Designs in Endlicher Geometrie

RWTH Publications (RWTH Aachen)

Abstract

This thesis is concerned with the study of Delsarte designs in symmetric association schemes, particularly in the context of finite geometry. We prove that m-ovoids of regular near polygons satisfying certain conditions must be hemisystems, and as a consequence, that for d ≥ 3 m-ovoids of DH(2d−1, q^2), DW(2d−1, q), and DQ(2d, q) are hemisystems. We also construct an infinite family of hemisystems of Q(2d, q), for q an odd prime power and d ≥ 2, the first known family for d ≥ 4. We generalise the AB-Lemma to constructions of m-covers other than just hemisystems. In the context of general Delsarte designs, we show that either the size of a design, or the strata in which it lies, may be constrained when certain Krein parameters vanish, and explore various consequences of this result. We also study the concept of a “witness” to the non-existence of a design, in particular by considering projection and inclusion of association schemes, and the implications this has on the existence of designs when the strata of a projected design is constrained. We furthermore introduce strong semi-canonicity and use it in a black-box pruned orderly algorithm for effective generation of designs and combinatorial objects. We use these techniques to find new computational results on various m-ovoids, partial ovoids, and hemisystems.

Authors 1

1. Jesse Lansdown corresponding Aachen

   RWTH Aachen University

   Affiliation as printed

   RWTH Aachen

   Mathematics and Statistics