Abstract
Arguably, the most important open problem in the theory of qanalogs of designs is the question regarding the existence of a qanalog 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 \(\alpha \)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\(\alpha \)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 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.
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 qanalogs 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 qanalog 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 RayChaudhuri [3, Problem 28]. Despite considerable investigations, its existence remains undecided for every single order q of the base field.
A qanalog 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 qanalog 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 qanalog D of the Fano plane. A derived design exists for each point \(P\in {{\,\mathrm{PG}\,}}(6,q)\) and is always a qdesign 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 qanalog 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 qanalog 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.
As a consequence, we get the following generalization of the result of [13].
Theorem 2
Let D be a qanalog 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 ksubspace. The set of all ksubspaces 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 1subspaces of V points, the 2subspaces lines, the 3subspaces planes, the 4subspaces solids and the \((v1)\)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 \({{\,\mathrm{PG}\,}}(v1,q) = {{\,\mathrm{PG}\,}}(V)\).^{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 ksubspaces of V is given by the Gaussian binomial coefficient
The Gaussian binomial coefficient \(\genfrac[]{0.0pt}{}{v}{1}_{q}\) is also known as the qanalog 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
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 nondegenerate 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 (elementwise) 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 \(0 \le t \le k\le vt\) 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 tsubspace of V is contained in exactly \(\lambda \) elements (called blocks) of D. In the important case \(\lambda = 1\), D is called a qSteiner system.
The earliest reference for subspace designs is [10]. It is stated that “Several people have observed that the concept of a tdesign 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 qanalog of the theorem of Assmus and Mattson [9], and that a t\((v,k,1)_q\) Steiner system provides a \((v,2(kt+1);k)_q\) 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
In particular, the number of blocks in D equals
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 qanalog of a Steiner system with \(t\ge 2\) are 2\((7,3,1)_q\), which are the parameters of the qanalog of the Fano plane. This explains the significance of the question of its realizability.
The numbers \(\lambda _i\) can be refined as follows. Let i, j be nonnegative integers with \(i + j \le t\) and let \(I\in \genfrac[]{0.0pt}{}{V}{i}_{q}\) and \(J\in \genfrac[]{0.0pt}{}{V}{vj}_{q}\). 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. 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 qPascal triangle of the subspace design D.
For a qanalog 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
For a point \(P \le V\), the derived design of D in P is the set of blocks
in the ambient vector space V/P.^{Footnote 3} By [18], \({{\,\mathrm{Der}\,}}_P(D)\) is a subspace design with the parameters \((t1)\)\((v1,k1,\lambda )_q\). In the case of a qanalog of the Fano plane, \({{\,\mathrm{Der}\,}}_P(D)\) has the parameters 1\((6,2,1)_q\).
2.3 Spreads
A 1\((v,k,1)_q\) Steiner system \({\mathcal {S}}\) is just a partition of the point set of V into ksubspaces. These objects are better known under the name \((k1)\)spread and have been investigated in geometry well before the emergence of subspace designs. A 1spread is also called a line spread.
A set \({\mathcal {S}}\) of ksubspaces is called a partial \((k1)\)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 \((k1)\)spread of V, known as the Desarguesian spread.
A \((k1)\)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 \((k1)\)spread of \(B + B'\). In other words, \({\mathcal {S}}\) is geometric if every 2ksubspace 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 \((k1)\)spread is geometric if and only if it is isomorphic to a Desarguesian spreads.
The derived designs of a qanalog 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 qanalog 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].
2.4 Generalized quadrangles
Definition 2.2
A generalized quadrangle is an incidence structure \(Q = ({\mathcal {P}},{\mathcal {L}},I)\) with a nonempty set of points \({\mathcal {P}}\), a nonempty 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 nonincident pointlinepair (P, L) there is a unique incident pointlinepair \((P',L')\) 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 \(P\in {\mathcal {P}}\).^{Footnote 5} The nondegenerate finite projective generalized quadrangles have been classified in [8, Theorem 1], see also [23, 4.4.8]. These are exactly the socalled 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.

(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 nontangent 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 \(q(\mathbf {x}) = x_1 x_2 + x_3 x_4 + x_5^2\).
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 \(W(q)^\perp \cong Q(4,q)\).
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 qanalog 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 5space 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 5subspace 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 5subspace 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 \((q^2 + 1)\cdot \genfrac[]{0.0pt}{}{53}{43}_{q} = (q^2 + 1)(q + 1) = q^3 + q^2 + q + 1\).^{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 5subspace. 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 5subspace such that all points of F are \(\alpha \)points. Then F is a \(\beta \)flat.
Proof
The 5subspace 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 5subspace 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
are inverse to each other. So they provide a bijective correspondence between the points and the 5subspaces of H.
Lemma 3.9
Let B be a block in H.

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

(b)
For all 5subspaces 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 5subspace 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 5subspace of H contains a suitable solid S.^{Footnote 7} The set of \(\genfrac[]{0.0pt}{}{64}{54}_{q} = q+1\) intermediate 5subspaces 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 5subspace \(\beta (P)\). 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
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)]. \(\square \)
Change history
24 July 2022
Missing Open Access funding information has been added in the Funding Note.
Notes
As noticed by Daniel Heinlein, the \([7,4;3]_3\) code of size 6977 constructed in [16] can be extended trivially by adding a further codeword.
In established symbols like \({{\,\mathrm{PG}\,}}(v1,q)\), the geometric dimension \(v1\) is not altered.
The expressions V/P and B/P are quotients of \({\mathbb {F}}_q\)vector spaces. In this way, \(B/P = \{x + P \mid x\in B\}\) is an \({\mathbb {F}}_q\)subspace of \(V/P = \{x + P \mid x\in V\}\) for every block \(B\in D_P\).
The definition of an \(\alpha \)point in [13] does not use the notion of a geometric spread. Instead, the property of a geometric spread in the factor space V/P has been written down explicitly, so the definitions are equivalent.
As pointed out by a referee, the latter condition is indeed necessary, as otherwise the generalized quadrangle \(T_2^*(O)\) in [23, 3.1.3] would be projective.
Remember that a solid cannot contain 2 blocks.
Using the fact that the poor solids form a dual 2\((7,3,q^4)_q\) subspace design in V [19, Remark 4.2], the total number of poor solids S in H is \(q^4\cdot \lambda _{1,0} = q^8 + q^6 + q^4 = q^4 (q^2 + q + 1) (q^2  q + 1)\).
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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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Kiermaier, M. On \(\alpha \)points of qanalogs of the Fano plane. Des. Codes Cryptogr. 90, 1335–1345 (2022). https://doi.org/10.1007/s10623022010333
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DOI: https://doi.org/10.1007/s10623022010333
Keywords
 Subspace design
 qanalog
 Fano plane
 Steiner system
 Subspace code
Mathematics Subject Classification
 05B05
 05A30
 51E10
 51E12
 94B60