arXiv subjects

Geertrui Van de Voorde

Publications and source records attributed to Geertrui Van de Voorde.

At least 37 records · Page 2Linked to original sources

arXiv2022

The weight distributions of linear sets in PG(1,q^5)

In this paper, we study the weight distributions of $\mathbb{F}_q$-linear sets in $\mathrm{PG}(1,q^5)$. Our main theorem proves that a linear set $S$ of rank $5$, which is not scattered has the following weight distribution for its points with weight larger than 1: (i) one point of weight $4$ or $5$, (ii) one point of weight $3$ and $0$, $q$, $q^2$ points of weight two, (iii) $s$ points of weight $2$ where $s\in [q-2\sqrt{q}+1,q+2\sqrt{q}+1]\cup\{2q,2q+1,2q+2,3q,3q+1,q^2+1\}$. In particular, there are no $2$-clubs in $\mathrm{PG}(1,q^5)$.

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arXiv2021

Characterising elliptic and hyperbolic hyperplanes of the parabolic quadric \Q(2n,q)

We provide a natural characterisation for the sets of elliptic and hyperbolic hyperplanes of the parabolic quadric Q(2n,q) when q is even. This characterisation is based on the number of elements of these sets through points and codimension 2 spaces and generalises [S. Barwick, A. Hui, and W-A. Jackson. Characterising elliptic solids of Q(4,q), q even. Discrete Math., 343 (6) (2020), 111857] and [S. Barwick, A. Hui, W-A. Jackson, and J. Schillewaert. Characterising hyperbolic solids of Q(4,q), q even. Des. Codes Cryptogr., 88 (1) (2020), 33--39.].

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arXiv2021

The geometric field of linearity of linear sets

If an Fq-linear set LU in a projective space is defined by a vector subspace U which is linear over a proper superfield of Fq, then all of its points have weight at least 2. It is known that the converse of this statement holds for linear sets of rank h in PG(1,q^h) but for linear sets of rank k < h, the converse of this statement is in general no longer true. The first part of this paper studies the relation between the weights of points and the size of a linear set, and introduces the concept of the geometric field of linearity of a linear set. This notion will allow us to show the main theorem, stating that for particular linear sets without points of weight 1, the converse of the above statement still holds as long as we take the geometric field of linearity into account.

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arXiv2021

Quasi-polar spaces

Quasi-polar spaces are sets of points having the same intersection numbers with respect to hyperplanes as classical polar spaces. Non-classical examples of quasi-quadrics have been constructed using a technique called pivoting [5]. We introduce a more general notion of pivoting, called switching, and also extend this notion to Hermitian polar spaces. The main result of this paper studies the switching technique in detail by showing that, for q >= 4, if we modify the points of a hyperplane of a polar space to create a quasi-polar space, the only thing that can be done is pivoting. The cases q = 2 and q = 3 play a special role for parabolic quadrics and are investigated in detail. Furthermore, we give a construction for quasi-polar spaces obtained from pivoting multiple times. Finally, we focus on the case of parabolic quadrics in even characteristic and determine under which hypotheses the existence of a nucleus (which was included in the definition given in [5]) is guaranteed.

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arXiv2020

On linear sets of minimum size

An $\mathbb{F}_q$-linear set of rank $k$ on a projective line $\mathrm{PG}(1,q^h)$, containing at least one point of weight one, has size at least $q^{k-1}+1$ (see [J. De Beule and G. Van De Voorde, The minimum size of a linear set, J. Comb. Theory, Ser: A 164 (2019), 109-124.]). The classical example of such a set is given by a club. In this paper, we construct a broad family of linear sets meeting this lower bound, where we are able to prescribe the weight of the heaviest point to any value between $k/2$ and $k-1$. Our construction extends the known examples of linear sets of size $q^{k-1}+1$ in $\mathrm{PG}(1,q^h)$ constructed for $k=h=4$ [G. Bonoli and O. Polverino, $\mathbb{F}_q$-Linear blocking sets in $\mathrm{PG}(2,q^4)$, Innov. Incidence Geom. 2 (2005), 35--56.] and $k=h$ in [G. Lunardon and O. Polverino. Blocking sets of size $q^t+q^{t-1}+1$. J. Comb. Theory, Ser: A 90 (2000), 148-158.]. We determine the weight distribution of the constructed linear sets and describe them as the projection of a subgeometry. For small $k$, we investigate whether all linear sets of size $q^{k-1}+1$ arise from our construction. Finally, we modify our construction to define linear sets of size $q^{k-1}+q^{k-2}+\ldots+q^{k-l}+1$ in $\mathrm{PG}(l,q)$. This leads to new infinite families of small minimal blocking sets which are not of Rédei type.

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arXiv2020

A note on large Kakeya sets

A Kakeya set $\mathcal{K}$ in an affine plane of order $q$ is the point set covered by a set $\mathcal{L}$ of $q+1$ pairwise non-parallel lines. Large Kakeya sets were studied by Dover and Mellinger; in [6] they showed that Kakeya sets with size at least $q^2-3q+9$ contain a large knot (a point of $\mathcal{K}$ lying on many lines of $\mathcal{L}$). In this paper, we improve on this result by showing that Kakeya set of size at least $\approx q^2-q\sqrt{q}+\frac{3}{2}q$ contain a large knot. Furthermore, we obtain a sharp result for planes of square order containing a Baer subplane.

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arXiv2019

Locally repairable codes with high availability based on generalised quadrangles

Locally Repairable Codes (LRC's) based on generalised quadrangles were introduced by Pamies-Juarez, Hollmann and Oggier in \cite{PaHoOg2013}, and bounds on the repairability and availability were derived. In this paper, we determine the values of the repairability and availability of such LRC's for a large portion of the currently known generalised quadrangles. In order to do so, we determine the minimum weight of the codes of translation generalised quadrangles and characterise the codewords of minimum weight.

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arXiv2019

On the product of elements with prescribed trace

This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\mathrm{Tr}$, for which elements $z$ in $\mathbb{L}$, and $a$, $b$ in $\mathbb{K}$, is it possible to write $z$ as a product $x\cdot y$, where $x,y\in \mathbb{L}$ with $\mathrm{Tr}(x)=a, \mathrm{Tr}(y)=b$? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least $5$. We also have results for arbitrary fields and extensions of degrees $2,3$ or $4$. We then apply our results to the study of PN functions, semifields, irreducible polynomials with prescribed coefficients, and to a problem from finite geometry concerning the existence of certain disjoint linear sets.

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arXiv2019

Translation hyperovals and $\mathbb{F}_2$-linear sets of pseudoregulus type

In this paper, we study translation hyperovals in PG$(2,q^k)$. The main result of this paper characterises the point sets defined by translation hyperovals in the André/Bruck-Bose representation. We show that the affine point sets of translation hyperovals in PG$(2,q^k)$ are precisely those that have a scattered $\mathbb{F}_2$-linear set of pseudoregulus type in PG$(2k-1,q)$ as set of directions. This correspondence is used to generalise the results of Barwick and Jackson who provided a characterisation for translation hyperovals in PG$(2,q^2)$.

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arXiv2018

Rank-metric codes, linear sets, and their duality

In this paper we investigate connections between linear sets and subspaces of linear maps. We give a geometric interpretation of the results of [18, Section 5] on linear sets on a projective line. We extend this to linear sets in arbitrary dimension, giving the connection between two constructions for linear sets defined in [9]. Finally, we then exploit this connection by using the MacWilliams identities to obtain information about the possible weight distribution of a linear set of rank n on a projective line $PG(1, q^n)$.

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arXiv2018

A new lower bound for the size of an affine blocking set

A blocking set in an affine plane is a set of points $B$ such that every line contains at least one point of $B$. The best known lower bound for blocking sets in arbitrary (non-desarguesian) affine planes was derived in the 1980's by Bruen and Silverman. In this note, we improve on this result by showing that a blocking set of an affine plane of order $q$, $q\geq 25$, contains at least $q+\lfloor\sqrt{q}\rfloor+3$ points.

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arXiv2018

The minimum size of a linear set

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 subspace. We then use this result to find a lower bound on the number of points in an Fq- linear set of rank k in PG(2, q^n). In the case k = n, this confirms a conjecture by Sziklai in [9].

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arXiv2017

Elation KM-arcs

In this paper, we study KM-arcs in $PG(2, q)$, the Desarguesian projective plane of order $q$. A KM-arc A of type $t$ is a natural generalisation of a hyperoval: it is a set of $q + t$ points in $PG(2, q)$ such that every line of $PG(2,q)$ meets A in $0,2$ or $t$ points. We study a particular class of KM-arcs, namely, elation KM-arcs. These KM-arcs are highly symmetrical and moreover, many of the known examples are elation KM-arcs. We provide an algebraic framework and show that all elation KM-arcs of type $q/4$ in $PG(2,q)$ are translation KM-arcs. Using a result of [2], this concludes the classification problem for elation KM-arcs of type $q/4$. Furthermore, we construct for all $q = 2^h$, $h > 3$, an infinite family of elation KM-arcs of type $q/8$, and for $q = 2^h$, where $4, 6, 7 | h$ an infinite family of KM-arcs of type $q/16$. Both families contain new examples of KM-arcs.

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arXiv2016

Desarguesian spreads and field reduction for elements of the semilinear group

The goal of this note is to create a sound framework for the interplay between field reduction for finite projective spaces, the general semilinear groups acting on the defining vector spaces and the projective semilinear groups. This approach makes it possible to reprove a result of Dye on the stabiliser in PGL of a Desarguesian spread in a more elementary way, and extend it to PΓL(n, q). Moreover a result of Drudge [5] relating Singer cycles with Desarguesian spreads, as well as a result on subspreads (by Sheekey, Rottey and Van de Voorde [19]) are reproven in a similar elementary way. Finally, we try to use this approach to shed a light on Condition (A) of Csajbok and Zanella, introduced in the study of linear sets [4].

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arXiv2015

Constructing minimal blocking sets using field reduction

We present a construction for minimal blocking sets with respect to $(k-1)$-spaces in $\mathrm{PG}(n-1,q^t)$, the $(n-1)$-dimensional projective space over the finite field $\mathbb{F}_{q^t}$ of order $q^t$. The construction relies on the use of blocking cones in the {\em field reduced} representation of $\mathrm{PG}(n-1,q^t)$, extending the well-known construction of linear blocking sets. This construction is inspired by the construction for minimal blocking sets with respect to the hyperplanes by Mazzocca, Polverino and Storme ({\em the MPS-construction}); we show that for a suitable choice of the blocking cone over a planar blocking set, we obtain larger blocking sets than the ones obtained from planar blocking sets in \cite{pol}. Furthermore we show that every minimal blocking set with respect to the hyperplanes in $\mathrm{PG}PG(n-1,q^t)$ can be obtained by applying field reduction to a minimal blocking set with respect to $(nt-t-1)$-spaces in $\mathrm{PG}(nt-1,q)$. We end by relating these constructions to the linearity conjecture for small minimal blocking sets. We show that if a small minimal blocking set is constructed from the MPS-construction, it is of Rédei-type whereas a small minimal blocking set arises from our cone construction if and only if it is linear.

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arXiv2015

Exending pseudo-arcs in odd characteristic

A {\em pseudo-arc} in $\mathrm{PG}(3n-1,q)$ is a set of $(n-1)$-spaces such that any three of them span the whole space. A pseudo-arc of size $q^n+1$ is a {\em pseudo-oval}. If a pseudo-oval $\mathcal{O}$ is obtained by applying field reduction to a conic in $\mathrm{PG}(2,q^n)$, then $\mathcal{O}$ is called a {\em pseudo-conic}. We first explain the connection of (pseudo-)arcs with Laguerre planes, orthogonal arrays and generalised quadrangles. In particular, we prove that the Ahrens-Szekeres GQ is obtained from a $q$-arc in $\mathrm{PG}(2,q)$ and we extend this construction to that of a GQ of order $(q^n-1,q^n+1)$ from a pseudo-arc of $\mathrm{PG}(3n-1,q)$ of size $q^n$. The main theorem of this paper shows that if $\mathcal{K}$ is a pseudo-arc in $\mathrm{PG}(3n-1,q)$, $q$ odd, of size larger than the size of the second largest complete arc in $\mathrm{PG}(2,q^n)$, where for one element $K_i$ of $\mathcal{K}$, the partial spread $\mathcal{S}=\{K_1,\ldots,K_{i-1},K_{i+1},\ldots,K_{s}\}/K_i$ extends to a Desarguesian spread of $\mathrm{PG}(2n-1,q)$, then $\mathcal{K}$ is contained in a pseudo-conic. The main result of \cite{Casse} also follows from this theorem.

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arXiv2015

A linear set view on KM-arcs

In this paper, we study KM-arcs of type t, i.e. point sets of size q + t in PG(2, q) such that every line contains 0, 2 or t of its points. We use field reduction to give a different point of view on the class of translation arcs. Starting from a particular F2-linear set, called an i-club, we reconstruct the projective triads, the translation hyperovals as well as the translation arcs constructed by Korchmaros-Mazzocca, Gacs-Weiner and Limbupasiriporn. We show the KM-arcs of type q/4 recently constructed by Vandendriessche are translation arcs and fit in this family. Finally, we construct a family of KM-arcs of type q/4. We show that this family, apart from new examples that are not translation KM-arcs, contains all translation KM-arcs of type q/4.

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arXiv2015

Unitals with many Baer secants through a fixed point

We show that a unital $U$ in $\mathrm{PG}(2,q^2)$ containing a point $P$, such that at least $q^2-ε$ of the secant lines through $P$ intersect $U$ in a Baer subline, is an ovoidal Buekenhout-Metz unital (where $ε\approx 2q$ for $q$ even and $ε\approx q^{3/2}/2$ for $q$ odd).

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