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Michel Lavrauw

Publications and source records attributed to Michel Lavrauw.

At least 37 records · Page 2Linked to original sources
arXiv2019

Arcs and tensors

To an arc $\mathcal{A}$ of $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ we associate a tensor in $\langle ν_{k,t}(\mathcal{A})\rangle^{\otimes k-1}$, where $ν_{k,t}$ denotes the Veronese map of degree $t$ defined on $\mathrm{PG}(k-1,q)$. As a corollary we prove that for each arc $\mathcal{A}$ in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$, which is not contained in a hypersurface of degree $t$, there exists a polynomial $F(Y_1,\ldots,Y_{k-1})$ (in $k(k-1)$ variables) where $Y_j=(X_{j1},\ldots,X_{jk})$, which is homogeneous of degree $t$ in each of the $k$-tuples of variables $Y_j$, which upon evaluation at any $(k-2)$-subset $S$ of the arc $\mathcal{A}$ gives a form of degree $t$ on $\mathrm{PG}(k-1,q)$ whose zero locus is the tangent hypersurface of $\mathcal{A}$ at $S$, i.e. the union of the tangent hyperplanes of $\mathcal{A}$ at $S$. This generalises the equivalent result for planar arcs ($k=3$), proven in \cite{BaLa2018}, to arcs in projective spaces of arbitrary dimension. A slightly weaker result is obtained for arcs in $\mathrm{PG}(k-1,q)$ of size $q+k-1-t$ which are contained in a hypersurface of degree $t$. We also include a new proof of the Segre-Blokhuis-Bruen-Thas hypersurface associated to an arc of hyperplanes in $\mathrm{PG}(k-1,q)$.

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arXiv2018

Planar arcs

Let $p$ denote the characteristic of ${\mathbb F}_q$, the finite field with $q$ elements. We prove that if $q$ is odd then an arc of size $q+2-t$ in the projective plane over ${\mathbb F}_q$, which is not contained in a conic, is contained in the intersection of two curves, which do not share a common component, and have degree at most $t+p^{\lfloor \log_p t \rfloor}$, provided a certain technical condition on $t$ is satisfied. This implies that if $q$ is odd then an arc of size at least $q-\sqrt{q}+\sqrt{q}/p+3$ is contained in a conic if $q$ is square and an arc of size at least $q-\sqrt{q}+\frac{7}{2}$ is contained in a conic if $q$ is prime. This is of particular interest in the case that $q$ is an odd square, since then there are examples of arcs, not contained in a conic, of size $q-\sqrt{q}+1$, and it has long been conjectured that if $q \neq 9$ is an odd square then any larger arc is contained in a conic. These bounds improve on previously known bounds when $q$ is an odd square and for primes less than $1783$. The previously known bounds, obtained by Segre \cite{Segre1967}, Hirschfeld and Korchmáros \cite{HK1996} \cite{HK1998}, and Voloch \cite{Voloch1990b} \cite{Voloch1991}, rely on results on the number of points on algebraic curves over finite fields, in particular the Hasse-Weil theorem and the Stöhr-Voloch theorem, and are based on Segre's idea to associate an algebraic curve in the dual plane containing the tangents to an arc. In this paper we do not rely on such theorems, but use a new approach starting from a scaled coordinate-free version of Segre's lemma of tangents. Arcs in the projective plane over ${\mathbb F}_q$ of size $q$ and $q+1$, $q$ odd, were classified by Segre \cite{Segre1955b} in 1955. In this article, we complete the classification of arcs of size $q-1$ and $q-2$.

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arXiv2016

Classification of 8-dimensional rank two commutative semifields

We classify the rank two commutative semifields which are 8-dimensional over their center $\mathbb{F}_{q}$. This is done using computational methods utilizing the connection to linear sets in $\mathrm{PG}(2,q^{4})$. We then apply our methods to complete the classification of rank two commutative semifields which are 10-dimensional over $\mathbb{F}_{3}$. The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.

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arXiv2016

FinInG: a package for Finite Incidence Geometry

FinInG is a package for computation in Finite Incidence Geometry. It provides users with the basic tools to work in various areas of finite geometry from the realms of projective spaces to the flat lands of generalised polygons. The algebraic power of GAP is exploited, particularly in its facility with matrix and permutation groups.

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arXiv2016

Classification of subspaces in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3$ and orbits in ${\mathbb{F}}^2\otimes {\mathbb{F}}^3\otimes {\mathbb{F}}^r$

This paper contains the classification of the orbits of elements of the tensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3 \otimes{\mathbb{F}}^r$, $r\geq 1$, under the action of two natural groups, for all finite; real; and algebraically closed fields. For each of the orbits we determine: a canonical form; the tensor rank; the rank distribution of the contraction spaces; and a geometric description. The proof is based on the study of the contraction spaces in ${\mathrm{PG}}({\mathbb{F}}^2\otimes{\mathbb{F}}^3)$ and is geometric in nature. Although the main focus is on finite fields, the techniques are mostly field independent.

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arXiv2016

The BEL-rank of finite semifields

In this article we introduce the notion of the BEL-rank of a finite semifield, prove that it is an invariant for the isotopism classes, and give geometric and algebraic interpretations of this new invariant. Moreover, we describe an efficient method for calculating the BEL-rank, and present computational results for all known small semifields.

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arXiv2015

Canonical forms of $2\times 3 \times 3$ tensors over the real field, algebraically closed fields, and finite fields

We classify the orbits of elements of the tensor product spaces ${\mathbb{F}}^2\otimes {\mathbb{F}}^3 \otimes {\mathbb{F}}^3$ for all finite; real; and algebraically closed fields under the action of two natural groups. The result can also be interpreted as the classification of the orbits in the $17$-dimensional projective space of the Segre variety product of a projective line and two projective planes. This extends the classification of the orbits in the $7$-dimensional projective space of the Segre variety product of three projective lines [M. Lavrauw and J. Sheekey: Orbits of the stabiliser group of the Segre variety product of three projective lines, Finite Fields Appl. (2014)]. The proof is geometric in nature, relies on properties of the Segre embedding, and uses the terminology of projective spaces.

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arXiv2014

Subgeometries and linear sets on a projective line

We define the splash of a subgeometry on a projective line, extending the definition of \cite{BaJa13} to general dimension and prove that a splash is always a linear set. We also prove the converse: each linear set on a projective line is the splash of some subgeometry. Therefore an alternative description of linear sets on a projective line is obtained. We introduce the notion of a club of rank $r$, generalizing the definition from \cite{FaSz2006}, and show that clubs correspond to tangent splashes. We determine the condition for a splash to be a scattered linear set and give a characterization of clubs, or equivalently of tangent splashes. We also investigate the equivalence problem for tangent splashes and determine a necessary and sufficient condition for two tangent splashes to be (projectively) equivalent.

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arXiv2014

Subspaces intersecting each element of a regulus in one point, André-Bruck-Bose representation and clubs

In this paper results are proved with applications to the orbits of $(n-1)$-dimensional subspaces disjoint from a regulus $\cR$ of $(n-1)$-subspaces in $\PG(2n-1,q)$, with respect to the subgroup of $\PGL(2n,q)$ fixing $\cR$. Such results have consequences on several aspects of finite geometry. First of all, a necessary condition for an $(n-1)$-subspace $U$ and a regulus $\cR$ of $(n-1)$-subspaces to be extendable to a Desarguesian spread is given. The description also allows to improve results in \cite{BaJa12} on the André-Bruck-Bose representation of a $q$-subline in $\PG(2,q^n)$. Furthermore, the results in this paper are applied to the classification of linear sets, in particular clubs.

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arXiv2014

Field reduction and linear sets in finite geometry

Based on the simple and well understood concept of subfields in a finite field, the technique called `field reduction' has proved to be a very useful and powerful tool in finite geometry. In this paper we elaborate on this technique. Field reduction for projective and polar spaces is formalized and the links with Desarguesian spreads and linear sets are explained in detail. Recent results and some fundamental ques- tions about linear sets and scattered spaces are studied. The relevance of field reduction is illustrated by discussing applications to blocking sets and semifields.

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arXiv2014

The isotopism problem of a class of 6-dimensional rank 2 semifields and its solution

In [U. Dempwolff: \textit{More Translation Planes and Semifields from Dembowski-Ostrom Polynomials}, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81--103] three classes of rank two presemifields of order $q^{2n}$, with $q$ and $n$ odd, were exhibited, leaving as an open problem the isotopy issue. In [M. Lavrauw, G. Marino, O. Polverino, R. Trombetti: \textit{Solution to an isotopism question concerning rank 2 semifields}, J. Comb. Des., to appear. DOI: 10.1002/jcd.21382], the authors faced with this problem answering the question whether these presemifields are new for $n>3$. In this paper we complete the study solving the case $n=3$.

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arXiv2014

On BEL-configurations and finite semifields

The BEL-construction for finite semifields was introduced in \cite{BEL2007}; a geometric method for constructing semifield spreads, using so-called BEL-configurations in $V(rn,q)$. In this paper we investigate this construction in greater detail, and determine an explicit multiplication for the semifield associated with a BEL-configuration in $V(rn,q)$, extending the results from \cite{BEL2007}, where this was obtained only for $r=n$. Given a BEL-configuration with associated semifields spread $\mathcal{S}$, we also show how to find a BEL-configuration corresponding to the dual spread $\mathcal{S}^d$. Furthermore, we study the effect of polarities in $V(rn,q)$ on BEL-configurations, leading to a characterisation of BEL-configurations associated to symplectic semifields. We give precise conditions for when two BEL-configurations in $V(n^2,q)$ define isotopic semifields. We define operations which preserve the BEL property, and show how non-isotopic semifields can be equivalent under this operation. We also define an extension of the ```switching'' operation on BEL-configurations in $V(2n,q)$ introduced in \cite{BEL2007}, which, together with the transpose operation, leads to a group of order $8$ acting on BEL-configurations.

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arXiv2013

Geometry of the inversion in a finite field and partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves

Let $L=\mathbb F_{q^n}$ be a finite field and let $F=\mathbb F_q$ be a subfield of $L$. Consider $L$ as a vector space over $F$ and the associated projective space that is isomorphic to ${\mathrm{PG}}(n-1,q)$. The properties of the projective mapping induced by $x\mapsto x^{-1}$ have been studied in \cite{Cs13,Fa02,Ha83,He85,Bu95}, where it is proved that the image of any line is a normal rational curve in some subspace. In this note a more detailed geometric description is achieved. Consequences are found related to mixed partitions of the projective spaces; in particular, it is proved that for any positive integer $k$, if $q\ge2^k-1$, then there are partitions of ${\mathrm{PG}}(2^k-1,q)$ in normal rational curves of degree $2^k-1$. For smaller $q$ the same construction gives partitions in $(q+1)$-tuples of independent points.

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arXiv2013

Constant rank-distance sets of hermitian matrices and partial spreads in hermitian polar spaces

In this paper we investigate partial spreads of $H(2n-1,q^2)$ through the related notion of partial spread sets of hermitian matrices, and the more general notion of constant rank-distance sets. We prove a tight upper bound on the maximum size of a linear constant rank-distance set of hermitian matrices over finite fields, and as a consequence prove the maximality of extensions of symplectic semifield spreads as partial spreads of $H(2n-1,q^2)$. We prove upper bounds for constant rank-distance sets for even rank, construct large examples of these, and construct maximal partial spreads of $H(3,q^2)$ for a range of sizes.

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arXiv2013

Solution to an isotopism question concerning rank 2 semifields

In [U. Dempwolff: More Translation Planes and Semifields from Dembowski-Ostrom Polynomials, Designs, Codes, Cryptogr. \textbf{68} (1-3) (2013), 81-103], the author gives a construction of three classes of rank two semifields of order $q^{2n}$, with $q$ and $n$ odd, using Dembowski-Ostrom polynomials. The question whether these semifields are new, i.e. not isotopic to previous constructions, is left as an open problem. In this paper we solve this problem for $n>3$, in particular we prove that two of these classes, labeled $D_{A}$ and $D_{AB}$, are new for $n>3$, whereas presemifields in family $D_{B}$ are isotopic to Generalized Twisted Fields for each $n\geq 3$.

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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].

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