Designs in finite geometry

• Designs in endlicher Geometrie

Lansdown, Jesse; Bamberg, John (Thesis advisor); Niemeyer, Alice Catherine (Thesis advisor); Royle, Gordon F. (Thesis advisor)

Aachen : RWTH Aachen University (2020, 2021)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2020. - Dissertation, University of Western Australia, 2020

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.

Institutions

• Department of Mathematics [110000]
• Algebra Teaching and Research Area [115320]