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Jozefien D'haeseleer

Publications and source records attributed to Jozefien D'haeseleer.

At least 19 recordsLinked to original sources
arXiv2026

A note on the classification of classical distance-regular graphs of negative type and the non-existence of hemisystems

DISCLAIMER: Due to an error in the literature, we cannot be sure that the conclusions drawn in this paper are correct. The goal of this note is to connect some interesting results in the literature on algebraic graph theory and finite geometry. In 1999, Weng gave an almost complete classification of classical distance-regular graphs of negative type with diameter at least 4. He proved that these graphs are either dual polar graphs of Hermitian polar spaces, Hermitian forms graphs, or fall into a last category. It was recently proved by Yian et al. that the latter category does not exist when the diameter equals 3, which by Weng's results proves that they do not exist for bigger diameter. Using a result of Vanhove, this proves that certain hemisystems in Hermitian polar spaces cannot exist.

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arXiv2026

Generalized Hamming weights of additive codes and geometric counterparts

We consider the geometric problem of determining the maximum number $n_q(r,h,f;s)$ of $(h-1)$-spaces in the projective space $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ does contain at most $s$ elements. In coding theory terms we are dealing with additive codes that have a large $f$th generalized Hamming weight. We also consider the dual problem of the minimum number $b_q(r,h,f;s)$ of $(h-1)$-spaces in $\operatorname{PG}(r-1,q)$ such that each subspace of codimension $f$ contains at least $s$ elements. We fully determine $b_2(5,2,2;s)$ as a function of $s$. We additionally give bounds and constructions for other parameters. For the computational results we partially use extensive integer linear programming computations.

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arXiv2026

Chromatic Number of Grassmann Graphs and MRD codes

In this paper we investigate the chromatic number of the Grassmann graphs and of their powers, denoted $J_q(n,m,t)$. In this graph, the vertices correspond to the $m$-dimensional subspaces in $\mathbb{F}_q^n$ and two vertices are adjacent if the corresponding subspaces intersect in a subspace of dimension at least $t$. By generalizing the lifting technique of Silva, Kötter and Kschischang, we use \emph{maximum rank distance (MRD)} codes to establish that $χ(J_q(n, m, t)) \leq (1 +o(1))n^{m-t}q^{(n-m)(m-t)})$ when $n \geq 2m$. Given that $J_q(n, m, t)$ is isomorphic to $J_q(n,n-m,n-2m+t)$, this establishes a new upper bound on $J_q(n, m, t)$ for any valid choice of parameters. Furthermore, we observe that in the regime that $n, m $, and $t$ are fixed, our bound is asymptotically tight, implying that $ χ(J_q(n, m, t)) = Θ(q^{(m-t)\max(n-m, m)}). $

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arXiv2025

On the Chromatic Number of Grassmann Graphs

In this paper we study the chromatic number of the Grassmann graphs $J_q(n, m)$. We show that $\binom{n-m+1}{1}_q \leq χ(J_q(n, m)) \leq \binom{n}{1}_q$, which is analogous to the best-known bounds for the chromatic number of the Johnson graphs $J(n, m)$. When $m = 2$, determining $χ(J_q(n, 2))$ is equivalent to determining the smallest number of partial line parallelisms that one can partition the lines of PG$(n-1, q)$ into. We survey known results about line parallelisms and their implications for $χ(J_q(n, 2))$. Finally, we prove that when $q$ is any power of two, and $n$ is any even integer, then $χ(J_q(n, 2)) < 2\binom{n-1}{1}_q$.

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arXiv2025

Cameron-Liebler sets of generators in the Klein quadric $Q^+(5,q)$

We investigate Cameron-Liebler sets of planes in the Klein quadric $Q^+(5,q)$ in PG$(5,q)$. We prove that there are many examples of such Cameron-Liebler sets of planes in the Klein quadric. More specifically, we provide an incomplete list of examples of such Cameron-Liebler sets of planes. By doing so, we also provide some characteristic results regarding these sets in connection with the Klein quadric. These results contribute to an open conjecture posed in [21].

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arXiv2024

Blocking sets, minimal codes and trifferent codes

We prove new upper bounds on the smallest size of affine blocking sets, that is, sets of points in a finite affine space that intersect every affine subspace of a fixed codimension. We show an equivalence between affine blocking sets with respect to codimension-$2$ subspaces that are generated by taking a union of lines through the origin, and strong blocking sets in the corresponding projective space, which in turn are equivalent to minimal codes. Using this equivalence, we improve the current best upper bounds on the smallest size of a strong blocking set in finite projective spaces over fields of size at least $3$. Furthermore, using coding theoretic techniques, we improve the current best lower bounds on strong blocking set. Our main motivation for these new bounds is their application to trifferent codes, which are sets of ternary codes of length $n$ with the property that for any three distinct codewords there is a coordinate where they all have distinct values. Over the finite field $\mathbb{F}_3$, we prove that minimal codes are equivalent to linear trifferent codes. Using this equivalence, we show that any linear trifferent code of length $n$ has size at most $3^{n/4.55}$, improving the recent upper bound of Pohoata and Zakharov. Moreover, we show the existence of linear trifferent codes of length $n$ and size at least $\frac{1}{3}\left( 9/5 \right)^{n/4}$, thus (asymptotically) matching the best lower bound on trifferent codes. We also give explicit constructions of affine blocking sets with respect to codimension-$2$ subspaces that are a constant factor bigger than the best known lower bound. By restricting to~$\mathbb{F}_3$, we obtain linear trifferent codes of size at least $3^{23n/312}$, improving the current best explicit construction that has size $3^{n/112}$.

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arXiv2024

A common generalization of hypercube partitions and ovoids in polar spaces

We investigate what we call generalized ovoids, that is families of totally isotropic subspaces of finite classical polar spaces such that each maximal totally isotropic subspace contains precisely one member of that family. This is a generalization of ovoids in polar spaces as well as the natural $q$-analog of a subcube partition of the hypercube (which can be seen as a polar space with $q=1$). Our main result proves that a generalized ovoid of $k$-spaces in polar spaces of large rank does not exist. More precisely, for $q=p^h$, $p$ prime, and some positive integer $k$, a generalized ovoid of $k$-spaces in a polar space $\mathcal{P}$ with rank $r \geq r_0(k, p)$ in a vector space $V(n,q)$ does not exist.

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arXiv2023

Regular ovoids and Cameron-Liebler sets of generators in polar spaces

Cameron-Liebler sets of generators in polar spaces were introduced a few years ago as natural generalisations of the Cameron-Liebler sets of subspaces in projective spaces. In this article we present the first two constructions of non-trivial Cameron-Liebler sets of generators in polar spaces. Also regular m-ovoids of k-spaces are introduced as a generalization of m-ovoids of polar spaces. They are used in one of the aforementioned constructions of Cameron-Liebler sets.

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arXiv2023

Cameron-Liebler sets in permutation groups

Consider a group $G$ acting on a set $Ω$, the vector $v_{a,b}$ is a vector with the entries indexed by the elements of $G$, and the $g$-entry is 1 if $g$ maps $a$ to $b$, and zero otherwise. A $(G,Ω)$-Cameron-Liebler set is a subset of $G$, whose indicator function is a linear combination of elements in $\{v_{a, b}\ :\ a, b \in Ω\}$. We investigate Cameron-Liebler sets in permutation groups, with a focus on constructions of Cameron-Liebler sets for 2-transitive groups.

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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 $\text{PG}(5,q)$.

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arXiv2022

Degree 2 Boolean Functions on Grassmann Graphs

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 known for $n \geq 5$. This paper focusses on providing a list of examples on the case $d=2$ in general dimension and in particular for $(n, k)=(6,3)$ and $(n,k) = (8, 4)$. We also discuss connections to the analysis of Boolean functions, regular sets/equitable bipartitions/perfect 2-colorings in graphs, $q$-analogs of designs, and permutation groups. In particular, this represents a natural generalization of Cameron-Liebler line classes.

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arXiv2022

On the Chromatic Number of some generalized Kneser Graphs

We determine the chromatic number of the Kneser graph qΓ_{7,{3,4}} of flags of vectorial type {3, 4} of a rank 7 vector space over the finite field GF(q) for large q and describe the colorings that attain the bound. This result relies heavily, not only on the independence number, but also on the structure of all large independent sets. Furthermore, our proof is more general in the following sense: it provides the chromatic number of the Kneser graphs qΓ_{2d+1,{d,d+1}} of flags of vectorial type {d, d+1} of a rank 2d+1 vector space over GF(q) for large q as long as the large independent sets of the graphs are only the ones that are known.

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arXiv2021

On the sunflower bound for $k$-spaces, pairwise intersecting in a point

A $t$-intersecting constant dimension subspace code $C$ is a set of $k$-dimensional subspaces in a projective space PG(n,q), where distinct subspaces intersect in a $t$-dimensional subspace. A classical example of such a code is the sunflower, where all subspaces pass through the same $t$-space. The sunflower bound states that such a code is a sunflower if $|C| > \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right)^2 + \left( \frac {q^{k + 1} - q^{t + 1}}{q - 1} \right) + 1$. In this article we will look at the case $t=0$ and we will improve this bound for $q\geq 9$: a set $\mathcal{S}$ of $k$-spaces in PG(n,q), $q\geq 9$, pairwise intersecting in a point is a sunflower if $|\mathcal{S}|> \left(\frac{2}{\sqrt[6]{q}}+\frac{4}{\sqrt[3]{q}}-\frac{5}{\sqrt{q}}\right)\left(\frac {q^{k + 1} - 1}{q - 1}\right)^2$.

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arXiv2021

Cameron-Liebler line classes in AG(3,q)

The study of Cameron-Liebler line classes in PG($3,q$) arose from classifying specific collineation subgroups of PG($3,q$). Recently, these line classes were considered in new settings. In this point of view, we will generalize the concept of Cameron-Liebler line classes to AG($3,q$). In this article we define Cameron-Liebler line classes using the constant intersection property towards line spreads. The interesting fact about this generalization is the link these line classes have with Cameron-Liebler line classes in PG($3,q$). Next to giving this link, we will also give some equivalent ways to consider Cameron-Liebler line classes in AG($3,q$), some classification results and an example based on the example found in [3] and [6].

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arXiv2021

Cameron-Liebler $k$-sets in $\text{AG}(n,q)$

We study Cameron-Liebler $k$-sets in the affine geometry, so sets of $k$-spaces in $\text{AG}(n, q)$. This generalizes research on Cameron-Liebler $k$-sets in the projective geometry $\text{PG}(n, q)$. Note that in algebraic combinatorics, Cameron-Liebler $k$-sets of $\text{AG}(n, q)$ correspond to certain equitable bipartitions of the Association scheme of $k$-spaces in $\text{AG}(n, q)$, while in the analysis of Boolean functions, they correspond to Boolean degree $1$ functions of $\text{AG}(n, q)$. We define Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ by intersection properties with $k$-spreads and show the equivalence of several definitions. In particular, we investigate the relationship between Cameron-Liebler $k$-sets in $\text{AG}(n, q)$ and $\text{PG}(n, q)$. As a by-product, we calculate the character table of the association scheme of affine lines. Furthermore, we characterize the smallest examples of Cameron-Liebler $k$-sets. This paper focuses on $\text{AG}(n, q)$ for $n > 3$, while the case for Cameron-Liebler line classes in $\text{AG}(3, q)$ was already treated separately.

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arXiv2020

Cameron-Liebler sets of k-spaces in PG(n,q)

Cameron-Liebler sets of k-spaces were introduced recently by Y. Filmus and F. Ihringer. We list several equivalent definitions for these Cameron-Liebler sets, by making a generalization of known results about Cameron-Liebler line sets in PG(n, q) and Cameron-Liebler sets of k-spaces in PG(2k + 1, q). We also present a classification result.

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arXiv2020

Hilton-Milner results in projective and affine spaces

In this article, we analyse maximal sets of $k$-spaces, in PG(n,q) and AG(n,q), $n>2k+t+2$, that pairwise meet in at least a $t$-space. It is known that for both PG(n,q) and AG(n,q), the largest example is a $t$-pencil, i.e. the set of all $k$-spaces containing a fixed $t$-space. In this paper, we analyse the structure of the second largest maximal example in both PG(n,q) and AG(n,q).

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