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# On $\alpha$-points of $q$-analogs of the Fano plane

Michael Kiermaier
_Universität Bayreuth_ Universität Bayreuth, Mathematisches Institut, D-95440
Bayreuth, Germany
email<EMAIL_ADDRESS>
homepage: http://www.mathe2.uni-bayreuth.de/michaelk/

###### Abstract

Arguably, the most important open problem in the theory of $q$-analogs of
designs is the question for the existence of a $q$-analog $D$ of the Fano
plane. It is undecided for every single prime power value $q\geq 2$.

A point $P$ is called an $\alpha$-point of $D$ if the derived design of $D$ in
$P$ is a geometric spread. In 1996, Simon Thomas has shown that there must
always exist at least one non-$\alpha$-point. For the binary case $q=2$, Olof
Heden and Papa Sissokho have improved this result in 2016 by showing that the
non-$\alpha$-points must form a blocking set with respect to the hyperplanes.

In this article, we show that a hyperplane consisting only of $\alpha$-points
implies the existence of a partiton of the symplectic generalized quadrangle
$W(q)$ into spreads. As a consequence, the statement of Heden and Sissokho is
generalized to all primes $q$ and all even values of $q$.

##  1 Introduction

Due to the connection to network coding, the theory of subspace designs has
gained a lot of interest recently. Subspace designs are the $q$-analogs of
combinatorial designs and arise by replacing the subset lattice of the finite
ambient set $V$ by the subspace lattice of a finite ambient vector space $V$.
Arguably the most important open problem in this field is the question for the
existence of a $q$-analog of the Fano plane, which is a subspace design with
the parameters $2$-$(7,3,1)_{q}$. This problem has already been stated in 1972
by Ray-Chaudhuri [Berge-RayChaudhuri-1974-LNMa411:278-287, Problem 28].
Despite considerable investigations, its existence remains undecided for every
single order $q$ of the base field.

It has been shown that the smallest instance $q=2$, the binary $q$-analog of a
Fano plane, can have at most a single nontrivial automorphism [Braun-
Kiermaier-Nakic-2016-EuJC51:443-457, Kiermaier-Kurz-
Wassermann-2018-DCC86[2]:239-250]. A $q$-analog of the Fano plane would be the
largest possible $[7,4;3]_{q}$ constant dimension subspace code. However, the
hitherto best known sizes of such constant dimension subspace codes still
leave considerable gaps, namely $333$ vs. $381$ in the binary case [Heinlein-
Kiermaier-Kurz-Wassermann-2019-AiMoC13[3]:457-475] and $6978$ vs. $7651$ in
the ternary case [Honold-Kiermaier-2016].111The $[7,4;3]_{3}$ code of size
$6977$ in [Honold-Kiermaier-2016] can be extended trivially by a further
codeword.

Another approach has been the investigation of the derived designs of a
putative $q$-analog $D$ of the Fano plane. A derived design exists for each
point $P\in\operatorname{PG}(6,q)$ and is always a $q$-design with the
parameters $1$-$(6,2,1)_{q}$, which is the same as a line spread of
$\operatorname{PG}(5,q)$. Following the notation of [Heden-
Sissokho-2016-ArsComb124:161-164], a point $P$ is called an _$\alpha$_ -point
of $D$ if the derived design in $P$ is the geometric spread, which is the most
symmetric and natural one among the $131044$ isomorphism types [Mateva-
Topalova-2009-JCD17[1]:90-102] of such spreads. For highest possible
regularity, one would expect all points to be $\alpha$-points. However, this
has been shown to be impossible, so there must always be at least one
non-$\alpha$-point of $D$ [Thomas-1996-GeomDed63[3]:247-253]. For the binary
case $q=2$, this result has been improved to the statement that each
hyperplane contains at least a non-$\alpha$-point Heden-
Sissokho-2016-ArsComb124:161-164, In other words, the non-$\alpha$-points of
of a binary $q$-analog of the Fano plane form a blocking set with respect to
the hyperplanes.

In this article, $\alpha$-points will be investigated for general values of
$q$, which will lead to the following theorem.

######  Theorem 1.

Let $D$ be a $q$-analog of the Fano plane and assume that there exists a
hyperplane $H$ such that all points of $H$ are $\alpha$-points of $D$. Then
the following equivalent statements hold:

  1. (a)

The line set of the symplectic generalized quadrangle $W(q)$ is partitionable
into spreads.

  2. (b)

The point set of the parabolic quadric $Q(4,q)$ is partitionable into ovoids.

As a consequence, we will get the following generalization of the result of
[Heden-Sissokho-2016-ArsComb124:161-164].

######  Theorem 2.

Let $D$ be a $q$-analog of the Fano plane and $q$ be prime or even. Then each
hyperplane contains a non-$\alpha$-point. In other words, the
non-$\alpha$-points form a blocking set with respect to the hyperplanes.

##  2 Preliminaries

Throughout the article, $q\neq 1$ is a prime power and $V$ is a vector space
over $\mathbb{F}_{q}$ of finite dimension $v$.

###  2.1 The subspace lattice

For simplicity, a subspace $U$ of $V$ of dimension
$\dim_{\mathbb{F}_{q}}(U)=k$ will be called a _$k$ -subspace_. The set of all
$k$-subspaces of $V$ is called the _Graßmannian_ and will be denoted by
$\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}$. Picking the “best of two worlds”, we will
prefer the algebraic dimension $\dim_{\mathbb{F}_{q}}(U)$ over the geometric
dimension $\dim_{\mathbb{F}_{q}}(U)-1$, but we will otherwise make heavy use
of geometric notions, such as calling the $1$-subspaces of $V$ _points_ , the
$2$-subspaces _lines_ , the $3$-subspaces _planes_ , the $4$-subspaces
_solids_ and the $(v-1)$-subspaces _hyperplanes_. In fact, the _subspace
lattice_ $\mathcal{L}(V)$ consisting of all subspaces of $V$ ordered by
inclusion is nothing else than the finite projective geometry
$\operatorname{PG}(v-1,q)=\operatorname{PG}(V)$.222In established symbols like
$\operatorname{PG}(v-1,q)$, the geometric dimension $v-1$ is not altered.
There are good reasons to consider the subset lattice as a subspace lattice
over the unary “field” $\mathbb{F}_{1}$ [Cohn-2004-AMM111[6]:487-495].

The number of all $k$-subspaces of $V$ is given by the Gaussian binomial
coefficient

$\\#\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}=\genfrac{[}{]}{0.0pt}{}{v}{k}_{q}=\begin{cases}\frac{(q^{v}-1)\cdots(q^{v-k+1}-1)}{(q^{k}-1)\cdots(q-1)}&\text{if
}k\in\\{0,\ldots,v\\}\text{;}\\\ 0&\text{otherwise.}\end{cases}$

The Gaussian binomial coefficient $\genfrac{[}{]}{0.0pt}{}{v}{1}_{q}$ is also
known as the $q$-analog of the number $v$ and will be abbreviated as
$[v]_{q}$.

For $S\subseteq\mathcal{L}(V)$ and $U,W\in\mathcal{L}(V)$, we will use the
abbreviations

$\displaystyle S|_{U}$ $\displaystyle=\\{B\in S\mid U\leq B\\}\text{,}$
$\displaystyle S|^{W}$ $\displaystyle=\\{B\in S\mid B\leq W\\}\quad\text{and}$
$\displaystyle S|_{U}^{W}$ $\displaystyle=\\{B\in S\mid U\leq B\leq
W\\}\text{.}$

For a point $P$ in a plane $E$, the set of all lines in $E$ passing through
$P$ is known as a _line pencil_.

The subspace lattice $\mathcal{L}(V)$ is isomorphic to its dual, which arises
from $\mathcal{L}(V)$ by reversing the order. Fixing a non-degenerate bilinear
form $\beta$ on $V$, a concrete isomorphism is given by $U\mapsto U^{\perp}$,
where $U^{\perp}=\\{\mathbf{x}\in V\mid\beta(\mathbf{x},\mathbf{u})=0\text{
for all }\mathbf{u}\in U\\}$. When addressing the dual of some geometric
object in $\operatorname{PG}(V)$, we mean its (element-wise) image under this
map. Up to isomorphism, the image does not depend on the choice of $\beta$.

###  2.2 Subspace designs

######  Definition 2.1.

Let $t,v,k$ be integers with $0\leq t\leq k\leq v-t$ and $\lambda$ another
positive integer. A set $D\subseteq\genfrac{[}{]}{0.0pt}{}{V}{k}_{q}$ is
called a $t$-$(v,k,\lambda)_{q}$ _subspace design_ if each $t$-subspace of $V$
is contained in exactly $\lambda$ elements (called _blocks_) of $D$. In the
important case $\lambda=1$, $D$ is called a _$q$ -Steiner system_.

The earliest reference for subspace designs is [Cameron-1974]. However, the
idea is older, since it is stated that “Several people have observed that the
concept of a $t$-design can be generalised […]”. They have also been mentioned
in a more general context in [Delsarte-1976-JCTSA20[2]:230-243]. The first
nontrivial subspace designs with $t\geq 2$ has been constructed in
[Thomas-1987-GeomDed24[2]:237-242], and the first nontrivial Steiner system
with $t\geq 2$ in [Braun-Etzion-Ostergard-Vardy-
Wassermann-2016-ForumMathPi:e7]. An introduction to the theory of subspace
designs can be found at [Braun-Kiermaier-Wassermann-2018-COST1], see also
[Suzuki-1989-Designs, Day 4].

Subspace designs are interlinked to the theory of network coding in various
ways. To this effect we mention the recently found $q$-analog of the theorem
of Assmus and Mattson [Byrne-Ravagnani-SIAMJDM33[3]:1242-1260], and that a
$t$-$(v,k,1)_{q}$ Steiner systems provides a $(v,2(k-t+1);k)_{q}$ constant
dimension network code of maximum possible size.

Classical combinatorial designs can be seen as the limit case $q=1$ of
subspace designs. Indeed, quite a few statements about combinatorial designs
have a generalization to subspace designs, such that the case $q=1$ reproduces
the original statement [Kiermaier-Laue-2015-AiMoC9[1]:105-115, Kiermaier-
Pavcevic-2015-JCD23[11]:463-480, Nakic-Pavcevic-2015-DCC77[1]:49-60, Braun-
Kiermaier-Kohnert-Laue-2017-JCTSA147:155-185].

One example of such a statement is the following
[Suzuki-1990-EuJC11[6]:601-607, Lemma 4.1(1)], see also [Kiermaier-
Laue-2015-AiMoC9[1]:105-115, Lemma 3.6]: If $D$ is a $t$-$(v,k,\lambda)_{q}$
subspace design, then $D$ is also an $s$-$(v,k,\lambda_{s})_{q}$ subspace
design for all $s\in\\{0,\ldots,t\\}$, where

$\lambda_{s}\coloneqq\lambda\frac{\genfrac{[}{]}{0.0pt}{}{v-s}{t-s}_{q}}{\genfrac{[}{]}{0.0pt}{}{k-s}{t-s}_{q}}.$

In particular, the number of blocks in $D$ equals

$\\#D=\lambda_{0}=\lambda\frac{\genfrac{[}{]}{0.0pt}{}{v}{t}_{q}}{\genfrac{[}{]}{0.0pt}{}{k}{t}_{q}}.$

So, for a design with parameters $t$-$(v,k,\lambda)_{q}$, the numbers
$\lambda_{s}$ necessarily are integers for all $s\in\\{0,\ldots,t\\}$
(_integrality conditions_). In this case, the parameter set
$t$-$(v,k,\lambda)_{q}$ is called _admissible_. It is further called
_realizable_ if a $t$-$(v,k,\lambda)_{q}$ design actually exists. The smallest
admissible parameters of a $q$-analog of a Steiner system are
$2$-$(7,3,1)_{q}$, which are the parameters of the $q$-analog of the Fano
plane. This explains the significance of the question of its realizability.

The numbers $\lambda_{i}$ can be refined as follows. Let $i,j$ be non-negative
integers with $i+j\leq t$ and let $I\in\genfrac{[}{]}{0.0pt}{}{V}{i}_{q}$ and
$J\in\genfrac{[}{]}{0.0pt}{}{V}{v-j}_{q}$. By [Suzuki-1990-EuJC11[6]:601-607,
Lemma 4.1], see also [Braun-Kiermaier-Wassermann-2018-COST1, Lemma 5], the
number

$\lambda_{i,j}\coloneqq\\#D|_{I}^{J}=\lambda\frac{\genfrac{[}{]}{0.0pt}{}{v-i-j}{k-i}_{q}}{\genfrac{[}{]}{0.0pt}{}{v-t}{k-t}_{q}}$

only depends on $i$ and $j$, but not on the choice of $I$ and $J$. The numbers
$\lambda_{i,j}$ are important parameters of a subspace design. A further
generalization is given by the intersection numbers in [Kiermaier-
Pavcevic-2015-JCD23[11]:463-480].

A nice way to arrange the numbers $\lambda_{i,j}$ is the following triangle
form.

$\begin{array}[]{cccccccccc}&&&&\lambda_{0,0}\\\
&&&\lambda_{1,0}&&\lambda_{0,1}\\\
&&\lambda_{2,0}&&\lambda_{1,1}&&\lambda_{0,2}\\\
&\iddots&&\iddots&&\ddots&&\ddots\\\
\lambda_{t,0}&&\lambda_{t-1,1}&&\ldots&&\lambda_{1,t-1}&&\lambda_{0,t}\end{array}$

For a $q$-analog of the Fano plane, we get:

$\displaystyle\begin{array}[]{ccccc}&\lx@intercol\hfil\lambda_{0,0}=q^{8}+q^{6}+q^{5}+q^{4}+q^{3}+x^{2}+1\hfil\lx@intercol\\\
&\lambda_{1,0}=q^{4}+q^{2}+1&&\lambda_{0,1}=q^{5}+q^{3}+q^{2}+1\\\
\lambda_{2,0}=1&&\lambda_{1,1}=q^{2}+1&&\lambda_{0,2}=q^{2}+1\\\ \end{array}$

As a consequence of the numbers $\lambda_{i,j}$, the _dual_ design
$D^{\perp}=\\{B^{\perp}\mid B\in D\\}$ is a subspace design with the
parameters

$t\text{-}\Big{(}v,v-k,\frac{\genfrac{[}{]}{0.0pt}{}{v-t}{k}_{q}}{\genfrac{[}{]}{0.0pt}{}{v-t}{k-t}_{q}}\Big{)}\text{.}$

For a point $P\leq V$, the _derived_ design of $D$ in $P$ is

$\operatorname{Der}_{P}(D)=\\{B/P\mid B\in D|_{P}\\}\text{.}$

By [Kiermaier-Laue-2015-AiMoC9[1]:105-115], $\operatorname{Der}_{P}(D)$ is a
subspace design with the parameters $(t-1)$-$(v-1,k-1,\lambda)_{q}$. In the
case of a $q$-analog of the Fano plane, $\operatorname{Der}_{P}(D)$ has the
parameters $1$-$(6,2,1)$.

###  2.3 Spreads

A $1$-$(v,k,1)_{q}$ Steiner system $\mathcal{S}$ is just a partition of the
point set of $V$ into $k$-subspaces. These objects are better known under the
name $(k-1)$-_spread_ and have been investigated in geometry well before the
emergence of subspace designs. A $1$-spread is also called _line spread_.

A set $\mathcal{S}$ of $k$-subspaces is called a _partial $(k-1)$-spread_ if
each point is covered by at most one element of $\mathcal{S}$. The points not
covered by any element are called _holes_. A recent survey on partial spreads
is found in [Honold-Kiermaier-Kurz-2018-COST].

The parameters $1$-$(v,k,1)_{q}$ are admissible if and only $v$ is divisible
by $k$. In this case, spreads do always exist [Segre-1964-AnnDMPA64[1]:1-76,
§VI]. An example can be constructed via field reduction: We consider $V$ as a
vector space over $\mathbb{F}_{q^{k}}$ and set
$\mathcal{S}=\genfrac{[}{]}{0.0pt}{}{V}{1}_{q^{k}}$. Switching back to vector
spaces over $\mathbb{F}_{q}$, the set $\mathcal{S}$ is a $(k-1)$-spread of
$V$, known as the _Desarguesian_ spread.

A $(k-1)$-spread $\mathcal{S}$ is called _geometric_ or _normal_ if for two
distinct blocks $B,B^{\prime}\in\mathcal{S}$, the set
$\mathcal{S}|^{B+B^{\prime}}$ is always a $(k-1)$-spread of $B+B^{\prime}$. In
other words, $\mathcal{S}$ is geometric if every $2k$-subspace of $V$ contains
either $0$, $1$ or $\frac{[2k]_{q}}{[k]_{q}}=q^{k}+1$ blocks of $\mathcal{S}$.
It is not hard to see that the Desarguesian spread is geometric. In fact, it
follows from [Barlotti-Cofman-1974-AmSUHamb40:231-241, Theorem 2] that a
$(k-1)$-spread is geometric if and only if it is isomorphic to a Desarguesian
spreads.

The derived designs of a $q$-analog of the Fano plane $D$ are line spreads in
$\operatorname{PG}(5,2)$. These spreads have been classified in [Mateva-
Topalova-2009-JCD17[1]:90-102] into $131044$ isomorphism types. The most
symmetric one among these spreads is the Desarguesian spread. Following the
notation of [Heden-Sissokho-2016-ArsComb124:161-164], a point $P$ is called an
_$\alpha$_ -point of the $q$-analog of the Fano plane $D$ if the derived
design in $P$ is the geometric spread.

###  2.4 Generalized quadrangles

######  Definition 2.2.

A _generalized quadrangle_ is an incidence structure
$Q=(\mathcal{P},\mathcal{L},I)$ with a non-empty set of _points_
$\mathcal{P}$, a non-empty set of _lines_ $\mathcal{L}$, and an incidence
relation $I\subseteq\mathcal{P}\times\mathcal{L}$ such hat

  1. (i)

Two distinct points are incident with at most a line.

  2. (ii)

Two distinct lines are incident with at most one point.

  3. (iii)

For each non-incident point-line-pair $(P,L)$ there is a unique incident
point-line-pair $(P^{\prime},L^{\prime})$ with $P\mathrel{I}L^{\prime}$ and
$P^{\prime}\mathrel{I}L$.

Generalized quadrangles have been introduced in the more general setting of
generalized polygons in [Tits-1959-InstHautesEtudesSciPublMath2:13-60], as a
tool in the theory of finite groups.

A generalized quadrangle $Q=(\mathcal{P},\mathcal{L},I)$ is called
_degenerate_ if there is a point $P$ such that each point of $Q$ is incident
with a line through $P$. If each line of $Q$ is incident with $t+1$ points,
and each point is incident with $s+1$ lines, we say that $Q$ is of _order_
$(s,t)$. The _dual_ $Q^{\perp}$ arises from $Q$ by interchanging the role of
the points and the lines. It is again a generalized quadrangle. Clearly,
$(Q^{\perp})^{\perp}=Q$, and $Q$ is of order $(s,t)$ if and only if
$Q^{\perp}$ is of order $(t,s)$.

Furthermore, $Q$ is said to be _projective_ if it is embeddable in some
Desarguesian projective geometry. This means that there is a Desarguesian
projective geometry with point set $\mathcal{P}^{\prime}$, line set
$\mathcal{L}^{\prime}$, and point-line incidence relation $I^{\prime}$ such
that $\mathcal{P}\subseteq\mathcal{P}^{\prime}$,
$\mathcal{L}\subseteq\mathcal{L^{\prime}}$ and for all
$(P,L)\in\mathcal{P}\times\mathcal{L}$ we have $P\mathrel{I}L$ if and only if
$P^{\prime}\mathrel{I^{\prime}}L^{\prime}$. The non-degenerate finite
projective generalized quadrangles have been classified in [Buekenhout-
Lefevre-1974-ArchMath25[1]:540-552, Th. 1], see also [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 4.4.8.]. These are exactly the so-
called _classical generalized quadrangles_ which are associated to a quadratic
form or a symplectic or Hermitean polarity on the ambient geometry, see
[Payne-Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.1.1.].

In this article, two of these classical generalized quadrangles will appear.

  1. (i)

The symplectic generalized quadrangle $W(q)$ [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.1.1 (iii)], consisting of the set
of points of $\operatorname{PG}(3,q)$ together with the totally isotropic
lines with respect to a symplectic polarity. Taking the geometry as
$\operatorname{PG}(\mathbb{F}_{q}^{4})$, the symplectic polarity can be
represented by the alternating bilinear form
$\beta(\mathbf{x},\mathbf{y})=x_{1}y_{2}-x_{2}y_{1}+x_{3}y_{4}-x_{4}y_{3}$.
The configuration of the lines $\mathcal{L}$ in $\operatorname{PG}(3,q)$ is
also known as a (general) linear complex of lines, see [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.1.1 (iii)] or
[Hirschfeld-1985-FiniteProjectiveSpacesOfThreeDimensions, Th. 15.2.13]. Under
the Klein correspondence, $\mathcal{L}$ is a non-tangent hyperplane section of
the Klein quadric.

  2. (ii)

The second one is the parabolic quadric $Q(4,q)$, whose points $\mathcal{P}$
are the zeroes of a parabolic quadratic form in $\operatorname{PG}(4,q)$, and
whose lines are all the lines contained in $\mathcal{P}$. Taking the geometry
as $\operatorname{PG}(\mathbb{F}_{q}^{5})$, the parabolic quadratic form can
be represented by $q(\mathbf{x},\mathbf{y})=x_{1}x_{2}+x_{3}x_{4}+x_{5}^{2}$.

Both $W(q)$ and $Q(4,q)$ are of order $(q,q)$. By [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.2.1] they are duals of each
other, meaning that $W(q)^{\top}\cong Q(4,q)$.

Let $Q=(\mathcal{P},\mathcal{L},I)$ be a generalized quadrangle. As in
projective geometries, a set $\mathcal{S}\subseteq\mathcal{L}$ is called a
_spread_ of $Q$ if each point of $Q$ is incident with a unique line in
$\mathcal{S}$. Dually, a set $\mathcal{O}\subseteq\mathcal{P}$ is called an
_ovoid_ of $Q$ if each line of $Q$ is incident with a unique point in
$\mathcal{O}$. Clearly, the spreads of $Q$ bijectively correspond to the
ovoids of $Q^{\perp}$. This already shows the equivalence of Part a and b in
Theorem 1.

##  3 Proof of the theorems

For the remainder of the article, we fix $v=7$ and assume that
$D\subseteq\genfrac{[}{]}{0.0pt}{}{V}{7}_{q}$ is a $q$-analog of the Fano
plane. The numbers $\lambda_{i,j}$ are defined as in Section 2.2.

By the design property, the intersection dimension of two distinct blocks
$B,B^{\prime}\in D$ is either $0$ or $1$. So by the dimension formula,
$\dim(B+B^{\prime})\in\\{5,6\\}$. Therefore two distinct blocks contained in a
common $5$-space always intersect in a point. Moreover, a solid $S$ of $V$
contains either a single block of no block at all. We will call $S$ a _rich
solid_ in the former case and a _poor solid_ in the latter.

######  Remark 3.1.

By [Kiermaier-Pavcevic-2015-JCD23[11]:463-480, Remark 4.2], the poor solids
form a dual $2$-$(7,3,q^{4})_{q}$ subspace design. By the above discussion,
the $\lambda_{0,2}=q^{2}+1$ blocks in any $5$-subspace $F$ form dual partial
spread in $F$. The poor solids contained in $F$ are exactly the holes of that
partial spread.

We will call a $5$-subspace $F$ such that all the $\lambda_{0,2}=q^{2}+1$
blocks in $F$ pass through a common point $P$ a _$\beta$ -flat_ with focal
point $P$.

######  Lemma 3.2.

The focal point of a $\beta$-flat is uniquely determined.

###### Proof.

Assume that $P\neq Q$ are focal points of a $\beta$-flat $F$. Then all
$\lambda_{0,2}=q^{2}+1>1$ blocks in $F$ pass through the line $P+Q$,
contradicting the Steiner system property.

######  Lemma 3.3.

Let $H$ be a hyperplane and $P$ a point in $H$. Then $P$ is the focal point of
at most one $\beta$-flat in $H$.

###### Proof.

There are $\lambda_{1,1}=q^{2}+1$ blocks in $H$ passing through $P$. For any
$\beta$-flat $F<H$ with focal point $P$, all these blocks are contained in
$F$.

Now assume that there are two such $\beta$-flats $F\neq F^{\prime}$. Then the
$q^{2}+1>1$ blocks in $D|_{P}^{H}$ are contained in $F\cap F^{\prime}$. This
is a contradiction, since $\dim(F\cap F^{\prime})\leq 4$ and any solid
contains at most a single block.

######  Lemma 3.4.

Let $F\in\genfrac{[}{]}{0.0pt}{}{V}{5}_{q}$ be a $\beta$-flat with focal point
$P$.

  1. (a)

Each point $Q\neq P$ is covered by a unique block in $F$.
(In other words: $D|^{F}/P$ is a line spread of
$F/P\cong\operatorname{PG}(3,q)$)

  2. (b)

A solid $S$ of $F$ is poor if and only if it does not contain $P$.

  3. (c)

For all poor solids $S$ of $F$, the set $\\{B\cap S\mid B\in D|^{F}\\}$ is a
line spread of $S$.

###### Proof.

Part a: As the blocks in $D|^{F}$ intersect each other only in the point $P$,
the number of points in $\genfrac{[}{]}{0.0pt}{}{F}{1}_{q}\setminus\\{P\\}$
covered by these blocks is
$(q^{2}+1)(\genfrac{[}{]}{0.0pt}{}{3}{1}_{q}-1)=q^{4}+q^{3}+q^{2}+q=\genfrac{[}{]}{0.0pt}{}{5}{1}_{q}-1$.
Therefore, all points in $F$ different from $P$ are covered by a single block
in $D_{F}$.

Part b: The number of solids in $F$ containing one of the $q^{2}+1$ blocks in
$F$ is
$(q^{2}+1)\cdot\genfrac{[}{]}{0.0pt}{}{5-3}{4-3}_{q}=(q^{2}+1)(q+1)=q^{3}+q^{2}+q+1$.333Remember
that a solid cannot contain $2$ blocks. These solids are rich. Moreover, the
$q^{4}$ solids in $F$ not containing $P$ do not contain a block, so they are
poor. As $q^{4}+(q^{3}+q^{2}+q+1)=\genfrac{[}{]}{0.0pt}{}{5}{4}_{q}$ is
already the total number of solids in $F$, the poor solids in $F$ are
precisely those not containing $P$.

Part c: Let $S$ be a poor solid of $F$. For every block $B$ in $F$ we have
$\dim(B\cap S)\leq 2$ as $S$ is poor, and moreover $\dim(B\cap
S)\geq\dim(B)+\dim(S)-\dim(F)=3+4-5=2$ by the dimension formula. So for all
blocks $B$ in $F$ we get that $B+S=F$ and $B\cap S$ is a line. By parts a and
b, every point of the poor solid $S$ is contained in a unique block in $F$.
Hence $\\{B\cap S\mid B\in D\text{ and }B+S=F\\}$ is a line spread of $S$.

######  Lemma 3.5.

Let $B,B^{\prime}$ be two distinct blocks which intersect in an $\alpha$-point
$P$. Then $P$ is the focal point of the $\beta$-flat $B+B^{\prime}$.

###### Proof.

Since $P=B\cap B^{\prime}$ is a point, $F=B+B^{\prime}$ is a $5$-subspace.
Since $P$ is an $\alpha$-point, we have that $\\{B^{\prime\prime}/P\mid
B^{\prime\prime}\in D|^{F}_{P}\\}$ is a line spread of
$F/P\cong\mathbb{F}_{q}^{4}$. Such a line spread contains
$\frac{[4]_{q}}{[2]_{q}}=q^{2}+1$ lines, so $F$ contains $q^{2}+1$ blocks
passing through $P$. However, the total number of blocks contained in $F$ is
only $\lambda_{0,2}=q^{2}+1$, so $F$ is a $\beta$-flat with focal point $P$.

######  Lemma 3.6.

Let $F$ be a $5$-subspace such that all points of $F$ are $\alpha$-points.
Then $F$ is a $\beta$-flat.

###### Proof.

The $5$-subspace $F$ contains $\lambda_{0,2}=q^{2}+1>1$ blocks. Let
$B,B^{\prime}$ be two distinct blocks in $F$. Then $P=B\cap B^{\prime}$ is a
point and $F=B+B^{\prime}$. By the precondition, $P$ is an $\alpha$-point, so
by Lemma 3.5, $P$ is the focal point of the $\beta$-flat $F$.

######  Remark 3.7.

The statement of Lemma 3.6 is still true if $F$ contains a single
non-$\alpha$-point $Q$. Then either all blocks contained in $F$ pass through
$Q$, or there are two distinct blocks $B$, $B^{\prime}$ in $F$ such that
$P=B\cap B^{\prime}\neq Q$. In the latter case, all blocks pass through the
$\alpha$-point $P$ as in the proof of Lemma 3.6.

######  Lemma 3.8.

Let $H$ be a hyperplane and $P$ an $\alpha$-point contained in $H$. Then $H$
contains a unique $\beta$-flat whose focal point is $P$.

###### Proof.

There are $\lambda_{1,1}=q^{2}+1>1$ blocks in $H$ containing $P$. Let
$B,B^{\prime}\in D|^{H}_{P}$. Then $P=B\cap B^{\prime}$. By Lemma 3.5, the
$\alpha$-point $P$ is the focal point of the $\beta$-flat $F=B+B^{\prime}$. By
Lemma 3.3, the $\beta$-flat $F$ is unique.

Now we fix a hyperplane $H$ of $V$ and assume that all its points are
$\alpha$-points.

By Lemma 3.6, every $5$-subspace $F$ is a $\beta$-flat. We denote its unique
focal point by $\alpha(F)$. Moreover by Lemma 3.8, each point $P$ of $H$ is
the focal point of a unique $\beta$-flat $F$ in $H$. We will denote this
$\beta$-flat by $\beta(P)$. Clearly, the mappings

$\alpha:\genfrac{[}{]}{0.0pt}{}{H}{5}_{q}\to\genfrac{[}{]}{0.0pt}{}{H}{1}_{q}\quad\text{and}\quad\beta:\genfrac{[}{]}{0.0pt}{}{H}{1}_{q}\to\genfrac{[}{]}{0.0pt}{}{H}{5}_{q}$

are inverse to each other. So they provide a bijective correspondence between
the points and the $5$-subspaces of $V$.

######  Lemma 3.9.

Let $B$ be a block in $H$.

  1. (a)

For all points $P$ of $B$, $B\leq\beta(P)$.

  2. (b)

For all $5$-subspaces $F$ in $H$ containing $B$, $\alpha(F)\leq B$.

###### Proof.

For part a, let $P$ be a point on $B$. There are $\lambda_{1,1}=q^{2}+1$
blocks in $H$ passing through $P$, which equals the number $\lambda_{0,2}$ of
blocks in $\beta(P)$ (which all pass through $P$). Therefore, $B\leq\beta(P)$.

For part b, let $F$ be a $5$-subspace containing $B$. All blocks in $F$ pass
through its focal point $\alpha(F)$.

By Lemma b, each $5$-subspace and a fortiori the hyperplane $H$ contains a
poor solid $S$. This poor solid $S$ is fixed for the remainder of this
article.

The set of $\genfrac{[}{]}{0.0pt}{}{6-4}{5-4}_{q}=q+1$ intermediate
$5$-subspaces $F$ with $S<F<H$ will be denoted by $\mathcal{F}$. For each
$F\in\mathcal{F}$, the set $\mathcal{L}_{F}:=\\{B\cap S\mid B\in D\text{ and
}B\in D|^{F}\\}$ is a line spread of $S$ by Lemma 3.4c.

######  Lemma 3.10.

The line spreads $\mathcal{L}_{F}$ with $F\in\mathcal{F}$ are pairwise
disjoint.

###### Proof.

Let $L\in\mathcal{L}_{F}\cap\mathcal{L}_{F^{\prime}}$ with
$F,F^{\prime}\in\mathcal{F}$. Then $L=B\cap S=B^{\prime}\cap S$ with
$B\in\mathcal{L}_{F}$ and $B^{\prime}\in\mathcal{L}_{F^{\prime}}$. So $B$ and
$B^{\prime}$ are two blocks passing through the same line $L$. The Steiner
system property gives $B=B^{\prime}$. Hence $F=B+S=B^{\prime}+S=F^{\prime}$.

Now let $\mathcal{L}=\bigcup_{F\in\mathcal{F}}\mathcal{L}_{F}$.

######  Lemma 3.11.

The set $\mathcal{L}$ consists of $q^{3}+q^{2}+q+1$ lines of $S$ and is
partitionable into $q+1$ line spreads of $S$.

###### Proof.

By Lemma 3.10, the sets $\mathcal{L}_{F}$ are pairwise disjoint, so
$\mathcal{L}$ is a set of
$\\#\mathcal{F}\cdot\\#D|^{F}=(q+1)(q^{2}+1)=q^{3}+q^{2}+q+1$ lines in $S$
which admits a partition into the $q+1$ line spreads $\mathcal{L}_{F}$ with
$F\in\mathcal{F}$.

######  Lemma 3.12.

For each point $P$ of $S$, $\mathcal{L}|_{P}$ is a line pencil in the plane
$E_{P}=\beta(P)\cap S$.

###### Proof.

Let $P$ be a point in $S$.

By Lemma 3.4b, the poor solid $S$ is not contained in the $5$-subspace
$\beta(P)$. Therefore, $\dim(\beta(P)\cap S)\leq 3$. On the other hand, as
both $S$ and $\beta(P)$ are contained in $H$, we have
$\dim(\beta(P)+S)\leq\dim(H)=6$ and therefore by the dimension formula
$\dim(\beta(P)\cap S)=\dim(\beta(P))+\dim(S)-\dim(\beta(P)+S)\geq 3$. Hence
$E_{P}=\beta(P)\cap S$ is a plane.

Let $L\in\mathcal{L}|_{P}$. Then there is a block $B\in D|^{H}$ with $B\cap
S=L$. By Lemma 3.9a, $B\leq\beta(P)$. So $L=B\cap S\leq\beta(P)\cap S=E_{P}$.
As the disjoint union of $q+1$ line spreads of $S$, $\mathcal{L}$ contains
$q+1$ lines passing through $P$. Therefore, these lines form a line pencil in
$E_{P}$ through $P$.

######  Lemma 3.13.

The incidence structure
$(\genfrac{[}{]}{0.0pt}{}{S}{1}_{q},\mathcal{L},\subseteq)$ is a projective
generalized quadrangle of order $(s,t)=(q,q)$.

###### Proof.

Clearly, every line in $\mathcal{L}$ contains $q+1$ points in $S$. By Lemma
3.11, through every point in $S$ there pass $q+1$ lines in $\mathcal{L}$. Now
let $P$ be a point in $S$ and $L\in\mathcal{L}$ not containing $P$.

By Lemma 3.10, there is a unique $F\in\mathcal{F}$ with $L\in\mathcal{L}_{F}$,
and there is a line $L^{\prime\prime}\in\mathcal{L}_{F}$ passing through $P$.
By Lemma 3.12, $L^{\prime\prime}<E_{P}$, so we get $L\not<E_{P}$ as otherwise
$L$ and $L^{\prime\prime}$ would be distinct intersecting lines in the spread
$\mathcal{L}_{F}$. Moreover, $L$ and $E_{P}$ are both contained in $S$, so
they cannot have trivial intersection. Therefore $L\cap E_{P}$ is a point.

Now let $P^{\prime}\in\genfrac{[}{]}{0.0pt}{}{S}{1}_{q}$ and
$L^{\prime}\in\mathcal{L}$ with $L\cap L^{\prime}=P^{\prime}$ and
$P+P^{\prime}=L^{\prime}$. Then $L^{\prime}$ is a line through $P$, so
$L^{\prime}<E_{P}$. So necessarily $P^{\prime}=E_{P}\cap L$ and
$L^{\prime}=P+P^{\prime}$, showing that $P^{\prime}$ and $L^{\prime}$ are
unique.

By Lemma 3.12 indeed $L^{\prime}\in\mathcal{L}$, as $P+P^{\prime}$ is a line
in $E_{P}$ containing $P$. This shows that $P^{\prime}$ and $L^{\prime}$ do
always exist and therefore, the incidence structure
$(\genfrac{[}{]}{0.0pt}{}{S}{1}_{,}\mathcal{L})$ is a generalized quadrangle
of order $(q,q)$.

######  Lemma 3.14.

$(\genfrac{[}{]}{0.0pt}{}{S}{1}_{q},\mathcal{L})$ is isomorphic to $W(q)$.

###### Proof.

By Lemma 3.13 we know that
$Q=(\genfrac{[}{]}{0.0pt}{}{S}{1}_{q},\mathcal{L},\subseteq)$ is a finite
generalized quadrangle of order $(s,t)=(q,q)$ embedded in
$\operatorname{PG}(S)$. By the classification in [Buekenhout-
Lefevre-1974-ArchMath25[1]:540-552, Theorem 1] (see also [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 4.4.8]), we know that $Q$ is a
finite classical generalized quadrangle which are listed in [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.1.2]. Comparing the orders and
the dimension of the ambient geometry, the only possibility for $Q$ is the
symplectic generalized quadrangle $W(q)$.

Now we can proof our main result.

###### Proof of Theorem 1.

Part a is a direct consequence of Lemma 3.14 and Lemma 3.11. The equivalence
of Part a and b has already been discussed in Section 2.4.

Theorem 2 is now a direct consequence.

###### Proof of Theorem 2.

We show that the statement in Theorem 1b is not satisfied.

For $q$ prime, the only ovoids of $Q(4,q)$ are the elliptic quadrics
$Q^{-}(3,q)$ [Ball-Govaerts-Storme-2006-DCC38[1]:131-145, Cor. 1]. As any two
such quadrics have nontrivial intersection, there is no partition of $Q(4,q)$
into ovoids.

For $q$ even, $Q(4,q)$ does not admit a partition into ovoids by [Payne-
Thas-2009-FiniteGeneralizedQuadrangles2nd, 3.4.1 (i)].

## Acknowledgement

I would like to thank Jan de Beule for his patient help with the theory of
generalized quadrangles during a research stay at the Vrije Universiteit
Brussel (VUB) in the framework of WOG Coding Theory and Cryptography.