arXiv subjects

Leo Storme

Publications and source records attributed to Leo Storme.

At least 19 recordsLinked to original sources

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].

math.CO↗

arXiv2024

The minimum weight of the code of intersecting lines in ${\rm PG}(3,q)$

We characterise the minimum weight codewords of the $p$-ary linear code of intersecting lines in ${\rm PG}(3,q)$, $q=p^h$, $q\geq19$, $p$ prime, $h\geq 1$. If $q$ is even, the minimum weight equals $q^3+q^2+q+1$. If $q$ is odd, the minimum weight equals $q^3+2q^2+q+1$. For $q$ even, we also characterise the codewords of second smallest weight.

math.CO↗

arXiv2024

On two non-existence results for Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$

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 of this collection. One of the main research questions remains the (non-)existence of Cameron-Liebler $k$-sets with parameter $x$. This paper improves two non-existence results. First we show that the parameter of a non-trivial Cameron-Liebler $k$-set in $\mathrm{PG}(n,q)$ should be larger than $q^{n-\frac{5k}{2}-1}$, which is an improvement of an earlier known lower bound. Secondly, we prove a modular equality on the parameter $x$ of Cameron-Liebler $k$-sets in $\mathrm{PG}(n,q)$ with $x<\frac{q^{n-k}-1}{q^{k+1}-1}$, $n\geq 2k+1$, $n-k+1\geq 7$ and $n-k$ even. In the affine case we show a similar result for $n-k+1\geq 3$ and $n-k$ even. This is a generalization of earlier known modular equalities in the projective and affine case.

math.CO↗

arXiv2021

On sets of subspaces with two intersection dimensions and a geometrical junta bound

In this article, constant dimension subspace codes whose codewords have subspace distance in a prescribed set of integers, are considered. The easiest example of such an object is a {\it junta}; i.e. a subspace code in which all codewords go through a common subspace. We focus on the case when only two intersection values for the codewords, are assigned. In such a case we determine an upper bound for the dimension of the vector space spanned by the elements of a non-junta code. In addition, if the two intersection values are consecutive, we prove that such a bound is tight, and classify the examples attaining the largest possible dimension as one of four infinite families.

math.CO↗

arXiv2021

Cameron-Liebler k-sets in subspaces and non-existence conditions

In this article we generalize the concepts that were used in the PhD thesis of Drudge to classify Cameron-Liebler line classes in PG$(n,q), n\geq 3$, to Cameron-Liebler sets of $k$-spaces in PG$(n,q)$ and AG$(n,q)$. In his PhD thesis, Drudge proved that every Cameron-Liebler line class in PG$(n,q)$ intersects every $3$-dimensional subspace in a Cameron-Liebler line class in that subspace. We are using the generalization of this result for sets of $k$-spaces in PG$(n,q)$ and AG$(n,q)$. Together with a basic counting argument this gives a very strong non-existence condition, $n\geq 3k+3$. This condition can also be improved for $k$-sets in AG$(n,q)$, with $n\geq 2k+2$.

math.CO↗

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].

math.CO↗

arXiv2020

Maximal sets of $k$-spaces pairwise intersecting in at least a $(k-2)$-space

In this paper, we analyze the structure of maximal sets of $k$-dimensional spaces in $\mathrm{PG}(n,q)$ pairwise intersecting in at least a $(k-2)$-dimensional space, for $3 \leq k\leq n-2$. We give an overview of the largest examples of these sets with size more than $f(k,q)=\max\{3q^4+6q^3+5q^2+q+1,θ_{k+1}+q^4+2q^3+3q^2\}$.

math.CO↗

arXiv2020

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.

math.CO↗

arXiv2019

Small weight code words arising from the incidence of points and hyperplanes in PG($\boldsymbol{n,q}$)

Let $C_{n-1}(n,q)$ be the code arising from the incidence of points and hyperplanes in the Desarguesian projective space PG($n,q$). Recently, Polverino and Zullo proved that within this code, all non-zero code words of weight at most $2q^{n-1}$ are scalar multiples of either the incidence vector of one hyperplane, or the difference of the incidence vectors of two distinct hyperplanes. We improve this result, proving that when $q>17$ and $q\notin\{25,27,29,31,32,49,121\}$, all code words of weight at most $(4q-\sqrt{8q}-\frac{33}{2})q^{n-2}$ are linear combinations of incidence vectors of hyperplanes through a fixed $(n-3)$-space. Depending on the omitted value for $q$, we can lower the bound on the weight of $c$ to obtain the same results.

math.CO↗

arXiv2019

Cameron-Liebler sets of generators in finite classical polar spaces

Cameron-Liebler sets were originally defined as collections of lines (`line classes') in $\mathrm{PG}(3,q)$ sharing certain properties with line classes of symmetric tactical decompositions. While there are many equivalent characterisations, these objects are defined as sets of lines whose characteristic vector lies in the image of the transpose of the point-line incidence matrix of $\mathrm{PG}(3,q)$, and so combinatorially they behave like a union of pairwise disjoint point-pencils. Recently, the concept of a Cameron-Liebler set has been generalised to several other settings. In this article we introduce Cameron-Liebler sets of generators in finite classical polar spaces. For each of the polar spaces we give a list of characterisations that mirrors those for Cameron-Liebler line sets, and also prove some classification results.

math.CO↗

arXiv2018

Optimal subspace codes in ${\rm PG}(4,q)$

We investigate subspace codes whose codewords are subspaces of ${\rm PG}(4,q)$ having non-constant dimension. In particular, examples of optimal mixed-dimension subspace codes are provided, showing that ${\cal A}_q(5,3) = 2(q^3+1)$.

math.CO↗

arXiv2017

On the independence number of graphs related to a polarity

We investigate the independence number of two graphs constructed from a polarity of $\mathrm{PG}(2,q)$. For the first graph under consideration, the Erdős-Rényi graph $ER_q$, we provide an improvement on the known lower bounds on its independence number. In the second part of the paper we consider the Erdős-Rényi hypergraph of triangles $\mathcal{H}_q$. We determine the exact magnitude of the independence number of $\mathcal{H}_q$, $q$ even. This solves a problem posed by Mubayi and Williford.

math.CO↗

arXiv2016

Identifying codes in vertex-transitive graphs and strongly regular graphs

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V|)+1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that reach both bounds. We exhibit infinite families of vertex-transitive graphs with integer and fractional identifying codes of order |V|^a with a in {1/4,1/3,2/5}. These families are generalized quadrangles (strongly regular graphs based on finite geometries). They also provide examples for metric dimension of graphs.

math.CO↗

arXiv2016

The Cameron-Liebler problem for sets

Cameron-Liebler line classes and Cameron-Liebler k-classes in PG(2k+1,q) are currently receiving a lot of attention. Links with the Erdős-Ko-Rado results in finite projective spaces occurred. We introduce here in this article the similar problem on Cameron-Liebler classes of sets, and solve this problem completely, by making links to the classical Erdős-Ko-Rado result on sets. We also present a characterisation theorem for the Cameron-Liebler classes of sets.

math.CO↗

arXiv2013

Generalised Veroneseans

In \cite{ThasHVM}, a characterization of the finite quadric Veronesean $\mathcal{V}_{n}^{2^{n}}$ by means of properties of the set of its tangent spaces is proved. These tangent spaces form a {\em regular generalised dual arc}. We prove an extension result for regular generalised dual arcs. To motivate our research, we show how they are used to construct a large class of secret sharing schemes.

math.MG↗

arXiv2012

Partial covers of PG(n,q)

In this paper, we show that a set of q+a hyperplanes, q>13, a<(q-10)/4, that does not cover PG(n,q), does not cover at least q^(n-1)-aq^(n-2) points, and show that this lower bound is sharp. If the number of non- covered points is at most q^(n-1), then we show that all non-covered points are contained in one hyperplane. Finally, using a recent result of Blokhuis, Brouwer, and Szonyi [3], we remark that the bound on a for which these results are valid can be improved to a<(q-2)/3 and that this upper bound on a is sharp

math.CO↗

arXiv2012

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.

math.CO↗

arXiv2012

On the code generated by the incidence matrix of points and hyperplanes in PG(n,q) and its dual

In this paper, we study the p-ary linear code C(PG(n, q)), q = p^h, p prime, h >= 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n, q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n, q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that find- ing the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2, q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar's lower bound [12].

math.CO↗