Jan De Beule
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View article: The largest sets of non-opposite chambers in spherical buildings of type $B$
The largest sets of non-opposite chambers in spherical buildings of type $B$ Open
The investigation into large families of non-opposite flags in finite spherical buildings has been a recent addition to a long line of research in extremal combinatorics, extending classical results in vector and polar spaces. This line of…
View article: Strongly regular graphs in hyperbolic quadrics
Strongly regular graphs in hyperbolic quadrics Open
Let $Q^+(2n+1,q)$ be a hyperbolic quadric of $\PG(2n+1,q)$. Fix a generator $Π$ of the quadric. Define $\cG_n$ as the graph with vertex set the points of $Q^+(2n+1,q)\setminus Π$ and two vertices adjacent if they either span a secant to $Q…
View article: An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings II
An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings II Open
We continue our investigation of Erdős-Ko-Rado (EKR) sets of flags in spherical buildings. In previous work, we used the theory of buildings and Iwahori-Hecke algebras to obtain upper bounds on their size. As the next step towards the clas…
View article: On two non-existence results for Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$
On two non-existence results for Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$ Open
This paper focuses on non-existence results for Cameron-Liebler $k$-sets. A Cameron-Liebler $k$-set is a collection of $k$-spaces in $\mathrm{PG}(n,q)$ or $\mathrm{AG}(n,q)$ admitting a certain parameter $x$, which is dependent on the size…
View article: On $m$-ovoids of $Q^+(7,q)$ with $q$ odd
On $m$-ovoids of $Q^+(7,q)$ with $q$ odd Open
In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection prop…
View article: Some non-existence results on $m$-ovoids in classical polar spaces
Some non-existence results on $m$-ovoids in classical polar spaces Open
In this paper we develop non-existence results for $m$-ovoids in the classical polar spaces $Q^-(2r+1,q), W(2r-1,q)$ and $H(2r,q^2)$ for $r>2$. In [4] a lower bound on $m$ for the existence of $m$-ovoids of $H(4,q^2)$ is found by using the…
View article: Degree 2 Boolean Functions on Grassmann Graphs
Degree 2 Boolean Functions on Grassmann Graphs Open
We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are k…
View article: On large partial ovoids of symplectic and Hermitian polar spaces
On large partial ovoids of symplectic and Hermitian polar spaces Open
In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces , odd square, , and of the Hermitian polar spaces , even or odd square, , , .
View article: On large partial ovoids of symplectic and Hermitian polar spaces
On large partial ovoids of symplectic and Hermitian polar spaces Open
In this paper we provide constructive lower bounds on the sizes of the largest partial ovoids of the symplectic polar spaces ${\cal W}(3, q)$, $q$ odd square, $q \not\equiv 0 \pmod{3}$, ${\cal W}(5, q)$ and of the Hermitian polar spaces ${…
View article: Degree 2 Boolean Functions on Grassmann Graphs
Degree 2 Boolean Functions on Grassmann Graphs Open
We investigate the existence of Boolean degree $d$ functions on the Grassmann graph of $k$-spaces in the vector space $\mathbb{F}_q^n$. For $d=1$ several non-existence and classification results are known, and no non-trivial examples are k…
View article: A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension
A modular equality for Cameron-Liebler line classes in projective and affine spaces of odd dimension Open
In this article we study Cameron-Liebler line classes in PG$(n,q)$ and AG$(n,q)$, objects also known as boolean degree one functions. A Cameron-Liebler line class $\mathcal{L}$ is known to have a parameter $x$ that depends on the size of $…
View article: An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings
An algebraic approach to Erdős-Ko-Rado sets of flags in spherical buildings Open
In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bound…
View article: An algebraic approach to Erd\H{o}s-Ko-Rado sets of flags in spherical buildings
An algebraic approach to Erd\H{o}s-Ko-Rado sets of flags in spherical buildings Open
In this paper, oppositeness in spherical buildings is used to define an EKR-problem for flags in projective and polar spaces. A novel application of the theory of buildings and Iwahori-Hecke algebras is developed to prove sharp upper bound…
View article: The minimum size of a linear set
The minimum size of a linear set Open
In this paper, we first determine the minimum possible size of an Fq-linear set of rank k in PG(1, q^n). We obtain this result by relating it to the number of directions determined by a linearized polynomial whose domain is restricted to a…
View article: Towards Generic Scalable Parallel Combinatorial Search
Towards Generic Scalable Parallel Combinatorial Search Open
Combinatorial search problems in mathematics, e.g. in finite geometry, are notoriously hard; a state-of-the-art backtracking search algorithm can easily take months to solve a single problem. There is clearly demand for parallel combinator…
View article: On the Smallest Non-Trivial Tight Sets in Hermitian Polar Spaces
On the Smallest Non-Trivial Tight Sets in Hermitian Polar Spaces Open
We show that an $x$-tight set of the Hermitian polar spaces $\mathrm{H}(4,q^2)$ and $\mathrm{H}(6,q^2)$ respectively, is the union of $x$ disjoint generators of the polar space provided that $x$ is small compared to $q$. For $\mathrm{H}(4,…
View article: On Subsets of the Normal Rational Curve
On Subsets of the Normal Rational Curve Open
A normal rational curve of the (k-1) -dimensional projective space over Fq is an arc of size q+1 , since any k points of the curve span the whole space. In this paper, we will prove that if q is odd, then a subset of size 3k-6 of a normal …
View article: FinInG: a package for Finite Incidence Geometry
FinInG: a package for Finite Incidence Geometry Open
FinInG is a package for computation in Finite Incidence Geometry. It provides users with the basic tools to work in various areas of finite geometry from the realms of projective spaces to the flat lands of generalised polygons. The algebr…
View article: Blocking and Double Blocking Sets in Finite Planes
Blocking and Double Blocking Sets in Finite Planes Open
In this paper, by using properties of Baer subplanes, we describe the construction of a minimal blocking set in the Hall plane of order $q^2$ of size $q^2+2q+2$ admitting $1-$, $2-$, $3-$, $4-$, $(q+1)-$ and $(q+2)-$secants. As a corollary…
View article: On subsets of the normal rational curve
On subsets of the normal rational curve Open
A normal rational curve of the $(k-1)$-dimensional projective space over ${\mathbb F}_q$ is an arc of size $q+1$, since any $k$ points of the curve span the whole space. In this article we will prove that if $q$ is odd then a subset of siz…