On $\alpha$-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 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 $q$.


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 [3, 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 [6,18]. 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 [14] and 6978 vs. 7651 in the ternary case [16]. 1 Another approach has been the investigation of the derived designs of a putative qanalog 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 131044 isomorphism types [21] of such spreads. For highest possible regularity, one would expect all points to be α-points. However, this has been shown to be impossible, so there must always be at least one non-α-point of D [28]. For the binary case q = 2, this result has been improved to the statement that each hyperplane contains at least a non-α-point Heden-Sissokho-2016-ArsComb124:161-164, In other words, the non-α-points of 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 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 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 will 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 Fq (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 Fq (U ) over the geometric dimension dim Fq (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 [v] q .
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 U → U ⊥ , where U ⊥ = {x ∈ V | β(x, u) = 0 for all u ∈ U }. 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
In the important case λ = 1, D is called a q-Steiner system.
The earliest reference for subspace designs is [10]. 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 [12]. The first nontrivial subspace designs with t ≥ 2 has 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 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 [19,20,22,5].
One example of such a statement is the following [26, Lemma 4.1 (1)], see also [19, 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 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 λ i can be refined as follows. Let i, j be non-negative integers with i + j ≤ t and let I ∈ V i q and J ∈ V v−j q . 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. The numbers λ i,j are important parameters of a subspace design. A further generalization is given by the intersection numbers in [20]. A nice way to arrange the numbers λ i,j is the following triangle form.
For a q-analog of the Fano plane, we get: As a consequence of the numbers λ i,j , the dual design In the case of a q-analog of the Fano plane, Der P (D) has the parameters 1-(6, 2, 1).

Spreads
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 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,§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 (k − 1)-spread S is called geometric or normal if for two distinct blocks B, B ∈ S, the set S| B+B is always a (k −1)-spread of B +B . In other words, S is geometric if every 2k-subspace of V contains either 0, 1 or [2k]q [k]q = q k + 1 blocks of 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 PG(5, 2). These spreads have been classified in [21] into 131044 isomorphism types. 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.

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 hat (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-linepair (P , L ) with P I L and P 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 = (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. This means that there is a Desarguesian projective geometry with point set P , line set L , and point-line incidence relation I such that P ⊆ P , L ⊆ L and for all (P, L) ∈ P ×L we have P I L if and only if P I L . The non-degenerate finite projective generalized quadrangles have been classified in [8, Th. 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 Hermitean 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 zeroes 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 q(x, y) = x 1 x 2 + x 3 x 4 + x 2 5 .
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 Part (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 7 q is a q-analog of the Fano plane. The numbers λ i,j are defined as in Section 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 of no block at all. We will call S a rich solid in the former case and a poor solid in the latter. [20,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 such that all the λ 0,2 = q 2 + 1 blocks in F pass through a common point P a β-flat with focal point P . 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. 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. 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, 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) · 5−3 4−3 q = (q 2 + 1)(q + 1) = q 3 + q 2 + q + 1. 3 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  Proof. Since P = B ∩ B is a point, F = B + B is a 5-subspace. Since P is an α-point, we have that {B /P | B ∈ D| F P } is a line spread of F/P ∼ = F 4 q . 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 F is a β-flat with focal point 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, B be two distinct blocks in F . Then P = B ∩ B is a point and F = B + B . By the precondition, 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.
Lemma 3.8 Let H be a hyperplane and P an α-point contained in H. Then H contains a unique β-flat whose focal point is P .
Proof. There are λ 1,1 = q 2 + 1 > 1 blocks in H containing P . Let B, B ∈ D| H P . Then P = B ∩ B . By Lemma 3.5, the α-point P is the focal point of the β-flat F = B + B . By Lemma 3.3, the β-flat F is unique. 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 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 V .

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 which admits 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 . Lemma 3.13 The incidence structure ( S 1 q , L, ⊆) is a projective generalized quadrangle of order (s, t) = (q, q).
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, p. 4.4.8]), we know that Q is a finite classical generalized quadrangle which are listed in [23, p. 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 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.