CanaDAM 2013
Université Memorial de Terre-Neuve, 10 - 13 juin 2013

Galois Geometries and Applications II
Président: Petr Lisonek (Simon Fraser University)
Org: Jan De Beule (Ghent University) et Petr Lisonek (Simon Fraser University)

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}

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.


Atlantic Association for Research in the Mathematical Sciences Centre de recherches mathmatiques The Fields Institute Pacific Institute for the Mathematical Sciences Socit mathmatique du Canada Université Memorial de Terre-Neuve