Direction problems in affine spaces
This paper is a survey paper on old and recent results on direction problems in finite dimensional affine spaces over a finite field.
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arXiv subjects
Jan De Beule
Publications and source records attributed to Jan De Beule.
A new family of tight sets in $\mathcal{Q}^{+}(5,q)$
In this paper, we describe a new infinite family of $\frac{q^{2}-1}{2}$-tight sets in the hyperbolic quadrics $\mathcal{Q}^{+}(5,q)$, for $q \equiv 5 \mbox{ or } 9 \bmod{12}$. Under the Klein correspondence, these correspond to Cameron--Liebler line classes of ${\rm PG}(3,q)$ having parameter $\frac{q^{2}-1}{2}$. This is the second known infinite family of nontrivial Cameron--Liebler line classes, the first family having been described by Bruen and Drudge with parameter $\frac{q^{2}+1}{2}$ in ${\rm PG}(3,q)$ for all odd $q$. The study of Cameron--Liebler line classes is closely related to the study of symmetric tactical decompositions of ${\rm PG}(3,q)$ (those having the same number of point classes as line classes). We show that our new examples occur as line classes in such a tactical decomposition when $q \equiv 9 \bmod 12$ (so $q = 3^{2e}$ for some positive integer $e$), providing an infinite family of counterexamples to a conjecture made by Cameron and Liebler in 1982; the nature of these decompositions allows us to also prove the existence of a set of type $\left(\frac{1}{2}(3^{2e}-3^{e}), \frac{1}{2}(3^{2e}+3^{e}) \right)$ in the affine plane ${\rm AG}(2,3^{2e})$ for all positive integers $e$. This proves a conjecture made by Rodgers in his PhD thesis.
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On the structure of the directions not determined by a large affine point set
Given a point set $U$ in an $n$-dimensional affine space of size $q^{n-1}-\varepsilon$, we obtain information on the structure of the set of directions that are not determined by $U$, and we describe an application in the theory of partial ovoids of certain partial geometries.
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On large maximal partial ovoids of the parabolic quadric $\q(4,q)$
We use the representation $T_2(O)$ for $\q(4,q)$ to show that maximal partial ovoids of $\q(4,q)$ of size $q^2-1$, $q=p^h$, $p$ odd prime, $h > 1$, do not exist. Although this was known before, we give a slightly alternative proof, also resulting in more combinatorial information of the known examples for $q$ prime.
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Sets of generators blocking all generators in finite classical polar spaces
We introduce generator blocking sets of finite classical polar spaces. These sets are a generalisation of maximal partial spreads. We prove a characterization of these minimal sets of the polar spaces Q(2n,q), Q-(2n+1,q) and H(2n,q^2), in terms of cones with vertex a subspace contained in the polar space and with base a generator blocking set in a polar space of rank 2.
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The known maximal partial ovoids of size $q^2-1$ of Q(4,q)
We present a description of maximal partial ovoids of size $q^2-1$ of the parabolic quadric $\q(4,q)$ as sharply transitive subsets of $\SL(2,q)$ and show their connection with spread sets. This representation leads to an elegant explicit description of all known examples. We also give an alternative representation of these examples which is related to root systems.
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On sets of vectors of a finite vector space in which every subset of basis size is a basis II
This article contains a proof of the MDS conjecture for $k \leq 2p-2$. That is, that if $S$ is a set of vectors of ${\mathbb F}_q^k$ in which every subset of $S$ of size $k$ is a basis, where $q=p^h$, $p$ is prime and $q$ is not and $k \leq 2p-2$, then $|S| \leq q+1$. It also contains a short proof of the same fact for $k\leq p$, for all $q$.
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