Date & Time: Friday, August 27, 2010 , 16:00-17:00
Venue: Ramanujan Hall
Title: The use of blocking sets in Galois geometries
Speaker: Prof. Leo Storme, Ghent University, Belgium
Abstract:
In Galois geometries, blocking sets play a central role. An s-fold blocking set with respect to the k-spaces of PG(n, q) is a set of points of PG(n, q) intersecting every k-subspace in at least s points. Their importance follows not only from their geometrical interest, but also greatly from their appearance in many other types of problems. This includes applications in coding theory, such as the extendability of linear codes and the characterization of linear codes meeting the Griesmer bound.
In this talk, we present the main results on different types of blocking sets in Galois geometries. Here, the focus mainly lies on the characterization of minimal s-fold blocking sets with respect to the k-spaces of PG(n, q), i.e., s-fold blocking sets B such that no proper subset of B still is an s-fold blocking set. Then we focus on the use of blocking sets in solving many different other geometrical problems and problems in related research areas. As a concrete geometrical problem, we discuss the use of blocking sets in the problem of the extendability of maximal partial t-spreads in PG(n, q), with t + 1 a divisor of n + 1, of small deficiency δ to t-spreads in PG(n, q). In this discussion of the use of blocking sets for solving other problems, sometimes small differences with the geomet- rical aim of classifying minimal s-fold blocking sets appear. For instance, the characterization problem for linear codes meeting the Griesmer bound involves in several cases the characterization of non-minimal s-fold blocking sets, where both the minimal and the non-minimal part within the s-fold blocking set needs to be characterized completely.
About the speaker:
Prof. Leo Storme is a well-known researcher in the field of finite projective geometry, with links to coding theory. He is a member of the research group on Incidence Geometry led by Prof. J. A. Thas at Ghent University, Belgium. He has published over 100 research articles, and is on the editorial board of several journals including Finite Fields and Their Applications (Elsevier) and the Journal of Combinatorial Designs (Wiley).