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Geertrui Van de Voorde

Publications and source records attributed to Geertrui Van de Voorde.

At least 19 recordsLinked to original sources
arXiv2026

Weak arcs and applications to the DNA-based storage access problem

Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.

math.CO↗
arXiv2026

Intersection numbers of the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3)

In this paper, we study and characterise the natural embedding of the twisted triality hexagon T(q^3,q) in PG(7,q^3). We begin by describing the possible intersections of subspaces of PG(7,q^3) with T(q^3,q). Then, we provide conditions on a set of lines L which ensures that L forms the line set of a naturally embedded twisted triality hexagon. This work follows up on similar results for the split Cayley hexagon by J. A. Thas and H. Van Maldeghem (2008) and F. Ihringer (2014).

math.CO↗
arXiv2025

On the weight distribution of linear sets with complementary weights and related constructions

In this paper, we continue the study of linear sets with complementary weights. We find criteria to determine the set of points of any fixed weight and use this to present particular linear sets with few points of weight more than one. We also present a product-type construction for linear sets of complementary type arising from any linear set, allowing us to control the weight distribution of the obtained linear set. Finally, we use this construction to create linear sets with many different weights, along with point sets of even type with many distinct intersection numbers.

math.CO↗
arXiv2025

A Completion Result for Partial Affine and Inversive Spaces

A partial affine plane of order $n$ is a point-line incidence structure with $n^2$ points and $n$ points on each line, such that every two lines meet in at most one point. In this paper, we show that a partial affine plane of order $n$, $n$ sufficiently large, in which parallelism is an equivalence relation, containing more than $n^2-\sqrt{n}$ lines, can be completed to an affine plane, thus improving the $40$-year old bound of [S. Dow. A completion problem for finite affine planes. Combinatorica, 6:321--325, 1986.] Furthermore, we derive a higher-dimensional result about the completion of $2$-$(n^d,n,1)$-designs, as well as for partial inversive spaces. In particular, we show that a partial $3$-$(n^2+1,n+1,1)$-design for which in every derived structure, parallelism is an equivalence relation, and there are at least $n^2+n-\sqrt{n}$ lines, can be completed to an inversive plane.

math.CO↗
arXiv2025

Subsets of P^4 with no four points on a plane

We describe a new construction of a subset of P^4 with no four points on a plane over any finite field of order q in which 3 is not a square. This set has size 2q + 1, is maximal with respect to inclusion, and is the largest known such set.

math.CO↗
arXiv2025

On the Hermitian Veronesean

The Hermitian Veronesean in $PG(3,q^2)$, given by $\mathcal{V}:=\{ (1,x,x^q,x^{q+1}):x\in\mathbb{F}_q\}\cup\{(0,0,0,1)\}$, is a well-studied rational curve, and forms a {\em special} set of the Hermitian surface $H(3,q^2)$. In this paper, we give two local characterisations of the Hermitian Veronesean, based on sublines and triples of points in perspective.

math.CO↗
arXiv2024

Anzahl theorems for disjoint subspaces generating a non-degenerate subspace II: quadratic forms

In this paper, we solve a classical counting problem for non-degenerate quadratic forms defined on a vector space in odd characteristic; given a subspace $π$, we determine the number of non-singular subspaces that are trivially intersecting with $π$ and span a non-singular subspace with $π$. Lower bounds for the quantity of such pairs where $π$ is non-singular were first studied in `Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)', which was later improved for even-dimensional subspaces in `Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)' and generalised in `Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)'. The explicit formulae, which allow us to give the exact proportion and improve the known lower bounds were derived in the symplectic and Hermitian case in `De Boeck and Van de Voorde (Linear Algebra Appl. 2024)'. This paper deals with the more complicated quadratic case.

math.CO↗
arXiv2024

How to survive the Squid Games using probability theory

In this paper, we consider how probability theory can be used to determine the survival strategy in two of the ``Squid Game" and ``Squid Game: The Challenge" challenges: the Hopscotch and the Warships. We show how Hopscotch can be easily tackled with the knowledge of the binomial distribution, taught in introductory statistics courses, while Warships is a much more complex problem, which can be tackled at different levels.

stat.OT↗
arXiv2024

Anzahl theorems for trivially intersecting subspaces generating a non-singular subspace I: symplectic and hermitian forms

In this paper, we solve a classical counting problem for non-degenerate forms of symplectic and hermitian type defined on a vector space: given a subspace $π$, we find the number of non-singular subspaces that are trivially intersecting with $π$ and span a non-singular subspace with $π$. Lower bounds for the quantity of such pairs where $π$ is non-singular were first studied in ``Glasby, Niemeyer, Praeger (Finite Fields Appl., 2022)'', which was later improved in ``Glasby, Ihringer, Mattheus (Des. Codes Cryptogr., 2023)'' and generalised in ``Glasby, Niemeyer, Praeger (Linear Algebra Appl., 2022)''. In this paper, we derive explicit formulae, which allow us to give the exact proportion and improve the known lower bounds.

math.CO↗
arXiv2024

Triangle-free graphs with diameter 2

There are finitely many graphs with diameter $2$ and girth 5. What if the girth 5 assumption is relaxed? Apart from stars, are there finitely many triangle-free graphs with diameter $2$ and no $K_{2,3}$ subgraph? This question is related to the existence of triangle-free strongly regular graphs, but allowing for a range of co-degrees gives the question a more extremal flavour. More generally, for fixed $s$ and $t$, are there infinitely many twin-free triangle-free $K_{s,t}$-free graphs with diameter 2? This paper presents partial results regarding these questions, including computational results, potential Cayley-graph and probabilistic constructions.

math.CO↗
arXiv2024

Upper bounds for the number of substructures in finite geometries from the container method

We use techniques from algebraic and extremal combinatorics to derive upper bounds on the number of independent sets in several (hyper)graphs arising from finite geometry. In this way, we obtain asymptotically sharp upper bounds for partial ovoids and EKR-sets of flags in polar spaces, line spreads in $\mathrm{PG}(2r-1,q)$ and plane spreads in $\mathrm{PG}(5,q)$, and caps in $\mathrm{PG}(3,q)$. The latter result extends work due to Roche-Newton and Warren and Bhowmick and Roche-Newton. Finally, we investigate caps in $p$-random subsets of $\mathrm{PG}(r,q)$, which parallels recent work for arcs in projective planes by Bhowmick and Roche-Newton, and by Roche-Newton and Warren, and arcs in projective spaces by Chen, Liu, Nie and Zeng.

math.CO↗
arXiv2024

Characterising ovoidal cones by their intersection numbers

In this paper, we characterise ovoidal cones by their intersection numbers. We first show that a set of points of $\mathrm{PG}(4,q)$ which intersects planes in $1$, $q+1$ or $2q+1$ points is either an ovoidal cone or a parabolic quadric, unless $q=3$, in which case also a sporadic example with automorphism group $M_{11}$ exists. We then show that a set of points of $\mathrm{PG}(4,q)$ which blocks all planes and intersects solids in $q+1$, $q^2+1$ or $q^2+q+1$ points is a plane or an ovoidal cone, and determine all examples that arise when the blocking condition is omitted.

math.CO↗
arXiv2024

A note on strong blocking sets and higgledy-piggledy sets of lines

This paper studies {\em strong blocking sets} in the $N$-dimensional finite projective space $\mathrm{PG}(N,q)$. We first show that certain unions of blocking sets cannot form strong blocking sets, which leads to a new lower bound on the size of a strong blocking set in $\mathrm{PG}(N,q)$. Our second main result shows that, for $q>\frac{2}{ln(2)}(N+1)$, there exists a subset of $2N-2$ lines of a Desarguesian line spread in $\mathrm{PG}(N,q)$, $N$ odd, in {\em higgledy-piggledy arrangement}; thus giving rise to a strong blocking set of size $(2N-2)(q+1)$.

math.CO↗
arXiv2023

On Bruen chains

It is known that a Bruen chain of the three-dimensional projective space $\mathrm{PG}(3,q)$ exists for every odd prime power $q$ at most $37$, except for $q=29$. It was shown by Cardinali et. al (2005) that Bruen chains do not exist for $41\le q\leq 49$. We develop a model, based on finite fields, which allows us to extend this result to $41\leqslant q \leqslant 97$, thereby adding more evidence to the conjecture that Bruen chains do not exist for $q>37$. Furthermore, we show that Bruen chains can be realised precisely as the $(q+1)/2$-cliques of a two related, yet distinct, undirected simple graphs.

math.CO↗
arXiv2022

A higgledy-piggledy set of planes based on the ABB-representation of linear sets

In this paper, we investigate the André/Bruck-Bose representation of certain $\mathbb{F}_q$-linear sets contained in a line of $\text{PG}(2,q^t)$. We show that scattered $\mathbb{F}_q$-linear sets of rank $3$ in $\text{PG}(1,q^3)$ correspond to particular hyperbolic quadrics and that $\mathbb{F}_q$-linear clubs in $\text{PG}(1,q^t)$ are linked to subspaces of a certain $2$-design based on normal rational curves; this design extends the notion of a circumscribed bundle of conics. Finally, we use these results to construct optimal higgledy-piggledy sets of planes in $\text{PG}(5,q)$.

math.CO↗
arXiv2022

Embedded antipodal planes and the minimum weight of the dual code of points and lines in projective planes of order $p^2$

The minimum weight of the code generated by the incidence matrix of points versus lines in a projective plane has been known for over 50 years. Surprisingly, finding the minimum weight of the dual code of projective planes of non-prime order is still an open problem, even in the Desarguesian case. In this paper, we focus on the case of projective planes of order $p^2$, where $p$ is prime, and we link the existence of small weight code words in the dual code to the existence of embedded subplanes and {\em antipodal planes}. In the Desarguesian case, we can exclude such code words by showing a more general result that no antipodal plane of order at least 3 can be embedded in a Desarguesian projective plane. Furthermore, we use combinatorial arguments to rule out the existence of code words in the dual code of points and lines of an arbitrary projective plane of order $p^2$, $p$ prime, of weight at most $2p^2-2p+4$ using more than two symbols. In particular, this leads to the result that the dual code of the Desarguesian projective plane $\mathrm{PG}(2,p^2)$, $p\geq 5$, has minimum weight at least $2p^2-2p+5$.

math.CO↗
arXiv2022

On the Equivalence, Stabilisers, and Feet of Buekenhout-Tits Unitals

This paper addresses a number of problems concerning Buekenhout-Tits unitals in $PG(2,q^2)$, where $q = 2^{e+1}$ and $e \geq 1$. We show that all Buekenhout-Tits unitals are $PGL$-equivalent (addressing an open problem in [S. Barwick and G. L. Ebert. Unitals in projective planes. Springer Monographs in Mathematics. Springer, New York, 2008.]), explicitly describe their $PΓL$-stabiliser (expanding Ebert's work in [G.L. Ebert. Buekenhout-Tits unitals. J. Algebraic. Combin. 6.2 (1997), 133-140], and show that lines meet the feet of points no on $\ell_\infty$ in at most four points. Finally, we show that feet of points not on $\ell_\infty$ are not always a $\{0,1,2,4\}$-set, in contrast to what happens for Buekenhout-Metz unitals [N. Abarzúa, R. Pomareda, and O. Vega. Feet in orthogonal-Buekenhout-Metz unitals. Adv. Geom. 18.2 (2018), 229-236].

math.CO↗