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].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].
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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:
(a)
The line set of the symplectic generalized quadrangle W(q) is partitionable into spreads.
(b)
The point set of the parabolic quadric Q(4, q) is partitionable into ovoids.
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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 \({{\,\mathrm{PG}\,}}(v-1,q) = {{\,\mathrm{PG}\,}}(V)\).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 coefficientThe 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 \([v]_{q}\).
$$\begin{aligned} \#\genfrac[]{0.0pt}{}{V}{k}_{q} = \genfrac[]{0.0pt}{}{v}{k}_{q} = {\left\{ \begin{array}{ll} \frac{(q^v-1)\cdots (q^{v-k+1}-1)}{(q^k-1)\cdots (q-1)} &{}\quad \text {if } k\in \{0,\ldots ,v\}\text {;}\\ 0 &{}\quad \text {otherwise.} \end{array}\right. } \end{aligned}$$
For \(S \subseteq {\mathcal {L}}(V)\) and \(U,W\in {\mathcal {L}}(V)\), we will use the abbreviationsFor a point P in a plane E, the set of all lines in E passing through P is known as a line pencil.
$$\begin{aligned} S|_U&= \{B\in S \mid U \le B\}\text {,} \\ S|^W&= \{B\in S\mid B\le W\}\quad \text {and} \\ S|_U^W&= \{B\in S\mid U \le B\le W\}\text {.} \end{aligned}$$
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 \(\beta \).
2.2 Subspace designs
Definition 2.1
Let t, v, k 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 \(s\in \{0,\ldots ,t\}\), whereIn particular, the number of blocks in D equalsSo, 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.
$$\begin{aligned} \lambda _s :=\lambda \frac{\genfrac[]{0.0pt}{}{v-s}{t-s}_{q}}{\genfrac[]{0.0pt}{}{k-s}{t-s}_{q}}. \end{aligned}$$
$$\begin{aligned} \#D = \lambda _0 = \lambda \frac{\genfrac[]{0.0pt}{}{v}{t}_{q}}{\genfrac[]{0.0pt}{}{k}{t}_{q}}. \end{aligned}$$
The numbers \(\lambda _i\) can be refined as follows. Let i, j 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 numberonly 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].
$$\begin{aligned} \lambda _{i,j} :=\# D|_I^J = \lambda \frac{\genfrac[]{0.0pt}{}{v-i-j}{k-i}_{q}}{\genfrac[]{0.0pt}{}{v-t}{k-t}_{q}} \end{aligned}$$
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 parametersFor a point \(P \le V\), the derived design of D in P is the set of blocksin the ambient vector space V/P.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-\((6,2,1)_q\).
$$\begin{aligned} t\text {-}\left( v,v-k,\frac{\genfrac[]{0.0pt}{}{v-t}{k}_{q}}{\genfrac[]{0.0pt}{}{v-t}{k-t}_{q}}\right) _{\!q}\text {.} \end{aligned}$$
$$\begin{aligned} {{\,\mathrm{Der}\,}}_P(D) = \{ B/P \mid B \in D|_P\} \end{aligned}$$
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 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.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].
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 that
(i)
Two distinct points are incident with at most a line.
(ii)
Two distinct lines are incident with at most one point.
(iii)
For each non-incident point-line-pair (P, L) 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 (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 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 \(P\in {\mathcal {P}}\).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].
In this article, two of these classical generalized quadrangles will appear. Both W(q) and Q(4, q) are of order (q, q). By [23, 3.2.1] they are duals of each other, meaning that \(W(q)^\perp \cong Q(4,q)\).
(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.
(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 \(q(\mathbf {x}) = x_1 x_2 + x_3 x_4 + x_5^2\).
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.
(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 \(F/P \cong {{\,\mathrm{PG}\,}}(3,q)\).
(b)
A solid S of F is poor if and only if it does not contain P.
(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 \((q^2 + 1)\cdot \genfrac[]{0.0pt}{}{5-3}{4-3}_{q} = (q^2 + 1)(q + 1) = q^3 + q^2 + q + 1\).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
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 mappingsare inverse to each other. So they provide a bijective correspondence between the points and the 5-subspaces of H.
$$\begin{aligned} \alpha : \genfrac[]{0.0pt}{}{H}{5}_{q} \rightarrow \genfrac[]{0.0pt}{}{H}{1}_{q} \quad \text {and}\quad \beta : \genfrac[]{0.0pt}{}{H}{1}_{q} \rightarrow \genfrac[]{0.0pt}{}{H}{5}_{q} \end{aligned}$$
Lemma 3.9
Let B be a block in H.
(a)
For all points P of B, \(B \le \beta (P)\).
(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.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 (q, q). \(\square \)
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
Theorem 2 is now a direct consequence.
Acknowledgements
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. Moreover, I would like to thank the two anonymous referees for their careful reading and their valuable remarks.
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