Symplectic translation planes, pseudo-ovals, and maximal 4-arcs

This thesis is comprised of the study of three distinct problems: the construction of symplectic translation planes, the characterisation of pseudo-ovals, and the classi cation of maximal arcs. Non-Desarguesian planes are the only finite projective spaces not to be governed by a finite field and, as such, hold a certain interest for finite geometers. In the '50s and '60s, Andre, Bruck and Bose, and Segre developed a method - now commonly referred to as the Andre/Bruck-Bose construction - to create translation planes, a subset of projective planes that includes many non-Desarguesian planes, from spreads of higher dimensional projective spaces. In Chapter 4, we outline previous computational work in this area, the linear programming methods we have used, and present eight new examples of translation planes. These examples correspond to spreads of W(5, 5) and W(5, 7). Appendix A contains the details required to reconstruct each of these spreads. Examples of pseudo-ovals can be constructed from ovals, using field reduction, and the existence of other types of pseudo-oval is unknown. Pseudo-ovals are represented, via geometric passages, as ovals in translation planes, as elements of translation generalised quadrangles, and, in some cases, as special sets in the Hermitian space H(3, q2). In Chapter 5 we go through relevant theorems in each situation and summarise the current status of the problem. The Mathon maximal arcs, generalised from the Denniston maximal arcs, are each formed by taking a union of regular hyperovals on a common nucleus. In Chapter 6 we prove that all maximal 4- arcs, in the Desarguesian plane PG(2, q), that are a union of regular hyperovals or that contain two conics that share exactly one point, are Mathon maximal 4-arcs. This is joint work with Nicola Durante from the Universit a degli Studi di Napoli Federico II. Background information on projective spaces and polar spaces is contained in Chapter 2. Chapter 3 describes, in more detail, the geometric passages between spaces that will be most useful for the remainder of the thesis and is, in particular, a prelude to Chapter 4. Chapters 4, 5 and 6 can be read independently of each other.