | 11 | 722 | 3
19 | 1 | 8 | 722 | 3
19 | 1 | 11 | 722 | 3

Table 5.  Data for the computer-assisted proof of Theorem 6.1. The second
column describes the type of geometry on $\mathbb{F}_{p}^{5}$, where $s=0$ for
square-type discriminant and $s=1$ otherwise. For $a$ as in the third column
we implemented the graph $\Gamma$ of Lemma 6.2 with the GRAPE package for GAP
[23, 27]. The number of vertices appears in the fourth column, and the last
column gives the clique number as found by GRAPE.

It remains to consider $p=5$. Turning to GAP again, we proceeded as above,
testing all possible values of $a\in\mathbb{F}_{5}$. Here again, we found the
clique number was less than ${n-2}=13$ in all but two cases: for the geometry
with square-type discriminant and $a\in\\{1,4\\}$, the clique number was $25$.
Hence, we are in one of these cases. Taking $V$, $\Gamma$, and $K$ as in Lemma
6.2, there must be a maximal clique $M$ that contains $K$, and then
$\Psi^{\prime}:=\\{\psi_{1},\psi_{2}\\}\cup M$ contains all the vectors of the
frame $\Psi=\\{\psi_{1},\psi_{2}\\}\cup K$, so that
$\operatorname{rank}\Psi^{\prime}=5$. However, for both of the remaining cases
we checked every maximal clique $M$ in $\Gamma$ of size $13\leq|M|\leq 25$,
and in each case we found $\operatorname{rank}\Psi^{\prime}\in\\{3,4\\}$.
(There were 16 such maximal cliques for $a=1$, and 156 for $a=4$.) This is a
contradiction. Therefore no such ETF exists. By Proposition 2.9, neither does
there exist an $(a,b)$-equiangular system of $15$ vectors with $a^{2}\neq b$
in any orthogonal geometry of dimension 5 over a finite field of odd
characteristic. ∎

## Acknowledgments

GRWG was partially supported by the Singapore Ministry of Education Academic
Research Fund (Tier 1); grant numbers: RG29/18 and RG21/20. JJ was supported
by NSF DMS 1830066. DGM was partially supported by AFOSR FA9550-18-1-0107 and
NSF DMS 1829955. This project began at the 2018 MFO Mini-Workshop on
Algebraic, Geometric, and Combinatorial Methods in Frame Theory. The authors
thank the other participants, Matt Fickus, and Steve Flammia for comments that
motivated the authors or provided insight.

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