On α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-points of q-analogs of the Fano plane

Arguably, the most important open problem in the theory of q-analogs of designs is the question regarding the existence of a q-analog D of the Fano plane. As of today, it remains undecided for every single prime power order q of the base field. A point P is called an α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-point of D if the derived design of D in P is a geometric spread. In 1996, Simon Thomas has shown that there always exists a non-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-point. For the binary case q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q = 2$$\end{document}, Olof Heden and Papa Sissokho have improved this result in 2016 by showing that the non-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-points must form a blocking set with respect to the hyperplanes. In this article, we show that a hyperplane consisting only of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-points implies the existence of a partition 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.

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].
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 ∈ 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 PG(5, q). Following the notation of [13], a point P is called an α-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 PG (5, q). For highest possible regularity, one would expect all points to be α-points.
However, this has been shown to be impossible, as there must always be at least one nonα-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-α-point [13]. In other words, the non-α-points of a binary q-analog of the Fano plane form a blocking set with respect to the hyperplanes.
In this article, α-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 α-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.
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-α-point. In other words, the non-α-points form a blocking set with respect to the hyperplanes.

Preliminaries
Throughout the article, q = 1 is a prime power and V is a vector space over F q of finite dimension v.

The subspace lattice
For simplicity, a subspace U of V of dimension dim 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 V k q . Picking the "best of two worlds", we will prefer the algebraic dimension dim F q (U ) over the geometric dimension dim 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 L(V ) consisting of all subspaces of V ordered by inclusion is nothing else than the finite projective geometry PG(v − 1, q) = PG(V ). 2 There are good reasons to consider the subset lattice as a subspace lattice over the unary "field" F 1 [11].
The number of all k-subspaces of V is given by the Gaussian binomial coefficient The Gaussian binomial coefficient v 1 q is also known as the q-analog of the number v and will be abbreviated as For S ⊆ L(V ) and U , W ∈ L(V ), we will use the abbreviations 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 L(V ) is isomorphic to its dual, which arises from L(V ) by reversing the order. Fixing a non-degenerate bilinear form β on V , a concrete isomorphism is given by When addressing the dual of some geometric object in PG(V ), we mean its (element-wise) image under this map. Up to isomorphism, the image does not depend on the choice of β.

Subspace designs
Definition 2.1 Let t, v, k be integers with 0 ≤ t ≤ k ≤ v −t and λ another positive integer. A set D ⊆ V k q is called a t-(v, k, λ) q subspace design if each t-subspace of V is contained in exactly λ elements (called blocks) of D. In the important case λ = 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 ≥ 2 have been constructed in [27], and the first nontrivial Steiner system with t ≥ 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 In particular, the number of blocks in D equals So, for a design with parameters t-(v, k, λ) q , the numbers λ s necessarily are integers for all s ∈ {0, . . . , t} (integrality conditions). In this case, the parameter set t-(v, k, λ) q is called admissible. It is further called realizable if a t-(v, k, λ) q design actually exists. The smallest admissible parameters of a nontrivial q-analog of a Steiner system with t ≥ 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 λ i can be refined as follows. Let i, j be non-negative integers with i + j ≤ t and let . By [26,Lemma 4.1], see also [7,Lemma 5], the number only depends on i and j, but not on the choice of I and J . Apparently, λ i,0 = λ i . The numbers λ 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 λ 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 λ 1,1 = λ 0,2 in the above triangle.
As a consequence of the numbers λ i, j , the dual design D ⊥ = {B ⊥ | B ∈ D} is a subspace design with the parameters For a point P ≤ V , the derived design of D in P is the set of blocks In the case of a q-analog of the Fano plane, Der P (D) has the parameters 1-(6, 2, 1) q .

Spreads
A 1-(v, k, 1) q Steiner system 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 S of k-subspaces is called a partial (k − 1)-spread if each point is covered by at most one element of 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 F q k and set S = V 1 q k . Switching back to vector spaces over F q , the set S is a (k − 1)-spread of V , known as the Desarguesian spread. A 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 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 α-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 PG(5, q) have been classified into 131 044 isomorphism types in [21].

Generalized quadrangles
Definition 2.2 A generalized quadrangle is an incidence structure Q = (P, L, I ) with a non-empty set of points P, a non-empty set of lines L, and an incidence relation I ⊆ P × L such that 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 = (P, 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 ⊥ arises from Q by interchanging the role of the points and the lines. It is again a generalized quadrangle. Clearly, (Q ⊥ ) ⊥ = Q, and Q is of order (s, t) if and only if Q ⊥ 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 (P, L,Ī ) such that P ⊆P, L ⊆L, for all (P, L) ∈ P × L we have P I L if and only if PĪ L, and for each point P ∈P with PĪ L for some line L ∈ L we have P ∈ 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. (ii) The second one is the parabolic quadric Q(4, q), whose points P are the zeros of a parabolic quadratic form in PG(4, q), and whose lines are all the lines contained in P.
Taking the geometry as PG(F 5 q ), the parabolic quadratic form can be represented by 5 . 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) ⊥ ∼ = Q(4, q).
Let Q = (P, L, I ) be a generalized quadrangle. As in projective geometries, a set S ⊆ L is called a spread of Q if each point of Q is incident with a unique line in S. Dually, a set O ⊆ P is called an ovoid of Q if each line of Q is incident with a unique point in O. Clearly, the spreads of Q bijectively correspond to the ovoids of Q ⊥ . This already shows the equivalence of parts (a) and (b) in Theorem 1.

Proof of the theorems
For the remainder of the article, we fix v = 7 and assume that D ⊆ V 3 q is a q-analog of the Fano plane. The numbers λ i, j are defined as in Sect. 2.2.
By the design property, the intersection dimension of two distinct blocks B, B ∈ D is either 0 or 1. So by the dimension formula, dim(B + B ) ∈ {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. [19,Remark 4.2], the poor solids form a dual 2-(7, 3, q 4 ) q subspace design. By the above discussion, the λ 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.

Remark 3.1 By
We will call a 5-subspace F a β-flat with focal point P ∈ F 1 q if all the λ 0,2 = q 2 + 1 blocks contained in F pass through P.

Lemma 3.2 The focal point of a β-flat is uniquely determined.
Proof Assume that P = Q are focal points of a β-flat F. Then all λ 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 β-flat in H .
Proof There are λ 1,1 = q 2 + 1 blocks in H passing through P. For any β-flat F < H with focal point P, all these blocks are contained in F. Now assume that there are two such β-flats F = F . Then the q 2 + 1 > 1 blocks in D| H P are contained in F ∩ F . This is a contradiction, since dim(F ∩ F ) ≤ 4 and any solid contains at most a single block.

Lemma 3.4 Let F ∈ V 5 q be a β-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 ∼ = 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 ∩ S | B ∈ 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 F 1 q \{P} covered by these blocks is (q 2 +1)( 3 1 q −1) = q 4 +q 3 +q 2 +q = 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) · 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) = 5 4 q is already the total number of solids in F, the poor solids in F are precisely those not containing P.
Part ( 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 λ 0,2 = q 2 + 1, so all the blocks contained in F pass through P. Lemma 3.6 Let F be a 5-subspace such that all points of F are α-points. Then F is a β-flat.
Proof The 5-subspace F contains λ 0,2 = q 2 + 1 > 1 blocks. Let B and B be two distinct blocks in F. Then P = B ∩ B is a point and F = B + B . By assumption, P is an α-point, so by Lemma 3.5, P is the focal point of the β-flat F.

Remark 3.7 The statement of Lemma 3.6 is still true if F contains a single non-α-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 ∩ B = Q. In the latter case, all blocks pass through the α-point P as in the proof of Lemma 3.6. Now we fix a hyperplane H of V and assume that all its points are α-points. By Lemma 3.6, every 5-subspace F of H is a β-flat. We denote its unique focal point by α(F). Moreover by Lemma 3.8, each point P of H is the focal point of a unique β-flat F in H . We will denote this β-flat by β(P). Clearly, the mappings α : are inverse to each other. So they provide a bijective correspondence between the points and the 5-subspaces of H . Proof For part (a), let P be a point on B. There are λ 1,1 = q 2 +1 blocks in H passing through P, which equals the number λ 0,2 of blocks in β(P) (which all pass through P). Therefore, B ≤ β(P). For part (b), let F be a 5-subspace containing B. All blocks in F pass through its focal point α(F).
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 6−4 5−4 q = q + 1 intermediate 5-subspaces F with S < F < H will be denoted by F . For each F ∈ F , the set L F := {B ∩ S | B ∈ D| F } is a line spread of S by Lemma 3.4(c).

Lemma 3.11
The set 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 L F are pairwise disjoint, so L is a set of #F · # 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 L F with F ∈ F . Lemma 3.12 For each point P of S, L| P is a line pencil in the plane E P = β(P) ∩ S.
Proof Let P be a point in S.
Let L ∈ L| P . Then there is a block B ∈ D| H with B ∩ S = L. By Lemma 3.9(a), B ≤ β(P). So L = B ∩ S ≤ β(P) ∩ S = E P . As the disjoint union of q + 1 line spreads of S, L contains q + 1 lines passing through P. Therefore, these lines form a line pencil in E P through P.
Proof Clearly, every line in L contains q + 1 points in S. By Lemma 3.11, through every point in S there pass q + 1 lines in L. Now let P be a point in S and L ∈ L not containing P.
By Lemma 3.10, there is a unique F ∈ F with L ∈ L F , and there is a line L ∈ L F passing through P. By Lemma 3.12, L < E P , so we get L < E P as otherwise L and L would be distinct intersecting lines in the spread L F . Moreover, L and E P are both contained in S, so they cannot have trivial intersection. Therefore L ∩ E P is a point. Now let P ∈ S 1 q and L ∈ L with L ∩ L = P and P + P = L . Then L is a line through P, so L < E P . So necessarily P = E P ∩ L and L = P + P , showing that P and L are unique.
By Lemma 3.12 indeed L ∈ 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 ( S 1 , L) is a generalized quadrangle of order (q, q). Lemma 3.14 ( S 1 q , L, ⊆) is isomorphic to W (q).
Proof By Lemma 3.13 we know that Q = ( S 1 q , L, ⊆) is a finite generalized quadrangle of order (s, t) = (q, q) embedded in 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).

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. 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.