CanaDAM 2013 Memorial University of Newfoundland, June 10 - 13, 2013 www.cms.math.ca//2013

Galois Geometries and Applications II
Chair: Petr Lisonek (Simon Fraser University)
Org: Jan De Beule (Ghent University) and Petr Lisonek (Simon Fraser University)
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MICHAEL BRAUN, University of Applied Sciences, Darmstadt, Germany
$q$-Analog of Packing Designs  [PDF] [SLIDES]

A $P_q(t,k,n)$ $q$-packing design is a selection of $k$-subspaces of ${\mathbb F}_q^n$ such that each $t$-subspace is contained in at most one element of the collection. A successful approach adopted from the Kramer-Mesner-method of prescribing a group of automorphisms was applied by Kohnert and Kurz to construct some constant dimension codes with moderate parameters which arise by $q$-packing designs. In this talk we recall this approach and give a version of the Kramer-Mesner-method breaking the condition that the whole $q$-packing design must admit the prescribed group of automorphisms. Finally, we give some improvements on the size of $P_2(2,3,n)$ $q$-packing designs.

JAN DE BEULE, Ghent University
Constructing Cameron-Liebler line classes with large parameter  [PDF] [SLIDES]

We report on joint work with Jeroen Demeyer, Klaus Metsch and Morgan Rodgers to construct Cameron-Liebler line classes with parameter $x \in \mathcal{O}(q^2)$. The geometrical understanding of the orbits of the points of $\mathrm{PG}(3,q)$ under a group of order $q^2+q+1$ preserving the desired Cameron-Liebler line class, together with the representation of $\mathrm{AG}(3,q)$ as $\mathbb{F}_{q^3}$ plays a central role. We overview the state of the art of the currently known examples, \cite{Rodgers1,Rodgers2}.

\begin{thebibliography}{99} \bibitem{Rodgers1} M.~Rodgers. \newblock Private communication.

\bibitem{Rodgers2} M.~Rodgers. \newblock Some new examples of cameron-liebler line classes in $\mathrm{PG}(3,q)$. \newblock {\em Des. Codes Cryptogr.}, to appear, DOI: 10.1007/s10623-011-9581-2 \end{thebibliography}

MAARTEN DE BOECK, UGent
The Erd\H{o}s-Ko-Rado problem for geometries  [PDF] [SLIDES]

An Erd\H{o}s-Ko-Rado set of a finite geometry is a set of $k$-dimensional subspaces such that any two subspaces have a non-empty intersection. It is maximal if it is non-extendable regarding this condition. The general Erd\H{o}s-Ko-Rado problem asks for the size and the classification of the (large) maximal Erd\H{o}s-Ko-Rado sets. In this talk we will focus on finite projective spaces (finite vector spaces), finite polar spaces and designs. I will present recent results on Erd\H{o}s-Ko-Rado sets of generators of a polar space, on Erd\H{o}s-Ko-Rado sets of planes in projective and polar spaces and on Erd\H{o}s-Ko-Rado sets of blocks in a unital.

SARA ROTTEY, Vrije Universiteit Brussel
The automorphism group of linear representations  [PDF]

We discuss the automorphism group of linear representations of projective point sets. A linear representation $T_n^*(K)$ of a point set $K$ is a point-line geometry embedded in $PG(n+1,q)$. The common misconception was that for $T_n^*(K)$ every automorphism is induced by a collineation of its ambient space. This is not true in general. We prove that every automorphism is induced by an automorphism of $T_n^*(S)$, where $S$ is the smallest subgeometry containing $K$. By use of field reduction, we uncover the full automorphism group of $T_n^*(S)$. This is joint work with Stefaan De Winter and Geertrui Van de Voorde.

ALFRED WASSERMANN, Department of Mathematics
Construction of $q$-analogs of Steiner systems  [PDF] [SLIDES]

The notion of $t$-designs and Steiner systems has been extended to vector spaces by Cameron and Delsarte in the 1970s. In projective geometry, $q$-analogs of Steiner systems are called $(s,r)$-spreads. Metsch (1999) conjectured that $q$-analogs of Steiner systems do not exist for $t\geq 2$. Here, we show how we constructed the first examples of $S_2[2,3,13]$ $q$-analogs of Steiner systems. For the search we prescribed the normalizer of a Singer cycle as automorphism group and solved the resulting system of Diophantine linear equations. This is joint work with M. Braun, T. Etzion, P. {\"O}sterg{\aa}rd, and A. Vardy.