# Switching equivalence of strongly regular polar graphs

Gábor P. Nagy  Bolyai Institute, University of Szeged, Aradi Vértanúk tere 1,
H-6720 Szeged, Hungary; and Institute of Mathematics, Budapest University of
Technology and Economics, Műegyetem rkp. 3, H-1111 Budapest, Hungary.
<EMAIL_ADDRESS><EMAIL_ADDRESS>and  Valentino Smaldore
Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università degli
Studi di Padova, Stradella S. Nicola 3, 36100 Vicenza, Italy.
<EMAIL_ADDRESS>
###### Abstract.

We prove the switching equivalence of the strongly regular polar graphs
$NO^{\pm}(4m,2)$, $NO^{\mp}(2m+1,4)$, and $\Gamma(O^{\mp}(4m,2))$ plus an
isolated vertex by giving an analytic description for them and their
associated two-graphs.

###### Contents

  1. 1 Symplectic and orthogonal groups in characteristic two
    1. 1.1 Quadratic forms and alternating bilinear forms
    2. 1.2 Symplectic and orthogonal transformations
  2. 2 Strongly regular graphs, Seidel switching, two-graphs
    1. 2.1 Preliminaries on projective quadrics
    2. 2.2 Strongly regular graphs on the non-singular points of polar spaces
    3. 2.3 Two-graphs and switchings
  3. 3 Analytic description of strongly regular polar graphs
  4. 4 Two-graphs of strongly regular polar graphs
    1. 4.1 The symplectic two-graphs
    2. 4.2 The descendants of symplectic two-graphs
    3. 4.3 The main result on the switching equivalence
  5. 5 Acknowledgement
  6. A Symplectic groups in even characteristic
    1. A.1 Symplectic transvections
    2. A.2 Proofs of the theorems on the structure of symplectic groups

## Introduction

A strongly regular graph with parameters $(v,k,\lambda,\mu)$ is a graph with
$v$ vertices where each vertex is incident with $k$ edges, any two adjacent
vertices have $\lambda$ common neighbors, and any two non-adjacent vertices
have $\mu$ common neighbors. Two strongly regular graphs have the same
parameters if and only if they are cospectral. Strongly regular graphs were
introduced by R. C. Bose in [1] in 1963, and ever since then they have been
intensively investigated.

Many interesting and well-studied examples of strongly regular graphs belong
to polar spaces. The tangent graph of a polar space $\mathcal{P}$ embedded in
a projective space is defined as the graph in which the vertices are non-
isotropic points of the polarity that defines $\mathcal{P}$, and two vertices
are adjacent if and only if they lie on the same tangent line to
$\mathcal{P}$. In [4] it is shown that in some cases tangent graphs are
strongly regular. Here we focus on the graph $NO^{\pm}(2m,2)$, whose vertices
are non-isotropic points w.r.t. a non-degenerate hyperbolic ($+1$ case) or
elliptic ($-1$ case) quadric $Q^{\pm}(2m-1,2)$ in a projective space of odd
dimension over the binary field. See [4] for more details.

The graph $NO^{\pm}(2m+1,q)$ is the graph defined on the non-singular
hyperplanes of one type with respect to a parabolic quadric $Q(2m,q)$ in a
projective space of even dimension, where two vertices are adjacent if the
intersection of the corresponding hyperplanes meets the quadric in a
degenerate section. This construction belongs to an idea of Wilbrink, reported
later in [3]. This paper aims to show the switching equivalence between the
graph $NO^{\pm}(4m,2)$ and the graph $NO^{\mp}(2m+1,4)$.

Two graphs are called switching equivalent if we can obtain one from the other
by applying a series of switching of edges and non-edges between subsets of
vertices. We will be mainly interested in the Seidel switching, introduced by
Seidel in [8], related to the concept of two-graph. Seidel switching is a
method to modify the adjacency of a given graph to obtain cospectral graphs.

A two-graph is a set of vertices and an incidence relation for triples of
vertices. A two-graph is regular if each pair of vertices is contained in a
constant number of triples. See [2, 9] for more information on two-graphs,
while in [5] the authors made a list of two-graphs of small order. Two graphs
are switching equivalent if and only if they have the same associated two-
graph.

The paper is organized as follows. In Section 1 we give some preliminaries on
symplectic and orthogonal polarities in characteristic two, while in Section 2
we focus on strongly regular polar graphs, Seidel switching, and two-graphs.
In Section 3 we give useful analytic descriptions of the graphs
$NO^{\pm}(2m,2)$, $NO^{\mp}(2m+1,q)$, $q=2^{h}$. In Section 4 we define the
classes $\mathcal{X}^{\pm}_{2m}$ of two-graphs with doubly transitive
automorphism group $Sp(2m,2)$, and prove the main theorem on the two-graphs of
$NO^{\pm}(4m,2)$, $NO^{\mp}(2m+1,4)$. Finally, Appendix A reports some useful
concepts and tools about symplectic groups and quadrics in characteristic two.

##  1\. Symplectic and orthogonal groups in characteristic two

###  1.1. Quadratic forms and alternating bilinear forms

Consider the $2m$-dimensional vector space $V=\mathbb{F}_{q}^{2m}$, $q=2^{h}$,
of row vectors. The map $\Theta:V\to\mathbb{F}_{q}$ is a quadratic form, if
$\Theta(\lambda u)=\lambda^{2}\Theta(u)$ for all $\lambda\in\mathbb{F}_{q}$,
$u\in V$, and

$f(u,v)=\Theta(u+v)+\Theta(u)+\Theta(v)$

is a bilinear form on $V$. In this case, we say that $\Theta$ linearizes to
$f$. In even dimension, the quadratic form $\Theta$ is non-degenerate, if its
associated bilinear form is non-degenerate. We use the notation
$u^{\perp},U^{\perp}$ for the orthogonal complement of $u\in V$, or $U\subset
V$. In characteristic two, quadratic forms always linearize to alternating
bilinear forms: $f(u,u)=0$ for all $u\in V$. A bilinear form is symplectic if
it is non-degenerate and alternating.

Take the matrices

$E=\Big{(}\begin{array}[]{cc}0&I_{m}\\\ 0&0\end{array}\Big{)}$

and

$F=E+E^{T}=\Big{(}\begin{array}[]{cc}0&I_{m}\\\ I_{m}&0\end{array}\Big{)}.$

We define the symplectic form $\langle.,.\rangle$ by

(1) $\langle u,v\rangle=uFv^{T}.$

Throughout this paper, we denote by $\Omega$ the set of quadratic forms
$V\to\mathbb{F}_{q}$ that linearize to $\langle.,.\rangle$. Then

$\vartheta_{0}(u)=uEu^{T}$

and

(2) $\vartheta_{a}(u)=\vartheta_{0}(u)+\langle a,u\rangle^{2}$

are both quadratic forms in $\Omega$. The map $\Theta+\vartheta_{0}$ is
additive and $(\Theta+\vartheta_{0})(\lambda
u)=\lambda^{2}(\Theta+\vartheta_{0})(u)$. As elements in $\mathbb{F}_{q}$ have
a unique square root, $(\Theta+\vartheta_{0})^{\frac{1}{2}}$ is a well-defined
linear map $V\to\mathbb{F}_{q}$. This implies $\Theta=\vartheta_{a}$ for some
$a\in\mathbb{F}_{q}^{2m}$, and

$\Omega=\\{\vartheta_{a}\mid a\in\mathbb{F}_{q}^{2m}\\}.$

Let $\Theta:\mathbb{F}_{q}^{2m}\to\mathbb{F}_{q}$ be a non-degenerate
quadratic form. Then $\Theta$ has Witt index $m-1$ or $m$. In the first case,
$\Theta$ is elliptic, and it is equivalent to

$f(x_{1},x_{2})+x_{3}x_{4}+\cdots+x_{n-1}x_{n},$

where $f(x,y)$ is an irreducible quadratic form in two variables. If the Witt
index is $m$, then $\Theta$ is hyperbolic, and it is equivalent to

$x_{1}x_{2}+x_{3}x_{4}+\cdots+x_{n-1}x_{n}.$

Elliptic or hyperbolic type quadratic forms are also called type $-1$ or type
$+1$, respectively. $\vartheta_{0}$ is of hyperbolic type.

In odd dimension, the behavior of quadratic and symplectic bilinear forms is
slightly different. The quadratic form
$\Theta:\mathbb{F}_{q}^{2m+1}\to\mathbb{F}_{q}$ has Witt index at most $m$.
Quadratic forms of maximum Witt index are non-degenerate. Such forms are
equivalent to

$x_{1}^{2}+x_{2}x_{3}+\cdots+x_{n-1}x_{n}.$

In this case, we say that $\Theta$ is parabolic, or that $\Theta$ is of type
$0$.

###  1.2. Symplectic and orthogonal transformations

As before, $V=\mathbb{F}_{q}^{2m}$, $q=2^{h}$. The linear map $A:V\to V$ is a
symplectic transformation w.r.t. the symplectic form $\langle.,.\rangle$
provided

$\langle uA,vA\rangle=\langle u,v\rangle$

holds for all $u,v\in V$. As $\langle.,.\rangle$ is non-degenerate, a
symplectic transformation is invertible; they form the group symplectic group
$Sp(2m,q)$. The affine symplectic group $ASp(2m,q)$ consists of maps $u\mapsto
uA+b$, where $A\in Sp(2m,q)$ and $b\in V$. Since $AFA^{T}=F$ we see that
$Sp(2m,q)\leq SL(2m,q)$, i.e., all symplectic transformations have determinant
$1$.

For any linear transformation $A:V\to V$ and quadratic form
$\Theta:V\to\mathbb{F}_{q}$, we define the map

$\Theta^{A}:V\to\mathbb{F}_{q},\qquad\Theta^{A}(u)=\Theta(uA^{-1}).$

$\Theta^{A}$ is a quadratic form, and $(\Theta,A)\mapsto\Theta^{A}$ is a group
action. If $A$ is symplectic w.r.t. $\langle.,.\rangle$, then $A$ preserves
$\Omega$. In particular, $Sp(2m,q)$ acts on $\Omega$.

Let $\Theta$ be a fixed quadratic form, and let $\varepsilon\in\\{-1,0,1\\}$
denote its type. The orthogonal group (associated to $\Theta$) is defined as

$O^{\varepsilon}(n,q)=\\{A\in GL(n,q)\mid\Theta(uA)=\Theta(u)\text{ for all
$u\in V$}\\}.$

Let $n=2m+1$ and $\Theta$ a parabolic quadratic form. If we linearize
$\Theta$, then we obtain a degenerate alternating form with $1$-dimensional
radical $V_{0}$. The factor space $V/V_{0}$ has a non-degenerate alternating
form. This induces the group isomorphism

$O(2m+1,q)\cong Sp(2m,q)$

for finite fields of characteristic two.

The following three theorems summarize common knowledge on the structure of
symplectic groups and their action on quadratic forms. Since it is hard to
find concise proof for these properties in the literature, we give the basic
ideas in the Appendix.

######  Theorem 1.1.

Let $q$ be a power of two, and $\langle.,.\rangle$ a symplectic bilinear form
on $\mathbb{F}_{q}^{2m}$. Let $\Omega$ be the set of quadratic forms
$V\to\mathbb{F}_{q}$ which linearize to $\langle.,.\rangle$.

  1. (i)

The symplectic group $Sp(2m,q)$ has two orbits

$\displaystyle\Omega^{+}$
$\displaystyle=\\{\vartheta_{a}\in\Omega\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=0\\},$
$\displaystyle\Omega^{-}$
$\displaystyle=\\{\vartheta_{a}\in\Omega\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=1\\}$

on $\Omega$.

  2. (ii)

$|\Omega^{+}|=\frac{1}{2}(q^{2m}+q^{m})$ and
$|\Omega^{-}|=\frac{1}{2}(q^{2m}-q^{m})$.

  3. (iii)

Let $R^{+}$ (resp. $R^{-}$) be the stabilizer subgroup of
$\Theta\in\Omega^{+}$ (resp. $\Theta\in\Omega^{-}$) in $Sp(2m,q)$. $R^{+}$
(resp. $R^{-}$) is isomorphic to $O^{+}(2m,q)$ (resp. $O^{-}(2m,q)$), and its
action on $\Omega$ is equivalent to the action of $O^{+}(2m,q)$ (resp.
$O^{-}(2m,q)$) on $V$.

The case $q=2$ has some further important properties.

######  Theorem 1.2.

$Sp(2m,2)$ acts $2$-transitively on $\Omega^{+}$ and on $\Omega^{-}$. $R^{+}$
(resp. $R^{-}$) acts transitively on $\Omega^{-}$ (resp. $\Omega^{+}$).

The translations $\tau_{d}:u\mapsto u+d$, $d\in V=\mathbb{F}_{2}^{2m}$, form
an elementary abelian group $N$, acting regularly on $V$. Any linear
transformation $A$ normalizes $N$. The affine symplectic group $ASp(2m,q)$ is
the closure group of $N$ and $Sp(2m,q)$.

######  Theorem 1.3.

In the affine symplectic group $ASp(2m,q)$, the regular normal subgroup $N$
has two complements $H_{1},H_{2}\cong Sp(2m,q)$ such that the orbits of
$H_{1}$ have length $1$ and $q^{2m}-1$ and the orbits of $H_{2}$ have length
$\frac{1}{2}(q^{2m}+q^{m})$, $\frac{1}{2}(q^{2m}-q^{m})$. The actions of
$H_{2}$ on the orbits are equivalent to the actions of $Sp(2m,q)$ on
$\Omega^{+}$ and $\Omega^{-}$.

##  2\. Strongly regular graphs, Seidel switching, two-graphs

###  2.1. Preliminaries on projective quadrics

We give a short overview of the notions related to the geometry of quadrics in
finite projective spaces; for more details, see [7]. We deal only with the
case of characteristic $2$.

Let $V(n+1,q)$ be a vector space of dimension $n+1$ over the finite field
$F_{q}$. The projective space $PG(n,q)$, or $PG(V)$, is the set of all
$1$-dimensional subspaces of $V$, called projective points. A non-zero vector
$a\in V$ determines the projective point $P(a)=\langle a\rangle$. Let $W$ be
an $(h+1)$-dimensional subspace of $V$. The $1$-dimensional subspaces of $W$
form a projective subspace $[W]=PG(h,q)$; we say that $[W]$ has projective
dimension $h$.

Let $\Theta:V\to\mathbb{F}_{q}$ be a quadratic form on $V$. Let
$\mathcal{V}(\Theta)$ denote the set of projective points $P(a)$ such that
$\Theta(a)=0$. This is a well-defined quadratic surface since $\Theta$ is
homogeneous of degree $2$. The quadric $Q=\mathcal{V}(\Theta)$ is non-
singular, if $\Theta$ is non-degenerate. We say that $Q$ is elliptic,
parabolic or hyperbolic, if $\Theta$ is a non-degenerate quadratic form of the
appropriate type. We also denote the elliptic, parabolic, and hyperbolic
quadrics of $PG(n,q)$ by $Q^{-}(n,q)$, $Q(n,q)$, and $Q^{+}(n,q)$,
respectively.

Let $Q$ be a non-singular quadric in $PG(n,q)$ and $S$ a projective subspace
of projective dimension $h>1$. $S\cap Q$ is a quadric in $S$. We say that $S$
is tangent or secant to $Q$ if $S\cap Q$ is singular or non-singular in $S$,
respectively. A line $\ell$ is tangent to $Q$ if and only if $|\ell\cap Q|=1$
or $\ell\subseteq Q$.

Let $Q=\mathcal{V}(\Theta)$ be a non-singular quadric in $PG(n,q)$, $n$ odd.
Then the underlying linear space $V$ has even dimension $n+1$. The alternating
form $\langle.,.\rangle$ associated with $\Theta$ is non-degenerate, and it
determines a symplectic polarity of $V$. The polar $S^{\perp}$ of a projective
subspace $S$ of projective dimension $h$ has dimension $n-h-1$. In particular,
the polar of a point is a hyperplane; $P\in Q$ if and only if $P^{\perp}$ is a
tangent hyperplane to $Q$.

Let $Q=\mathcal{V}(\Theta)$ be a non-singular quadric in $PG(n,q)$, $n$ even.
Then, $\langle.,.\rangle$ is singular with a $1$-dimensional kernel $\langle
u\rangle$, such that $\Theta(u)\neq 0$. The projective point $N=P(u)$ is
called the nucleus of $Q$. The hyperplane is tangent to $Q$ if and only if it
contains $N$. If the hyperplane $H$ does not contain $N$, then $H\cap Q$ is a
non-singular quadric in $PG(n-1,q)$, of type $+1$ (hyperbolic) or of type $-1$
(elliptic).

###  2.2. Strongly regular graphs on the non-singular points of polar spaces

In this subsection, we recall the definition of two infinite classes of
strongly regular graphs, see [4, Section 3.1] for the proofs of the Lemmas 2.2
and 2.4 and further detail.

Consider a non-singular quadric $Q^{\pm}(2m-1,2)$ in $PG(2m-1,2)$, of type
$\pm 1$. Then $|Q^{\pm}(2m-1,2)|=2^{2m-1}\pm 2^{m-1}-1$, and
$|PG(2m-1,2)\setminus Q^{\pm}(2m-1,2)|=2^{2m-1}\mp 2^{m-1}$.

######  Definition 2.1.

The graph $NO^{\pm}(2m,2)$ is the graph whose vertex set is
$PG(2m-1,2)\setminus Q^{\pm}(2m-1,2)$, and two vertices are adjacent if and
only if the two corresponding points lie on a line tangent to the quadric
$Q^{\pm}(2m-1,2)$.

######  Lemma 2.2.

$NO^{\pm}(2m,2)$ is a strongly regular graph with parameters

$\displaystyle v$ $\displaystyle=2^{2m-1}\mp 2^{m-1},$ $\displaystyle k$
$\displaystyle=2^{2m-2}-1,$ $\displaystyle\lambda$ $\displaystyle=2^{2m-3}-2,$
$\displaystyle\mu$ $\displaystyle=2^{2m-3}\pm 2^{m-2}.$

The automorphism group of $NO^{\pm}(2m,2)$ contains the orthogonal group
$O^{\pm}(2m,2)$.

Now consider a non-singular parabolic quadric $Q(2m,q)$ in $PG(2m,q)$, $q$
even. Let $\mathcal{H}^{\pm}$ the set of hyperplanes not containing the
nucleus $N$ and intersecting the quadric in a non-singular quadric of type
$\pm 1$.

######  Definition 2.3.

The graph $NO^{\pm}(2m+1,q)$ is the graph whose vertex set is
$\mathcal{H}^{\pm}$, and two vertices $H_{1}$ and $H_{2}$ are adjacent if and
only if $H_{1}\cap H_{2}\cap Q(2m,q)$ is degenerate.

######  Lemma 2.4.

$NO^{\pm}(2m+1,q)$ is a strongly regular graph with parameters

$\displaystyle v$ $\displaystyle=\frac{1}{2}q^{m}(q^{m}\pm 1),$ $\displaystyle
k$ $\displaystyle=(q^{m-1}\pm 1)(q^{m}\mp 1),$ $\displaystyle\lambda$
$\displaystyle=2(q^{2m-2}-1)\pm q^{m-1}(q-1),$ $\displaystyle\mu$
$\displaystyle=2q^{m-1}(q^{m-1}\pm 1).$

The automorphism group of $NO^{\pm}(2m+1,q)$ contains $O(2m+1,q)\cong
Sp(2m,q)$.

###  2.3. Two-graphs and switchings

We start this subsection by defining the concepts of two-graph, and of two-
graph associated with a given graph.

######  Definition 2.5 (Two-graph).

A two-graph is a pair $(X,T)$, where $T$ is a set of unordered triples of a
vertex set $X$, such that every (unordered) quadruple from $X$ contains an
even number of triples from $T$. In a regular two-graph, each pair of vertices
is in a constant number of triples.

######  Definition 2.6 (Associated two-graph).

Given a graph $G=(V,E)$, the set of triples $T$ of the vertex set $V$ whose
induced subgraph has an odd number of edges forms a two-graph on the set $V$.
The two-graph $\operatorname{Omega}(G)=(V,T)$ is the associated two-graph of
$G$.

Let $\Omega=(V,T)$ be a two-graph. The descendant of $\Omega$ w.r.t. the
vertex $w\in V$ is the graph $\Omega_{w}=(V,E)$ where $E$ consists of pairs
$(u,v)$ such that $\\{u,v,w\\}\in T$. By definition, $w$ is an isolated vertex
of the descendant. It is not hard to see that
$\operatorname{Omega}(\Omega_{w})=\Omega$, which shows that every two-graph
can be represented as an associated two-graph of a graph.

######  Definition 2.7 (Seidel switching [8]).

Given a graph $G=(V,E)$ and a subset $Y$ of the vertex set, the operation of
switching $G$ with respect to $Y$ consists of replacing all edges from $Y$ to
its complement by nonedges, and all nonedges by edges, while leaving the edges
within $Y$ or outside $Y$ unchanged.

Switching defines an equivalence relation on the set of all graphs with a
given $n$-element vertex set, each equivalence class containing $2^{n-1}$
graphs (since switching with respect to a set and its complement are the same
thing, and switching with respect to two sets is the same as switching with
respect to their symmetric difference). The equivalence classes are called
switching classes.

The following proposition is well-known; we give the proof for the sake of
completeness.

######  Proposition 2.8.

Two graphs are switching equivalent if and only if they have the same
associated two-graph.

###### Proof.

Consider a subset $Y\subseteq V$ as above. W.l.o.g. we say that each triple
$t=\\{u,v,z\\}$ either has all vertices in $Y$, or two vertices in $Y$ and one
in $V\setminus Y$. In the former case, the adjacencies in $t$ are fixed since
the adjacencies in a component $G_{i}$ are not involved in the switching. In
the latter case, we easily check that the switching preserves the parity of
the number of edges between $u$, $v$, and $z$ (we have 1, 2, or 0 edges
through $Y$ and $V\setminus Y$, and the Seidel switching deletes the edge and
adds another one, deletes both edges, or adds two edges, respectively).

Conversely, assume that $G_{1}=(V,E_{1})$ and $G_{2}=(V,E_{2})$ have the same
associated two-graph $\Omega$. Let $F$ be the symmetric difference of $E_{1}$
and $E_{2}$. It is straightforward to check that the graph $H=(V,F)$ has the
associated two-graph empty. This implies that the non-adjacency relation is an
equivalence relation on $V$, with at most $2$ equivalence classes. Choosing
$Y$ to be one of the equivalence classes, we find that $G_{2}$ is the
switching of $G_{1}$ w.r.t. $Y$. ∎

##  3\. Analytic description of strongly regular polar graphs

Throughout this section, notation will follow Sections 1 and 2.

######  Proposition 3.1 (Analytic description of $NO^{\pm}(2m,2)$).

Let $\Theta$ be a non-degenerate quadratic form on $\mathbb{F}_{2}^{2m}$ of
type $\pm 1$. Let $\langle.,.\rangle$ be the symplectic form associated to
$\Theta$. Define the graph $(V,E)$ with

$\displaystyle V$
$\displaystyle=\\{a\in\mathbb{F}_{2}^{2m}\mid\Theta(a)=1\\},$ $\displaystyle
E$ $\displaystyle=\\{(a,b)\in V\times V\mid\langle a,b\rangle=0\text{ and
}a\neq b\\}.$

Then $(V,E)\cong NO^{\pm}(2m,2)$.

###### Proof.

Let $Q^{\pm}(2m-1,2)=\mathcal{V}(\Theta)$ correspond to the quadratic form
$\Theta$. Projective points $PG(2m-1,2)\setminus Q^{\pm}(2m-1,2)$ have
coordinate vectors $a\in\mathbb{F}_{2}^{2m}$ with $\Theta(a)=1$. Hence, the
vertices of $NO^{\pm}(2m,2)$ are in $1$-to-$1$ correspondence with our set
$V$. For $a,b\in V$, the projective line $P(a)P(b)$ is consists of the three
points $\\{P(a),P(b),P(a+b)\\}$ and

$\displaystyle a\sim b$ $\displaystyle\Leftrightarrow P(a+b)\in
Q^{\pm}(2m-1,2)$ $\displaystyle\Leftrightarrow\Theta(a+b)=0$
$\displaystyle\Leftrightarrow\Theta(a)+\Theta(b)+\langle a,b\rangle=0$
$\displaystyle\Leftrightarrow\langle a,b\rangle=0.\qed$

Using quadratic forms, we aim to give an equivalent (analytic) description of
$NO^{\pm}(2m+1,q)$ using quadratic forms. We say that two non-singular
quadrics $Q_{1},Q_{2}$ of $PG(2m-1,q)$ are tangent if there is a hyperplane
$H$ such that $H$ is tangent to both $Q_{1},Q_{2}$, and $Q_{1}\cap H=Q_{2}\cap
H$. For the non-singular quadrics $Q_{1}=\mathcal{V}(\vartheta_{a})$,
$Q_{2}=\mathcal{V}(\vartheta_{b})$, being tangent has a simple analytic
characterization.

######  Lemma 3.2.

For $\vartheta_{a},\vartheta_{b}\in\Omega$, the following are equivalent:

  1. (i)

The non-singular quadrics $\mathcal{V}(\vartheta_{a})$,
$\mathcal{V}(\vartheta_{b})$ are tangent.

  2. (ii)

$\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})$ is a degenerate
quadric.

  3. (iii)

$P(a+b)\in\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})$.

  4. (iv)

$\vartheta_{a}(a+b)=0$.

###### Proof.

By the definition (2), we have

(3)
$\displaystyle\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})=\mathcal{V}(\vartheta_{a})\cap
H=\mathcal{V}(\vartheta_{b})\cap H,$

where $H=(a+b)^{\perp}$ is the hyperplane polar to $a+b$. Hence,
$\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})$ is degenerate if
and only if $H$ is tangent to both $\mathcal{V}(\vartheta_{a})$ and
$\mathcal{V}(\vartheta_{b})$. This holds if and only if $H^{\perp}=P(a+b)$ is
a point of both of them. This shows (ii)$\Leftrightarrow$(iii). The
equivalence (iii)$\Leftrightarrow$(iv) is obvious. (ii), (iii) and equation
(3) imply (i).

Assume (i): Let $K$ be a hyperplane which is tangent to both
$\mathcal{V}(\vartheta_{a})$, $\mathcal{V}(\vartheta_{b})$, and
$\mathcal{V}(\vartheta_{a})\cap K=\mathcal{V}(\vartheta_{b})\cap K$. Then
$\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})\subseteq K$.
However, $\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})$ spans $H$
by (3), therefore $H=K$. As $H$ is tangent to $\mathcal{V}(\vartheta_{a})$,
$\mathcal{V}(\vartheta_{b})$, we have
$P(a+b)=H^{\perp}\in\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})$,
and (iii) holds. ∎

Being tangent implies the type of quadratic forms.

######  Lemma 3.3.

  1. (i)

If $\mathcal{V}(\vartheta_{a})$ and $\mathcal{V}(\vartheta_{b})$ are tangent,
then $\vartheta_{a},\vartheta_{b}$ have the same type.

  2. (ii)

If $q=2$ and $\vartheta_{a}$ and $\vartheta_{b}$ have the same type, then
$\mathcal{V}(\vartheta_{a})$ and $\mathcal{V}(\vartheta_{b})$ are tangent.

###### Proof.

Since all $\vartheta_{a}$ linearize to $\langle.,.\rangle$, by definition we
have

$\displaystyle\vartheta_{a}(a+b)$
$\displaystyle=\vartheta_{a}(a)+\vartheta_{a}(b)+\langle a,b\rangle$
$\displaystyle=\vartheta_{0}(a)+\vartheta_{0}(b)+\langle a,b\rangle+\langle
a,b\rangle^{2}.$

As $\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\langle
a,b\rangle)=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\langle
a,b\rangle^{2})$, the implications

$\mathcal{V}(\vartheta_{a})$, $\mathcal{V}(\vartheta_{b})$ are tangent
$\displaystyle\Longleftrightarrow\vartheta_{a}(a+b)=0$
$\displaystyle\Longrightarrow\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{a}(a+b))=0$
$\displaystyle\Longleftrightarrow\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(b))$
$\displaystyle\Longleftrightarrow\text{$\vartheta_{a}$, $\vartheta_{b}$ have
the same type}$

hold by Lemma 3.2 and Theorem 1.1(i). If $q=2$, then the right implication
above becomes equivalence, as well. ∎

In the following lemma, we make a first attempt to interpret the graphs
$NO^{\pm}(2m+1,q)$ analytically.

######  Lemma 3.4.

Define the graph $(W^{\pm},F)$ with

$\displaystyle W^{+}$
$\displaystyle=\\{a\in\mathbb{F}_{q}^{2m}\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=0\\}$
$\displaystyle W^{-}$
$\displaystyle=\\{a\in\mathbb{F}_{q}^{2m}\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=1\\}$
$\displaystyle F$ $\displaystyle=\\{(a,b)\in W^{\pm}\times
W^{\pm}\mid\vartheta_{a}(a+b)=0\text{ and }a\neq b\\}.$

Then $(W^{\pm},F)\cong NO^{\pm}(2m+1,q)$.

###### Proof.

Firstly see that $W^{\pm}$ and $\Omega^{\pm}$ are in 1-to-1 correspondence by
associating $a\in W^{\pm}$ to $\vartheta_{a}\in\Omega^{\pm}$. Fix a hyperplane
$H_{0}$ not through the nucleus $N$ and define the central projection $\pi$
from $N$ to $H_{0}$, bijection between $Q$ and $H_{0}$. The restriction of
$\langle.,.\rangle$ to $H_{0}$ is non-singular, and let $\Omega^{\pm}$ be the
set of quadratic forms on $H_{0}$, linearizing to
$\langle.,.\rangle|_{H_{0}}$. Moreover, $\pi$ preserves the symplectic form,
and $\pi$ induces a bijection between $\mathcal{H}^{\pm}$ and $\Omega^{\pm}$.
Indeed, if the hyperplane $H$ does not contain $N$, then $Q\cap H$ is non-
singular in $H$ and $Q^{\prime}=\pi(Q\cap H)$ is non-singular in $H_{0}$.
There is a quadratic form $\Theta$ such that $Q^{\prime}=\mathcal{V}(\Theta)$,
and $\Theta$ linearizes to a multiple of $\langle.,.\rangle$ as $\pi$
preserves the symplectic structure. By multiplying with an appropriate scalar,
we can assume $\Theta\in\Omega$. In particular, $\Theta=\vartheta_{a}$ for
some $a\in\mathbb{F}_{q}^{2m}$.

Fix $H_{1},H_{2}\in\mathcal{H}^{\pm}$ with $\pi(Q\cap
H_{1})=\mathcal{V}(\vartheta_{a})$, $\pi(Q\cap
H_{2})=\mathcal{V}(\vartheta_{b})$, $a,b\in\mathbb{F}_{q}^{2m}$,
$\vartheta_{a},\vartheta_{b}\in\Omega^{\pm}$. Since $Q\cap H_{1}$ is non-
degenerate,

$\displaystyle H_{1}\sim H_{2}$ $\displaystyle\Longleftrightarrow Q\cap
H_{1}\cap H_{2}\mbox{ degenerate}$ $\displaystyle\Longleftrightarrow\pi(Q\cap
H_{1})\cap\pi(Q\cap H_{2})\mbox{ degenerate}$
$\displaystyle\Longleftrightarrow\mathcal{V}(\vartheta_{a})\cap\mathcal{V}(\vartheta_{b})\mbox{
degenerate}$
$\displaystyle\Longleftrightarrow\mathcal{V}(\vartheta_{a}),\mathcal{V}(\vartheta_{b})\mbox{
tangent}$ $\displaystyle\Longleftrightarrow\vartheta_{a}(a+b)=0.\qed$

We can now analytically describe the graphs $NO^{\pm}(2m+1,q)$.

######  Proposition 3.5 (Analytic description of $NO^{\pm}(2m+1,q)$).

Let $\Theta$ be a non-degenerate quadratic form on $\mathbb{F}_{q}^{2m}$ of
type $\mp 1$. Let $\langle.,.\rangle$ be the symplectic form associated to
$\Theta$. Define the graph $(V,E)$ with

$\displaystyle V$
$\displaystyle=\\{a\in\mathbb{F}_{q}^{2m}\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\Theta(a))=1\\}$
$\displaystyle E$ $\displaystyle=\\{(a,b)\in V\times V\mid\text{$a\neq b$ and
$\Theta(a+b)=\langle a,b\rangle^{2}$}\\}.$

Then $(V,E)\cong NO^{\pm}(2m+1,q)$.

###### Proof.

Let $\varepsilon=\pm 1$ be the sign of the graph; the quadratic form $\Theta$
has type $-\varepsilon$. Define $\lambda(\varepsilon)\in\mathbb{F}_{2}$ such
that $\varepsilon=(-1)^{\lambda(\varepsilon)}$. Write $\Theta=\vartheta_{d}$
for some $d\in\mathbb{F}_{q}^{2m}$. Since $\vartheta_{0}$ has type $+1$, we
have

$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(d))=1+\lambda(\varepsilon)$

by Theorem 1.1(i). This implies

$\displaystyle\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{d}(x))$
$\displaystyle=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(x)+\langle
x,d\rangle^{2})$
$\displaystyle=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(x)+\langle
x,d\rangle)$
$\displaystyle=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(x+d)+\vartheta_{0}(d))$
$\displaystyle=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(x+d))+1+\lambda(\varepsilon).$

In other words, for all $x\in\mathbb{F}_{q}^{2m}$

$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{d}(x))=1\Longleftrightarrow\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(x+d))=\lambda(\varepsilon).$

This means that $W^{\pm}=V+d$, and the translation $\tau_{d}:x\mapsto x+d$
induces a bijection

$V\longleftrightarrow W^{\pm}.$

Here,
$W^{\pm}=\\{a\in\mathbb{F}_{q}^{2m}\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=\lambda(\varepsilon)\\}$
is defined as in Lemma 3.4. For the pair $(a,b)\in V\times V$, $a\neq b$, we
have

$\displaystyle(a,b)\in E$
$\displaystyle\Longleftrightarrow\vartheta_{d}(a+b)=\langle a,b\rangle^{2}$
$\displaystyle\Longleftrightarrow\vartheta_{d}(a+b)=\langle a+b,a\rangle^{2}$
$\displaystyle\Longleftrightarrow\vartheta_{a+d}(a+b)+\vartheta_{a}(a+b)+\vartheta_{0}(a+b)+\langle
a+b,a\rangle^{2}=0$ $\displaystyle\Longleftrightarrow\vartheta_{a+d}(a+b)=0$
$\displaystyle\Longleftrightarrow(a+d,b+d)\in F.$

Again, $F$ is the set of edges of the graph in Lemma 3.4. What we see is that
$\tau_{d}$ induces an isomorphism between the graphs $(V,E)$ and
$(W^{\pm},F)$. ∎

##  4\. Two-graphs of strongly regular polar graphs

###  4.1. The symplectic two-graphs

Define the graph $\Sigma_{2m}$ with vertex set $\mathbb{F}_{2}^{2m}$ and
adjacency relation $\langle a,b\rangle=0$ for vectors $a\neq b$. This graph is
not regular, since the origin is adjacent to all nonzero vectors. The
automorphism group of $\Sigma_{2m}$ contains the linear symplectic group
$Sp(2m,2)$.

For non-adjacent vertices $a,b$ of $\Sigma_{2m}$, we have $\langle
a,b\rangle=1$. The triple $\\{a,b,c\\}$ has an odd number of edges if and only
if

(4) $\displaystyle\langle a,b\rangle+\langle a,c\rangle+\langle b,c\rangle=0.$

This means that $(\mathbb{F}_{2}^{2m},\mathcal{T}_{2m})$ is the two-graph of
$\Sigma_{2m}$, where $\mathcal{T}_{2m}$ is the set of triples that satisfy
(4). By some abuse of notation, we denote the two-graph by $\mathcal{T}_{2m}$
as well. As

$\displaystyle\langle a,b\rangle+\langle a,c\rangle+\langle b,c\rangle=\langle
a+d,b+d\rangle+\langle a+d,c+d\rangle+\langle b+d,c+d\rangle,$

the translation $\tau_{d}:x\mapsto x+d$ stabilizes $\mathcal{T}_{2m}$ for all
$d$. This implies that $ASp(2m,2)$ is a subgroup of
$\operatorname{Aut}(\mathcal{T}_{2m})$. In particular,
$\operatorname{Aut}(\mathcal{T}_{2m})$ is $2$-transitive and
$\mathcal{T}_{2m}$ is a regular two-graph.

By Theorem 1.3, there is a subgroup $H_{2}\cong Sp(2m,2)$ in $ASp(2m,2)$ with
orbits $X^{+}$, $X^{-}$ of size $2^{2m-1}-2^{m-1}$ and $2^{2m-1}+2^{m-1}$.
Moreover, $H_{2}$ acts $2$-transitively on $X^{+},X^{-}$. Let us denote by
$\mathcal{X}^{\pm}_{2m}$ the set of triples $\\{a,b,c\\}$ that satisfy (4).
Then $(X^{+},\mathcal{X}^{+}_{2m})$, $(X^{-},\mathcal{X}^{-}_{2m})$ are sub-
two-graphs of $\mathcal{T}_{2m}$. We use the notation
$\mathcal{X}^{+}_{2m},\mathcal{X}^{-}_{2m}$ for these two-graphs too. As their
automorphism group is $2$-transitive, $\mathcal{X}^{+}_{2m}$ and
$\mathcal{X}^{-}_{2m}$ are regular two-graphs.

The above definition is rather sloppy since the subgroup $H_{2}$ is defined
only up to conjugacy in $ASp(2m,2)$, and the sets $X^{+},X^{-}$ are defined up
to translations. One possibility is to choose the sets

$\displaystyle X^{+}$
$\displaystyle=\\{a\in\mathbb{F}_{2}^{2m}\mid\vartheta_{0}(a)=1\\},$
$\displaystyle X^{-}$
$\displaystyle=\\{a\in\mathbb{F}_{2}^{2m}\mid\vartheta_{0}(a)=0\\}$

according to Theorem 1.1. In the following proposition, we opt for a
representation that is in agreement with the analytic descriptions of our
strongly regular graphs (Propositions 3.1 and 3.5).

######  Proposition 4.1.

Let $\Theta$ be a non-degenerate quadratic form on $\mathbb{F}_{2}^{2m}$ of
type $\pm 1$. Let $\langle.,.\rangle$ be the symplectic form associated to
$\Theta$. Define the set of vectors

$X=\\{a\in\mathbb{F}_{2}^{2m}\mid\Theta(a)=1\\}$

and the set of triples

$T=\\{\\{a,b,c\\}\mid a\neq b\neq c\neq a,\;\langle a,b\rangle+\langle
a,c\rangle+\langle b,c\rangle=0\\}.$

Then the following hold:

  1. (i)

$\mathcal{X}^{\pm}_{2m}=(X,T)$ is a regular two-graph of degree $2^{2m-2}\mp
2^{m-1}-2$.

  2. (ii)

The automorphism group of $\mathcal{X}^{\pm}_{2m}$ is isomorphic to $Sp(2m,2)$
in its $2$-transitive action on $2^{2m-1}\mp 2^{m-1}$ points.

###### Proof.

We have seen in the proof of Proposition 3.5, that for some
$d\in\mathbb{F}_{2}^{2m}$, the translation $\tau_{d}$ induces a bijection
between $X^{\pm}$ and $X$. Translations are contained in $ASp(2m,2)$, and
preserve the relation (4), hence (ii) and the regularity holds. It only
remains to compute the degree. Let $a,b\in\mathbb{F}_{2}^{2m}$ be different
vectors such that their projective line $\ell$ is exterior to the projective
quadric $\Theta=0$. Since the third point of $\ell$ is $a+b$, we have

$\Theta(a)=\Theta(b)=\Theta(a+b)=1.$

Hence $\langle a,b\rangle=1$. Fix any point
$x\in\mathbb{F}_{2}^{2m}\setminus\\{a,b\\}$ with $\Theta(x)=1$.

$\displaystyle\\{a,b,x\\}\in T$ $\displaystyle\Longleftrightarrow\langle
a,x\rangle+\langle b,x\rangle+\langle a,b\rangle=0$
$\displaystyle\Longleftrightarrow\langle a+b,x\rangle=1$
$\displaystyle\Longleftrightarrow\text{$a+b$ and $x$ are non-adjacent in
$NO^{\pm}(2m,2)$.}$

By Lemma 2.2, the parameters of $NO^{\pm}(2m,2)$ are $v=2^{2m-1}\mp 2^{m-1}$
and $k=2^{2m-2}-1$. Therefore, the degree of the two-graph is

$\displaystyle|\\{x\in\mathbb{F}_{2}^{2m}\mid\\{a,b,x\\}\in T\\}|$
$\displaystyle=v-k-1-2$ $\displaystyle=(2^{2m-1}\mp 2^{m-1})-(2^{2m-2}-1)-3$
$\displaystyle=2^{2m-2}\mp 2^{m-1}-2.\qed$

###  4.2. The descendants of symplectic two-graphs

In this subsection, we introduce another structure of polar graph related to
quadrics in odd projective dimension, over the binary field, see [4, Sections
2.6.2 and 2.6.3].

######  Definition 4.2 (Orthogonal graphs).

The graph $\Gamma(O^{\pm}(2m,2))$ is the graph whose vertex set is
$Q^{\pm}(2m-1,2)$ and in which two vertices are adjacent if and only if the
two corresponding points lie on a totally isotropic line tangent to the
quadric $Q^{\pm}(2m-1,2)$.

######  Lemma 4.3.

$\Gamma(O^{\pm}(2m,2))$ is a strongly regular graph with parameters

$\displaystyle v$ $\displaystyle=(2^{m}\mp 1)(2^{m-1}\pm 1),$ $\displaystyle
k$ $\displaystyle=2(2^{m-1}\mp 1)(2^{m-2}\pm 1),$ $\displaystyle\lambda$
$\displaystyle=4(2^{m-2}\mp 1)(2^{m-3}\pm 1)+1,$ $\displaystyle\mu$
$\displaystyle=(2^{m-1}\mp 1)(2^{m-2}\pm 1).$

The automorphism group of $\Gamma(O^{\pm}(2m,2))$ contains the orthogonal
group $O^{\pm}(2m,2)$.

######  Proposition 4.4 (Analytic description of $\Gamma(O^{\pm}(2m,2))$).

Let $\Theta$ be a non-degenerate quadratic form on $\mathbb{F}_{2}^{2m}$ of
type $\pm 1$. Let $\langle.,.\rangle$ be the symplectic form associated to
$\Theta$. Define the graph $(V,E)$ with

$\displaystyle V$
$\displaystyle=\\{a\in\mathbb{F}_{2}^{2m}\setminus\\{0\\}\mid\Theta(a)=0\\}$
$\displaystyle E$ $\displaystyle=\\{(a,b)\in V\times V\mid\langle
a,b\rangle=0\text{ and }a\neq b\\}.$

Then $(V,E)\cong\Gamma(O^{\pm}(2m,2))$.

###### Proof.

The proof works as in Proposition 3.1. ∎

Recall that the descendant of a two-graph $\Omega=(X,T)$ w.r.t. the vertex
$w\in X$ is defined by considering the graph structure $\Omega_{w}=(X,F)$
where $(u,v)\in F$ if and only if $\\{u,v,w\\}\in T$. We can compute the
descendant of the two-graph $\mathcal{X}^{\pm}_{2m}$.

######  Proposition 4.5.

The descendant of $\mathcal{X}^{\pm}_{2m}$ is the graph
$\Gamma(O^{\mp}(2m,2))=(V,E)$, plus an isolated vertex.

###### Proof.

Take the two-graph $\mathcal{X}^{\pm}_{2m}=(X,T)$ as in Proposition 4.1, and
fix $d\in X$. Then $\Theta(d)=1$. The descendant of $\mathcal{X}^{\pm}_{2m}$
will be denoted as $(\mathcal{X}^{\pm}_{2m})_{d}=(X,F)$. As in the proof of
Proposition 3.5, the translation $\tau_{d}$ gives us the bijection between the
vertex sets, where $d$ plays the role of the isolated vertex. We need only to
prove the adjacency relation.

$\displaystyle(a,b)\in F$ $\displaystyle\Longleftrightarrow\\{a,b,d\\}\in T$
$\displaystyle\Longleftrightarrow\langle a,b\rangle+\langle a,d\rangle+\langle
b,d\rangle=0$ $\displaystyle\Longleftrightarrow\langle a+d,b+d\rangle=0$
$\displaystyle\Longleftrightarrow(a+d,b+d)\in E.$

Then $\tau_{d}$ induces an isomorphism between the graphs
$(\mathcal{X}^{\pm}_{2m})_{d}$ and $\Gamma(O^{\mp}(2m,2))\cup\\{0\\}$. ∎

###  4.3. The main result on the switching equivalence

The purpose of this section is to prove the switching equivalence of the
strongly regular graphs $NO^{\pm}(4m,2)$ and $NO^{\mp}(2m+1,4)$. For this, we
will need to identify the vector spaces $\mathbb{F}_{2}^{4m}$ and
$\mathbb{F}_{4}^{2m}$. We start with a technical lemmas.

######  Lemma 4.6.

Let $q=2^{h}$, and $V$ be an $\mathbb{F}_{q}$-linear space of dimension $2m$.
We consider $V$ as a $2^{hm}$ dimensional $\mathbb{F}_{2}$-linear space at the
same time.

  1. (i)

Any symplectic $\mathbb{F}_{q}$-bilinear form $\langle.,.\rangle$ determines a
symplectic $\mathbb{F}_{2}$-bilinear form $\langle.,.\rangle^{*}$ by

$\langle
u,v\rangle^{*}=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\langle
u,v\rangle).$

  2. (ii)

Similarly, if $\Theta:V\to\mathbb{F}_{q}$ is a quadratic form over
$\mathbb{F}_{q}$, then

$\Theta^{*}(u)=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\Theta(u))$

is a quadratic form over $\mathbb{F}_{2}$.

  3. (iii)

If $\Theta$ linearizes to $\langle.,.\rangle$, then $\Theta^{*}$ linearizes to
$\langle.,.\rangle^{*}$. In particular, $\Theta^{*}$ is non-singular if and
only if $\Theta$ is non-singular.

  4. (iv)

If $\Theta$ is non-singular, then $\Theta$ and $\Theta^{*}$ have the same
type.

  5. (v)

Fix the symplectic forms $\langle.,.\rangle$ and
$\langle.,.\rangle^{*}=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\langle.,.\rangle)$.
Then $\Theta\mapsto\Theta^{*}$ is a $1$-to-$1$ correspondence between the
$\mathbb{F}_{q}$-quadratic forms linearizing to $\langle.,.\rangle$ and the
$\mathbb{F}_{2}$-quadratic forms linearizing to $\langle.,.\rangle^{*}$.

###### Proof.

(i), (ii) and (iii) are obvious. We show (iv) by counting the solutions
$\Theta^{*}(x)=0$ with $x\in V$. We have

$|\Theta^{-1}(a)|=\begin{cases}q^{2m-1}\pm q^{m-1}(q-1)&\text{if $a=0$,}\\\
q^{2m-1}\mp q^{m-1}&\text{if $a\neq 0$.}\end{cases}$

In a field $\mathbb{F}_{q}$ of even characteristic there are $\frac{q}{2}-1$
non-zero elements of absolute trace $0$. This implies

$\displaystyle|\\{x\in
V\mid\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\Theta(x))=0\\}|$
$\displaystyle=q^{2m-1}\pm
q^{m-1}(q-1)+\Big{(}\frac{q}{2}-1\Big{)}(q^{2m-1}\mp q^{m-1})$
$\displaystyle=\frac{1}{2}(q^{2m}\pm q^{m})$ $\displaystyle=2^{2hm-1}\pm
2^{hm-1}.$

This is precisely the number of solutions of a quadratic equation of the
appropriate type, in a space of dimension $2hm$ over $\mathbb{F}_{2}$. (v) The
sets of both kinds of quadratic forms have cardinalities $q^{2m}=2^{2hm}$. As
$\Theta\mapsto\Theta^{*}$ is injective, (v) follows. ∎

We are now able to state our main result.

######  Theorem 4.7.

  1. (i)

The two-graph of $NO^{\pm}(2m,2)$ is isomorphic to $\mathcal{X}^{\pm}_{2m}$.

  2. (ii)

The two-graph of $NO^{\mp}(2m+1,4)$ is isomorphic to $\mathcal{X}^{\pm}_{4m}$.

  3. (iii)

The strongly regular polar graphs $NO^{\pm}(4m,2)$ and $NO^{\mp}(2m+1,4)$ are
switching equivalent.

###### Proof.

(i) We represent $NO^{\pm}(2m,2)$ and $\mathcal{X}^{\pm}_{2m}$ as in
Proposition 3.1 and Theorem 4.7 using a quandratic form $\Theta^{*}$ of type
$\pm$, which linearizes to the symplectic form $\langle.,.\rangle^{*}$ on
$\mathbb{F}_{2}^{2m}$. Then, $NO^{\pm}(2m,2)$ and $\mathcal{X}^{\pm}_{2m}$
have the same set of vertices. In $NO^{\pm}(2m,2)$, two vertices $a$ and $b$
are adjacent if and only if $\langle a,b\rangle^{*}=0$, Clearly, for all
$a,b\in V$, $\langle a,b\rangle^{*}\in\\{0,1\\}$. The triple $\\{a,b,c\\}$ is
in the two-graph associated with $NO^{\pm}(2m,2)$ if and only if there is an
odd number of adjacencies, that means there is an even number of 1’s in
$\\{\langle a,b\rangle^{*},\langle a,c\rangle^{*},\langle a,b\rangle^{*}\\}$.
Equivalently,

$\langle a,b\rangle^{*}+\langle a,c\rangle^{*}+\langle a,b\rangle^{*}=0.$

This proves the claim.

(ii) Let $q=4$, $V=\mathbb{F}_{4}^{2m}$, $\langle.,.\rangle$ and
$\langle.,.\rangle^{*}=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\langle.,.\rangle)$
symplectic forms over $\mathbb{F}_{4}$ and $\mathbb{F}_{2}$, respectively. Let
$\Theta$ be a quadratic form of type $\pm 1$ over $\mathbb{F}_{4}$, which
linearizes to $\langle.,.\rangle$. The quadratic form
$\Theta^{*}(x)=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\Theta(x))$
over $\mathbb{F}_{2}$ has the same type $\pm 1$. We represent
$NO^{\mp}(2m+1,4)$ and $\mathcal{X}^{\pm}_{4m}$ as in Proposition 3.5 and
Theorem 4.7. Then $NO^{\mp}(2m+1,4)$ and $\mathcal{X}^{\pm}_{4m}$ have the
same set of vertices

$X^{\pm}=\\{a\in V\mid\Theta^{*}(a)=1\\}.$

Moreover,

$a,b$ are adjacent in $NO^{\mp}(2m+1,4)$
$\displaystyle\Longleftrightarrow\Theta(a+b)+\langle a,b\rangle^{2}=0$
$\displaystyle\Longleftrightarrow\Theta(a)+\Theta(b)+\langle
a,b\rangle+\langle a,b\rangle^{2}=0$
$\displaystyle\Longleftrightarrow\Theta(a)+\Theta(b)+\langle
a,b\rangle^{*}=0.$

Since
$\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}(\Theta(a))=\Theta^{*}(a)=1$
and
$\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}(\Theta(b))=\Theta^{*}(b)=1$,
we have

$\operatorname{Tr}_{\mathbb{F}_{4}/\mathbb{F}_{2}}(\Theta(a)+\Theta(b))=0,$

that is, $\Theta(a)+\Theta(b)\in\mathbb{F}_{2}$. This implies

$\Theta(a+b)+\langle a,b\rangle^{2}\in\mathbb{F}_{2}$

for all $a,b\in X^{\pm}$. In particular, the triple $\\{a,b,c\\}$ has an even
number of non-edges if and only if

$\displaystyle\Theta(a+b)+\langle a,b\rangle^{2}+\Theta(a+c)+\langle
a,c\rangle^{2}+\Theta(b+c)+\langle b,c\rangle^{2}=0,$

or equivalently,

$\displaystyle\langle a,b\rangle^{*}+\langle a,c\rangle^{*}+\langle
b,c\rangle^{*}=0.$

This proves that the two-graph of $NO^{\mp}(2m+1,4)$ and
$\mathcal{X}^{\pm}_{4m}$ consist of the same set of triples.

(iii) The claim follows from (i), (ii), and Proposition 2.8. ∎

Theorem 4.7 shows that $NO^{\pm}(4m,2)$ and $NO^{\mp}(2m+1,4)$ have the same
associated two-graph, whose descendant is $\Gamma(O^{\mp}(4m,2))$. Our last
result will show the parameters of the Seidel switching between
$NO^{\pm}(4m,2)$ and $NO^{\mp}(2m+1,4)$.

######  Proposition 4.8.

The switching sets of the Seidel switching between $NO^{\pm}(4m,2)$ and
$NO^{\mp}(2m+1,4)$ are

$\displaystyle A$
$\displaystyle=\\{a\in\mathbb{F}_{4}^{2m}\mid\Theta(a)=\lambda\\},$
$\displaystyle B$
$\displaystyle=\\{a\in\mathbb{F}_{4}^{2m}\mid\Theta(a)=\lambda+1\\},$

where $\lambda\in\mathbb{F}_{4}\setminus\mathbb{F}_{2}$ is a root of
$X^{2}+X+1=0$. The switching sets have size $2^{4m-2}\mp 2^{2m-2}$ and are
regular of degree $2^{4m-3}\mp 2^{2m-2}-1$.

###### Proof.

Let $\Theta$ and $\Theta^{*}$ be defined as in Theorem 4.7. The vertices $a,b$
have the same adjacencies in both graphs if and only if $\Theta(a)=\Theta(b)$.
$\Theta^{*}(a)=\Theta^{*}(b)=1$ implies
$\Theta(a),\Theta(b)\in\\{\lambda,\lambda+1\\}$. Therefore, the switching sets
are the sets $A$, $B$ as given. Both graphs have $v=2^{4m-1}\mp 2^{2m-1}$
vertices. The degrees and the numbers of common neighbors of non-adjacent
vertices are $k=2^{4m-2}-1$, $\mu=2^{4m-3}\pm 2^{2m-2}$, and
$k^{\prime}=2^{4m-2}\mp 3\cdot 2^{2m-2}-1$, $\mu^{\prime}=2^{4m-3}\mp
2^{2m-1}$, respectively. By [4, Proposition 1.1.2], the switching sets have
size $v/2$ and are regular of degree $d=k-\mu=k^{\prime}-\mu^{\prime}$. ∎

##  5\. Acknowledgement

We thank Andries E. Brouwer and Gábor Korchmáros for stimulating discussion
and valuable remarks.

This research was supported by project TKP2021-NVA-09, implemented with the
support provided by the Ministry of Innovation and Technology of Hungary from
the National Research, Development and Innovation Fund, financed under the
TKP2021-NVA funding scheme. Partially supported by the NKFIH-OTKA Grant SNN
132625.

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##  Appendix A Symplectic groups in even characteristic

The theorems of Section 1 are well known, but it is hard to find a concise
proof for them in the literature. The purpose of the Appendix is to give the
most important steps of the proofs of Theorem 1.1, 1.2 and 1.3. We use the
terminology and notation of Section 1.

######  Lemma A.1.

Let $\Theta_{1},\Theta_{2}\in\Omega$ be non-degenerate quadratic forms
linearizing to the same symplectic form $\langle.,.\rangle$. Then
$\Theta_{1},\Theta_{2}\in\Omega$ are $GL(2m,q)$-equivalent if and only if they
are $Sp(2m,q)$-equivalent.

###### Proof.

Since $Sp(2m,q)\leq GL(2m,q)$, we only need to prove the right implication.
Let $A\in GL(2m,q)$. If $\Theta_{1}^{A}=\Theta_{2}$, then for all $u,v\in V$,

$\displaystyle\langle u,v\rangle$
$\displaystyle=\Theta_{1}(u+v)+\Theta_{1}(u)+\Theta_{1}(v)$
$\displaystyle=\Theta_{2}((u+v)A)+\Theta_{2}(uA)+\Theta_{2}(vA)$
$\displaystyle=\langle uA,vA\rangle.$

Hence $A\in Sp(2m,q)$. ∎

######  Lemma A.2.

The orders of the orthogonal groups are given by the formulas

$\displaystyle|O(2m+1,q)|$ $\displaystyle=q^{m^{2}}\prod_{i=1}^{m}(q^{2i}-1),$
$\displaystyle|O^{\pm}(2m,q)|$ $\displaystyle=2q^{m(m-1)}(q^{m}\mp
1)\prod_{i=1}^{m-1}(q^{2i}-1).$

###### Proof.

This follows from Theorems 3.12, 14.2 and 14.48 of [6]. ∎

Witt’s Extension Theorem [6, Theorem 5.2] implies the following.

######  Lemma A.3.

Let $\Theta$ be a quadratic form of type $\varepsilon$ on
$V=\mathbb{F}_{q}^{n}$. Let $a,b\in V$ be two non-zero vectors. There is an
element $A\in O^{\varepsilon}(n,q)$ if and only if $\Theta(a)=\Theta(b)$. ∎

Let $n=2m+1$ be an odd integer and $\Theta$ be a parabolic quadratic form. If
we linearize $\Theta$, then we obtain a degenerate alternating form with
$1$-dimensional radical $V_{0}$. The factor space $V/V_{0}$ has a non-
degenerate alternating form. This induces the group isomorphism

$O(2m+1,q)\cong Sp(2m,q)$

for finite fields of characteristic two, see [6, Theorem 14.2]. If $n=2m$ is
even, then the associated alternating form is non-singular. This embeds
$O^{\varepsilon}(2m,q)$ into $Sp(2m,q)$. In fact, in even dimension, the
orthogonal groups are maximal subgroups of the symplectic group. Their index
is

$|O(2m+1,q):O^{\pm}(2m,q)|=\frac{1}{2}q^{m}\frac{q^{2m}-1}{q^{m}\mp
1}=\frac{1}{2}q^{m}(q^{m}\pm 1).$

###  A.1. Symplectic transvections

We introduce now the subset of symplectic transvections, by defining

$\begin{array}[]{lccc}T_{a}:&V&\longrightarrow&V\\\ &u&\mapsto&u+\langle
u,a\rangle a,\\\ \end{array}$

see [6] for more information about transvections. The following lemma holds:

######  Lemma A.4.

  1. (i)

$T_{a}^{2}=1$, $\forall a\in V$;

  2. (ii)

$u$ is fixed by $T_{a}$ if and only if $u\in a^{\perp}$;

  3. (iii)

$A^{-1}T_{a}A=T_{aA}$, $A\in Sp(2m,q)$. ∎

######  Lemma A.5.

Let $a,b,c\in V$, $\gamma\in\mathbb{F}^{*}_{q}$. The following hold:

(5) $\displaystyle\vartheta_{a}^{T_{c}}(u)$
$\displaystyle=\vartheta_{a}(u)+\langle c,u\rangle^{2}(\vartheta_{a}(c)+1),$
(6) $\displaystyle\vartheta_{b}(u)$ $\displaystyle=\vartheta_{a}(u)+\langle
a+b,u\rangle^{2},$ (7) $\displaystyle\vartheta_{a}(a+b)$
$\displaystyle=\vartheta_{0}(a)+\vartheta_{0}(b)+\langle
a,b\rangle^{2}+\langle a,b\rangle,$ (8)
$\displaystyle\vartheta_{a}^{T_{\gamma(a+b)}}(u)$
$\displaystyle=\vartheta_{b}(u)+(1+\gamma^{2}+\gamma^{4}\vartheta_{a}(a+b))\langle
a+b,u\rangle^{2},$ (9) $\displaystyle
1+\gamma^{2}+\gamma^{4}\vartheta_{a}(a+b)$
$\displaystyle=\gamma^{4}(\frac{1}{\gamma^{4}}+\langle
a,b\rangle^{2}+\frac{1}{\gamma^{2}}+\langle
a,b\rangle+\vartheta_{0}(a)+\vartheta_{0}(b)).$

###### Proof.

For (5): $\displaystyle\vartheta_{a}^{T_{c}}(u)$
$\displaystyle=\vartheta_{a}(uT_{c})$ $\displaystyle=\vartheta_{a}(u+\langle
c,u\rangle c)$ $\displaystyle=\vartheta_{a}(u)+\langle
u,c\rangle^{2}\vartheta_{a}(c)+\langle u,\langle u,c\rangle c\rangle$
$\displaystyle=\vartheta_{a}(u)+\langle c,u\rangle^{2}(\vartheta_{a}(c)+1).$
For (6): $\displaystyle\vartheta_{b}(u)$
$\displaystyle=\vartheta_{0}(u)+\langle b,u\rangle^{2}$
$\displaystyle=\vartheta_{0}(u)+\langle a,u\rangle^{2}+\langle
a+b,u\rangle^{2}$ $\displaystyle=\vartheta_{a}(u)+\langle a+b,u\rangle^{2}.$
For (7): $\displaystyle\vartheta_{a}(a+b)$
$\displaystyle=\vartheta_{a}(a)+\vartheta_{a}(b)+\langle a,b\rangle$
$\displaystyle=\vartheta_{0}(a)+\vartheta_{0}(b)+\langle
a,b\rangle^{2}+\langle a,b\rangle.$

(8) follows from (5) and (6):

$\displaystyle\vartheta_{a}^{T_{\gamma(a+b)}}(u)$
$\displaystyle=\vartheta_{a}(u)+(1+\vartheta_{a}(\gamma(a+b))\langle
u,\gamma(a+b)\rangle^{2}$ $\displaystyle=\vartheta_{b}(u)+\langle
a+b,u\rangle^{2}+(1+\vartheta_{a}(\gamma(a+b))\gamma^{2}\langle
a+b,u\rangle^{2}$
$\displaystyle=\vartheta_{b}(u)+(1+\gamma^{2}+\gamma^{4}\vartheta_{a}(a+b))\langle
a+b,u\rangle^{2}.$

Finally, we obtain (9) by (6),

$\displaystyle 1+\gamma^{2}+\gamma^{4}\vartheta_{a}(a+b)$
$\displaystyle=1+\gamma^{2}+\gamma^{4}(\vartheta_{0}(a)+\vartheta_{0}(b)+\langle
a,b\rangle+\langle a,b\rangle^{2})$
$\displaystyle=\gamma^{4}(\frac{1}{\gamma^{4}}+\frac{1}{\gamma^{2}}+\langle
a,b\rangle^{2}+\langle a,b\rangle+\vartheta_{0}(a)+\vartheta_{0}(b)).\qed$

######  Lemma A.6.

Let $a,b\in V$, $a\neq b$. The following are equivalent:

  1. (i)

$\exists c\in V:\vartheta_{a}^{T_{c}}=\vartheta_{b}$;

  2. (ii)

$\exists\gamma\in\mathbb{F}^{*}_{q}:\vartheta_{a}^{T_{\gamma(a+b)}}=\vartheta_{b}$;

  3. (iii)

$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))=\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(b))$;

  4. (iv)

the polynomial $t^{2}+t+\vartheta_{0}(a)+\vartheta_{0}(b)$ is reducible over
$\mathbb{F}_{q}$.

###### Proof.

(ii) implies (i) trivially. It is well known that $t^{2}+t+\lambda$ is
reducible over $\mathbb{F}_{q}$ if and only if
$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\lambda)=0$, hence
(iii)$\Leftrightarrow$(iv). Assume (iv) and let $t\in\mathbb{F}_{q}$ such that
$t^{2}+t+\vartheta_{0}(a)+\vartheta_{0}(b)=0$. If
$\gamma=\frac{1}{\sqrt{\langle a,b\rangle+t}}$, then (8) implies (ii).
Conversely, if (ii) is true, then
$1+\gamma^{2}+\gamma^{4}\vartheta_{a}(a+b)=0$, and
$t=\frac{1}{\gamma^{2}}+\langle a,b\rangle$ is a root of
$t^{2}+t+\vartheta_{0}(a)+\vartheta_{0}(b)$ in $\mathbb{F}_{q}$. This proves
$(ii)\Leftrightarrow(iv)$. In order to show $(i)\Rightarrow(ii)$, let

$\displaystyle\vartheta_{a}^{T_{c}}(u)$
$\displaystyle=\vartheta_{a}(u)+\langle u,c\rangle^{2}(1+\vartheta_{a}(c))$
$\displaystyle=\vartheta_{0}(u)+\langle u,a\rangle^{2}+\langle
u,c\sqrt{\vartheta_{a}(c)+1}\rangle^{2}$
$\displaystyle=\vartheta_{a+c\sqrt{\vartheta_{a}(c)+1}}(u).$

By (i), $b=a+c\sqrt{\vartheta_{a}(c)+1}$, and since $a\neq b$, we have
$\vartheta_{a}(c)+1\neq 0$ and $c=\frac{a+b}{\sqrt{\vartheta_{a}(c)+1}}$. ∎

###  A.2. Proofs of the theorems on the structure of symplectic groups

###### Proof of Theorem 1.1.

(i) Let $\Omega^{+}$ and $\Omega^{-}$ denote the set of hyperbolic and
elliptic quadratic forms in $\Omega$. The elements in $\Omega^{\pm}$ are
$GL(2m,q)$-equivalent, hence by Proposition A.1, they are
$Sp(2m,q)$-equivalent. This means that $Sp(2m,q)$ has these two orbits in
$\Omega$. As
$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))$ can take
only two values $0$ and $1$, Lemma A.6 implies that the orbits are also
characterized by the equations

$\displaystyle\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))$
$\displaystyle=0$
$\displaystyle\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(a))$
$\displaystyle=1.$

Clearly, $\vartheta_{0}\in\Omega^{+}$ and
$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(0))=0$, which
proves the claim.

(ii) If $\beta\in\mathbb{F}_{q}\setminus\\{0\\}$, then the number of solutions
$a\in\mathbb{F}_{q}^{2m}$ of equation $\vartheta_{0}(a)=\beta$ is
$q^{2m-1}-q^{m-1}$. For $\beta=0$, the number of solutions is
$q^{2m-1}+q^{-1}(q-1)$. This allows us to compute the number of solutions of
$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(0))=0$ and
$\operatorname{Tr}_{\mathbb{F}_{q}/\mathbb{F}_{2}}(\vartheta_{0}(0))=1$.

(iii) For any $a,b\in\mathbb{F}_{q}^{2m}$, we have

$\vartheta_{a+b}=\vartheta_{0}(u)+\langle
a+b,u\rangle^{2}=\vartheta_{a}(u)+\langle b,u\rangle^{2}.$

Assume $\vartheta_{a}^{A}=\vartheta_{a}$. Then

$\displaystyle\vartheta_{a+b}^{A}(u)$ $\displaystyle=\vartheta_{a+b}(uA^{-1})$
$\displaystyle=\vartheta_{a}(uA^{-1})+\langle b,uA^{-1}\rangle^{2}$
$\displaystyle=\vartheta_{a}(u)+\langle bA,u\rangle^{2}$
$\displaystyle=\vartheta_{a+bA}(u).$

This means that $b\mapsto\vartheta_{a+b}$ induces an equivalence between the
linear action of the symplectic group on $\mathbb{F}_{q}^{2m}$, and the action
of $Sp(2m,q)$ on $\Omega$. ∎

###### Proof of Theorem 1.2.

The claim is equivalent to the fact that the stabilizers $R^{\pm}$ have three
orbits: $\\{\vartheta_{a}\\}$, $\Omega^{\pm}\setminus\\{\vartheta_{a}\\}$, and
$\Omega^{\mp}$. By Theorem 1.1(iii), the stabilizers are permutation
equivalent to the linear action of the orthogonal group $O^{\pm}(2m,2)$.
Indeed, for $q=2$, the orthogonal groups have three orbits $\\{0\\}$,
$\\{u\in\mathbb{F}_{2}^{2m}\setminus\\{0\\}\mid\vartheta_{a}(u)=0\\}$ and
$\\{u\in\mathbb{F}_{2}^{2m}\mid\vartheta_{a}(u)=1\\}$. ∎

###### Proof of Theorem 1.3.

The symplectic group $H_{1}=Sp(2m,q)$ itself is a complement to $N$ in
$ASp(2m,q)$, the orbits of $H_{1}$ have length $1$ and $2^{2m}-1$. To find the
second complement, we identify $\mathbb{F}_{q}^{2m}$ with $\Omega$ by
$a\mapsto\vartheta_{a}$. The action of $Sp(2m,q)$ induces an action on
$\mathbb{F}_{q}^{2m}$ which we denote by $a\mapsto a^{A}$ as well.
Straightforward computation shows

$\displaystyle\vartheta_{a+b+c}(u)$ $\displaystyle=\vartheta_{0}(u)+\langle
a+b+c,u\rangle^{2}$ $\displaystyle=\vartheta_{0}(u)+\langle
a,u\rangle^{2}+\vartheta_{0}(u)+\langle
b,u\rangle^{2}+\vartheta_{0}(u)+\langle c,u\rangle^{2},$
$\displaystyle=\vartheta_{a}(u)+\vartheta_{b}(u)+\vartheta_{c}(u),$

and

$\displaystyle\vartheta_{a}^{A}(u)$ $\displaystyle=\vartheta_{a}(uA^{-1})$
$\displaystyle=\vartheta_{0}(uA^{-1})+\langle a,uA^{-1}\rangle^{2}$
$\displaystyle=\vartheta_{0}^{A}(u)+\vartheta_{0}(u)+\vartheta_{0}(u)+\langle
aA,u\rangle^{2}$
$\displaystyle=\vartheta_{0}^{A}(u)+\vartheta_{0}(u)+\vartheta_{aA}(u).$

Fix $A\in Sp(2m,q)$ and define $b$ as $\vartheta_{0}^{A}=\vartheta_{b}$. Then,

$\vartheta_{a}^{A}=\vartheta_{aA+b}.$

In other words, $a\mapsto a^{A}=aA+b$ is an element of $ASp(2m,q)$. Obviously,
the maps $\\{a\mapsto a^{A}\mid A\in Sp(2m,q)\\}$ form a group $H_{2}\cong
Sp(2m,q)$. The only translation in $H_{2}$ is the trivial one; therefore,
$H_{2}$ is a complement of $N$. The orbits of $H_{2}$ have length
$\frac{1}{2}(q^{2m}+q^{m})$, $\frac{1}{2}(q^{2m}-q^{m})$. The identification
of $\mathbb{F}_{q}^{2m}$ and $\Omega$ is an equivalence between the actions of
$H_{2}$ on the orbits and the actions of $Sp(2m,q)$ on $\Omega^{+}$ and
$\Omega^{-}$. ∎