Galois Geometries and Applications II
Chair:
Petr Lisonek (Simon Fraser University)
Org:
Jan De Beule (Ghent University) and
Petr Lisonek (Simon Fraser University)
[
PDF]
 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 KramerMesnermethod 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 KramerMesnermethod 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 CameronLiebler line classes with large parameter [PDF] [SLIDES]

We report on joint work with Jeroen Demeyer, Klaus Metsch and Morgan Rodgers to construct CameronLiebler 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 CameronLiebler 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 cameronliebler line classes in
\(\mathrm{PG}(3,q)\).
\newblock {\em Des. Codes Cryptogr.}, to appear, DOI: 10.1007/s1062301195812
\end{thebibliography}
 MAARTEN DE BOECK, UGent
The Erd\H{o}sKoRado problem for geometries [PDF] [SLIDES]

An Erd\H{o}sKoRado set of a finite geometry is a set of $k$dimensional subspaces such that any two subspaces have a nonempty intersection. It is maximal if it is nonextendable regarding this condition. The general Erd\H{o}sKoRado problem asks for the size and the classification of the (large) maximal Erd\H{o}sKoRado 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}sKoRado sets of generators of a polar space, on Erd\H{o}sKoRado sets of planes in projective and polar spaces and on Erd\H{o}sKoRado 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 pointline 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.