## 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 regarding 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 [3, Problem 28]. Despite considerable investigations, its existence remains undecided for every single order q of the base field.

A q-analog of the Fano plane would be a $$[7,4;3]_q$$ constant dimension subspace code of size $$q^8 + q^6 + q^5 + q^4 + q^3 + q^2 + 1$$. However, the hitherto best known sizes of such constant dimension subspace codes still leave considerable gaps, namely 333 vs. 381 in the binary case [14] and 6978 vs. 7651 in the ternary case [16].Footnote 1 Furthermore, 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 [5, 20].

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 $$P\in {{\,\mathrm{PG}\,}}(6,q)$$ and is always a q-design with the parameters 1-$$(6,2,1)_q$$, which is the same as a line spread of $${{\,\mathrm{PG}\,}}(5,q)$$. Following the notation of [13], 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 line spreads of $${{\,\mathrm{PG}\,}}(5,q)$$. For highest possible regularity, one would expect all points to be $$\alpha$$-points.

However, this has been shown to be impossible, as there must always be at least one non-$$\alpha$$-point of D [28]. For the binary case $$q=2$$, this result has been improved to the statement that each hyperplane contains at least one non-$$\alpha$$-point [13]. In other words, the non-$$\alpha$$-points 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 leads 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 get the following generalization of the result of [13].

### 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, $$q \ne 1$$ 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 $${{\,\mathrm{PG}\,}}(v-1,q) = {{\,\mathrm{PG}\,}}(V)$$.Footnote 2 There are good reasons to consider the subset lattice as a subspace lattice over the unary “field” $${\mathbb {F}}_1$$ [11].

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

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

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

\begin{aligned} S|_U&= \{B\in S \mid U \le B\}\text {,} \\ S|^W&= \{B\in S\mid B\le W\}\quad \text {and} \\ S|_U^W&= \{B\in S\mid U \le B\le W\}\text {.} \end{aligned}

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 $${{\,\mathrm{PG}\,}}(V)$$, we mean its (element-wise) image under this map. Up to isomorphism, the image does not depend on the choice of $$\beta$$.

### Definition 2.1

Let tvk be integers with $$0 \le t \le k\le v-t$$ 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 [10]. It is stated that “Several people have observed that the concept of a t-design can be generalised [...]”, so the idea might been around before. Subspace designs have also been mentioned in a more general context in [12]. The first nontrivial subspace designs with $$t \ge 2$$ have been constructed in [27], and the first nontrivial Steiner system with $$t \ge 2$$ in [4]. An introduction to the theory of subspace designs can be found at [7], see also [25, 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 [9], and that a t-$$(v,k,1)_q$$ Steiner system 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 [6, 18, 19, 22].

One example of such a statement is the following [26, Lemma 4.1(1)], see also [18, 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

\begin{aligned} \lambda _s :=\lambda \frac{\genfrac[]{0.0pt}{}{v-s}{t-s}_{q}}{\genfrac[]{0.0pt}{}{k-s}{t-s}_{q}}. \end{aligned}

In particular, the number of blocks in D equals

\begin{aligned} \#D = \lambda _0 = \lambda \frac{\genfrac[]{0.0pt}{}{v}{t}_{q}}{\genfrac[]{0.0pt}{}{k}{t}_{q}}. \end{aligned}

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 nontrivial q-analog of a Steiner system with $$t\ge 2$$ 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 ij be non-negative integers with $$i + j \le t$$ and let $$I\in \genfrac[]{0.0pt}{}{V}{i}_{q}$$ and $$J\in \genfrac[]{0.0pt}{}{V}{v-j}_{q}$$. By [26, Lemma 4.1], see also [7, Lemma 5], the number

\begin{aligned} \lambda _{i,j} :=\# D|_I^J = \lambda \frac{\genfrac[]{0.0pt}{}{v-i-j}{k-i}_{q}}{\genfrac[]{0.0pt}{}{v-t}{k-t}_{q}} \end{aligned}

only depends on i and j, but not on the choice of I and J. Apparently, $$\lambda _{i,0} = \lambda _i$$. The numbers $$\lambda _{i,j}$$ are important parameters of a subspace design. A further generalization is given by the intersection numbers in [19].

A nice way to arrange the numbers $$\lambda _{i,j}$$ is the following triangle form, which may be called the q-Pascal triangle of the subspace design D.

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

The proof of the result of this article will make use of the equality $$\lambda _{1,1} = \lambda _{0,2}$$ in the above triangle.

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

\begin{aligned} t\text {-}\left( v,v-k,\frac{\genfrac[]{0.0pt}{}{v-t}{k}_{q}}{\genfrac[]{0.0pt}{}{v-t}{k-t}_{q}}\right) _{\!q}\text {.} \end{aligned}

For a point $$P \le V$$, the derived design of D in P is the set of blocks

\begin{aligned} {{\,\mathrm{Der}\,}}_P(D) = \{ B/P \mid B \in D|_P\} \end{aligned}

in the ambient vector space V/P.Footnote 3 By [18], $${{\,\mathrm{Der}\,}}_P(D)$$ 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, $${{\,\mathrm{Der}\,}}_P(D)$$ has the parameters 1-$$(6,2,1)_q$$.

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 a 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 [17].

The parameters 1-$$(v,k,1)_q$$ are admissible if and only v is divisible by k. In this case, spreads do always exist [24, Sect. 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 $$B,B'\in {\mathcal {S}}$$, the set $${\mathcal {S}}|^{B + B'}$$ is always a $$(k-1)$$-spread of $$B + B'$$. In other words, $${\mathcal {S}}$$ is geometric if every 2k-subspace of V contains either 0, 1 or $$[2k]_{q}/[k]_{q} = q^k + 1$$ blocks of $${\mathcal {S}}$$. It is not hard to see that the Desarguesian spread is geometric. In fact, it follows from [2, 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 $${{\,\mathrm{PG}\,}}(5,q)$$. The most symmetric one among these spreads is the Desarguesian spread. Following the notation of [13], 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.Footnote 4

We remark that in the binary case $$q=2$$, the line spreads of $${{\,\mathrm{PG}\,}}(5,q)$$ have been classified into $$131\,044$$ isomorphism types in [21].

### 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 that

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 (PL) there is a unique incident point-line-pair $$(P',L')$$ with $$P\mathrel {I} L'$$ and $$P' \mathrel {I} L$$.

Generalized quadrangles have been introduced in the more general setting of generalized polygons in [29], 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 (st). 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 (st) if and only if $$Q^\perp$$ is of order (ts).

Furthermore, Q is said to be projective if it is embeddable in some Desarguesian projective geometry in the following sense: There is a Desarguesian projective geometry $$({{\mathcal {P}}}, {{\mathcal {L}}}, {\bar{I}})$$ such that $${\mathcal {P}}\subseteq \bar{{\mathcal {P}}}$$, $${\mathcal {L}}\subseteq \bar{{\mathcal {L}}}$$, for all $$(P,L)\in {{\mathcal {P}}}\times {{\mathcal {L}}}$$ we have $$P\mathrel {I} L$$ if and only if $$P \mathrel {{\bar{I}}} L$$, and for each point $$P\in \bar{{\mathcal {P}}}$$ with $$P \mathrel {{\bar{I}}} L$$ for some line $$L\in {\mathcal {L}}$$ we have $$P\in {\mathcal {P}}$$.Footnote 5 The non-degenerate finite projective generalized quadrangles have been classified in [8, Theorem 1], see also [23, 4.4.8]. These are exactly the so-called classical generalized quadrangles which are associated to a quadratic form or a symplectic or Hermitian polarity on the ambient geometry, see [23, 3.1.1].

1. (i)

The symplectic generalized quadrangle W(q) consisting of the set of points of $${{\,\mathrm{PG}\,}}(3,q)$$ together with the totally isotropic lines with respect to a symplectic polarity. Taking the geometry as $${{\,\mathrm{PG}\,}}({\mathbb {F}}_q^4)$$, 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 $${{\,\mathrm{PG}\,}}(3,q)$$ is also known as a (general) linear complex of lines, see [23, 3.1.1 (iii)] or [15, Theorem 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 zeros of a parabolic quadratic form in $${{\,\mathrm{PG}\,}}(4,q)$$, and whose lines are all the lines contained in $${\mathcal {P}}$$. Taking the geometry as $${{\,\mathrm{PG}\,}}({\mathbb {F}}_q^5)$$, the parabolic quadratic form can be represented by $$q(\mathbf {x}) = x_1 x_2 + x_3 x_4 + x_5^2$$.

Both W(q) and Q(4, q) are of order (qq). By [23, 3.2.1] they are duals of each other, meaning that $$W(q)^\perp \cong Q(4,q)$$.

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 parts (a) and (b) in Theorem 1.

## 3 Proof of the theorems

For the remainder of the article, we fix $$v = 7$$ and assume that $$D \subseteq \genfrac[]{0.0pt}{}{V}{3}_{q}$$ is a q-analog of the Fano plane. The numbers $$\lambda _{i,j}$$ are defined as in Sect. 2.2.

By the design property, the intersection dimension of two distinct blocks $$B,B'\in D$$ is either 0 or 1. So by the dimension formula, $$\dim (B + B') \in \{5,6\}$$. 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 or 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 [19, 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 a $$\beta$$-flat with focal point $$P\in \genfrac[]{0.0pt}{}{F}{1}_{q}$$ if all the $$\lambda _{0,2} = q^2 + 1$$ blocks contained in F pass through P.

### Lemma 3.2

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

### Proof

Assume that $$P \ne Q$$ 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. $$\square$$

### 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 $$F \ne F'$$. Then the $$q^2 + 1 > 1$$ blocks in $$D|_P^H$$ are contained in $$F \cap F'$$. This is a contradiction, since $$\dim (F \cap F') \le 4$$ and any solid contains at most a single block. $$\square$$

### Lemma 3.4

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

1. (a)

Each point in F different from P is covered by a unique block in F.

In other words, $$D|^F/P$$ is a line spread of $$F/P \cong {{\,\mathrm{PG}\,}}(3,q)$$.

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, each point in F that is different from P is covered by a single point in $$D|^F$$.

Part (b): The number of solids in F containing one of the $$q^2 + 1$$ blocks in F is $$(q^2 + 1)\cdot \genfrac[]{0.0pt}{}{5-3}{4-3}_{q} = (q^2 + 1)(q + 1) = q^3 + q^2 + q + 1$$.Footnote 6 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 $$\dim (B \cap S) \le 2$$ as S is poor, and moreover $$\dim (B \cap S) \ge \dim (B) + \dim (S) - \dim (F) = 3 + 4 - 5 = 2$$ 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. $$\square$$

### Lemma 3.5

Let P be an $$\alpha$$-point and $$B, B'\in D$$ two blocks with $$B \cap B' = P$$. Then $$B + B'$$ is a $$\beta$$-flat with focal point P.

### Proof

Since $$P = B \cap B'$$ is a point, $$F = B + B'$$ is a 5-subspace. Since P is an $$\alpha$$-point, we have that $$\{B'' / P \mid B''\in D|^F_P\}$$ is a line spread of $$F / P \cong {\mathbb {F}}_q^4$$. Such a line spread contains $$[4]_q/[2]_q = q^2 + 1$$ 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 all the blocks contained in F pass through P. $$\square$$

### 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 and $$B'$$ be two distinct blocks in F. Then $$P = B\cap B'$$ is a point and $$F = B + B'$$. By assumption, P is an $$\alpha$$-point, so by Lemma 3.5, P is the focal point of the $$\beta$$-flat F. $$\square$$

### 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, $$B'$$ in F such that $$P = B\cap B' \ne Q$$. 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 $$B, B'\in D|^H_P$$. Then $$P = B \cap B'$$. By Lemma 3.5, the $$\alpha$$-point P is the focal point of the $$\beta$$-flat $$F = B + B'$$. By Lemma 3.3, the $$\beta$$-flat F is unique. $$\square$$

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 of H 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

\begin{aligned} \alpha : \genfrac[]{0.0pt}{}{H}{5}_{q} \rightarrow \genfrac[]{0.0pt}{}{H}{1}_{q} \quad \text {and}\quad \beta : \genfrac[]{0.0pt}{}{H}{1}_{q} \rightarrow \genfrac[]{0.0pt}{}{H}{5}_{q} \end{aligned}

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

### Lemma 3.9

Let B be a block in H.

1. (a)

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

2. (b)

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

### 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, $$B \le \beta (P)$$.

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

For the remainder of this article, we fix a poor solid S of H. Note that by Lemma 3.4(b), every 5-subspace of H contains a suitable solid S.Footnote 7 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 $${\mathcal {L}}_F :=\{B \cap S \mid B\in D|^F\}$$ is a line spread of S by Lemma 3.4(c).

### Lemma 3.10

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

### Proof

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

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 admitting a partition into the $$q+1$$ line spreads $${\mathcal {L}}_F$$ with $$F\in {\mathcal {F}}$$. $$\square$$

### 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 3.4(b), the poor solid S is not contained in the 5-subspace $$\beta (P)$$. Therefore, $$\dim (\beta (P)\cap S) \le 3$$. On the other hand, as both S and $$\beta (P)$$ are contained in H, we have $$\dim (\beta (P) + S) \le \dim (H) = 6$$ and therefore by the dimension formula $$\dim (\beta (P)\cap S) = \dim (\beta (P)) + \dim (S) - \dim (\beta (P)+ S) \ge 3$$. 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 3.9(a), $$B \le \beta (P)$$. So $$L = B \cap S \le \beta (P) \cap S = E_P$$. 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. $$\square$$

### 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 $$L''\in {\mathcal {L}}_F$$ passing through P. By Lemma 3.12, $$L'' < E_P$$, so we get $$L \not < E_P$$ as otherwise L and $$L''$$ 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 $$P'\in \genfrac[]{0.0pt}{}{S}{1}_{q}$$ and $$L'\in {\mathcal {L}}$$ with $$L \cap L' = P'$$ and $$P + P' = L'$$. Then $$L'$$ is a line through P, so $$L' < E_P$$. So necessarily $$P' = E_P \cap L$$ and $$L' = P + P'$$, showing that $$P'$$ and $$L'$$ are unique.

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

### Lemma 3.14

$$(\genfrac[]{0.0pt}{}{S}{1}_{q},{\mathcal {L}}, \subseteq )$$ 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 $${{\,\mathrm{PG}\,}}(S)$$. By the classification in [8, Theorem 1] (see also [23, 4.4.8]), we know that Q is a finite classical generalized quadrangle which are listed in [23, 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). $$\square$$

Now we can prove our main result.

### Proof of Theorem 1

Part (a) follows from Lemmas 3.14 and 3.11. The equivalence of parts (a) and (b) has already been discussed at the end of Sect. 2.4. $$\square$$

Theorem 2 is now a direct consequence.

### Proof of Theorem 2

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

For q prime, the only ovoids of Q(4, q) are the elliptic quadrics $$Q^{-}(3,q)$$ [1, 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 [23, 3.4.1 (i)]. $$\square$$