## A study of finite projective planes: coordinatisation and construction

##### Date

2023

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University of Delaware

##### Abstract

This dissertation consists of three parts. In the first part comprising of Chapters 2, 3, and 4, strategies for optimal coordinatisation of the finite projective planes are developed and illustrated with the coordinatisation of the planes of order 16. We have sought to establish the meaning or sense of optimality in the various coordinatisations. More than one aspect of optimality in different coordinatisations of the same plane are explored. Nice PTR polynomial representations of some finite projective planes are obtained as a consequence of optimal coordinatisations. We also present the PTR polynomials in forms that reveal some properties of the associated plane. ☐ In the second part, comprising of Chapters 5, 6, and 7, we obtain some special planar ternary ring (PTR) polynomial representations of two individual planes and a class of planes. A brief description of each work will be given shortly. ☐ In the third part, comprising of Chapter 8, a method of constructing some finite projective planes of Lenz-Barlotti (LB) types II.1 or above is given. The method constructs a PTR multiplication table of a plane with some given special subsets of the coordinatising set. The coordinatising set is taken to be the set of elements of a finite field and the special subsets correspond to various elation and homology groups admitted by the resulting plane. In this dissertation, the PTR addition table is always taken to be the finite field addition table which underlines our assumption that the resulting plane admits an elementary abelian transitive elation group. We have given examples of some planes of orders 16, 25, 32, 49, 64, and 81 constructed by our method. ☐ Chapter 1 gives the preliminaries and background for the following chapters. In Chapter 2, we develop the theory of optimal coordinatisation with a special focus on the planes of LB types II.1 or above. Chapter 3 illustrates the theory developed in Chapter 2 with detailed descriptions and results of the coordinatisations of the planes of order 16 LB type II.1 or above. We continue the development of the theory of optimal coordinatisation in Chapter 4 by extending it to the planes of LB type I.1. Some coordinatisations of the planes of order 16 LB type I.1 are given. The next three chapters concern special PTR polynomial representations of some planes. In Chapter 5, a special coordinatisation and a PTR polynomial representation of the Figueroa plane of order 27 is given. A special PTR polynomial representation of the Hall planes from their definition as derived planes is obtained in Chapter 6. In Chapter 7, we obtain a PTR polynomial for the SEMI4 plane of order 16 by analysing the semifield multiplication given in Knuth [26]. Chapter 8 describes a method of constructing PTR multiplication tables of planes of LB type II.1 or above admitting an elementary abelian transitive elation group. Some planes of small orders constructed by this method are given. We give the planes in the form of PTR polynomials and supplement the data with the order of its full collineation group, p-rank, and a multiset giving the orders of central collineation groups having axes the lines of the plane. ☐ Finally, some future directions are given in Chapter 9. Appendix A is the unpublished article [7]. The results in the article are used to develop the theory of optimal coordinatisation in this dissertation. Appendix B contains a polynomial which appears in the PTR polynomial of the Figueroa plane of order 27 given in Chapter 5.

##### Description

##### Keywords

Construction of planes, Coordinatisation, Projective plane, PTR polynomial, PTR tables, Representation