arXiv subjects

Sam Adriaensen

Publications and source records attributed to Sam Adriaensen.

25 records · Page 2Linked to original sources

arXiv2022

Incidence-free sets and edge domination in incidence graphs

A set of edges $Γ$ of a graph $G$ is an edge dominating set if every edge of $G$ intersects at least one edge of $Γ$, and the edge domination number $γ_e(G)$ is the smallest size of an edge dominating set. Expanding on work of Laskar and Wallis, we study $γ_e(G)$ for graphs $G$ which are the incidence graph of some incidence structure $D$, with an emphasis on the case when $D$ is a symmetric design. In particular, we show in this latter case that determining $γ_e(G)$ is equivalent to determining the largest size of certain incidence-free sets of $D$. Throughout, we employ a variety of combinatorial, probabilistic and geometric techniques, supplemented with tools from spectral graph theory.

math.CO↗

arXiv2022

Cones from maximum $h$-scattered linear sets and a stability result

This paper mainly focuses on cones whose basis is a maximum $h$-scattered linear set. We start by investigating the intersection sizes of such cones with the hyperplanes. Then we analyze two constructions of point sets with few intersection sizes with the hyperplanes. In particular, the second one extends the construction of translation KM-arcs in projective spaces, having as part at infinity a cone with basis a maximum $h$-scattered linear set. As an instance of the second construction we obtain cylinders with a hyperoval as basis, which we call hypercylinders, for which we are able to provide a stability result. The main motivation for these problems is related to the connections with both Hamming and rank distance codes. Indeed, we are able to construct codes with few weights and to provide a stability result for the codes associated with hypercylinders.

math.CO↗

arXiv2022

On additive MDS codes with linear projections

We support some evidence that a long additive MDS code over a finite field must be equivalent to a linear code. More precisely, let $C$ be an $\mathbb F_q$-linear $(n,q^{hk},n-k+1)_{q^h}$ MDS code over $\mathbb F_{q^h}$. If $k=3$, $h \in \{2,3\}$, $n > \max \{q^{h-1},h q -1\} + 3$, and $C$ has three coordinates from which its projections are equivalent to linear codes, we prove that $C$ itself is equivalent to a linear code. If $k>3$, $n > q+k$, and there are two disjoint subsets of coordinates whose combined size is at most $k-2$ from which the projections of $C$ are equivalent to linear codes, we prove that $C$ is equivalent to a code which is linear over a larger field than $\mathbb F_q$.

cs.IT↗

arXiv2022

Stability of Erdős-Ko-Rado Theorems in Circle Geometries

Circle geometries are incidence structures that capture the geometry of circles on spheres, cones and hyperboloids in 3-dimensional space. In a previous paper, the author characterised the largest intersecting families in finite ovoidal circle geometries, except for Möbius planes of odd order. In this paper we show that also in these Möbius planes, if the order is greater than 3, the largest intersecting families are the sets of circles through a fixed point. We show the same result in the only known family of finite non-ovoidal circle geometries. Using the same techniques, we show a stability result on large intersecting families in all ovoidal circle geometries. More specifically, we prove that an intersecting family $\mathcal F$ in one of the known finite circle geometries of order $q$, with $|\mathcal F| \geq \frac 1 {\sqrt2} q^2 + 2 \sqrt 2 q + 8$, must consist of circles through a common point, or through a common nucleus in case of a Laguerre plane of even order.

math.CO↗

arXiv2021

Erdős-Ko-Rado theorems for ovoidal circle geometries and polynomials over finite fields

In this paper we investigate Erdős-Ko-Rado theorems in ovoidal circle geometries. We prove that in Möbius planes of even order greater than 2, and ovoidal Laguerre planes of odd order, the largest families of circles which pairwise intersect in at least one point, consist of all circles through a fixed point. In ovoidal Laguerre planes of even order, a similar result holds, but there is one other type of largest family of pairwise intersecting circles. As a corollary, we prove that the largest families of polynomials over $\mathbb F_q$ of degree at most $k$, with $2 \leq k < q$, which pairwise take the same value on at least one point, consist of all polynomials $f$ of degree at most $k$ such that $f(x) = y$ for some fixed $x$ and $y$ in $\mathbb F_q$. We also discuss this problem for ovoidal Minkowski planes, and we investigate the largest families of circles pairwise intersecting in two points in circle geometries.

math.CO↗

arXiv2020

Small Weight Code Words of Projective Geometric Codes

We investigate small weight code words of the $p$-ary linear code $\mathcal C_{j,k}(n,q)$ generated by the incidence matrix of $k$-spaces and $j$-spaces of PG$(n,q)$ and its dual, with $q$ a prime power and $0 \leq j < k < n$. Firstly, we prove that all code words of $\mathcal C_{j,k}(n,q)$ up to weight $\left(3 - \mathcal{O}\left(\frac 1 q \right) \right) \genfrac{[}{]}{0pt}{}{k+1}{j+1}_q$ are linear combinations of at most two $k$-spaces (i.e. two rows of the incidence matrix). As for the dual code $\mathcal C_{j,k}(n,q)^\perp$, we manage to reduce both problems of determining its minimum weight (1) and characterising its minimum weight code words (2) to the case $\mathcal C_{0,1}(n,q)^\perp$. This implies the solution to both problem (1) and (2) if $q$ is prime and the solution to problem (1) if $q$ is even.

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↗