Abstract

The goal of this paper is to present results of the work I have completed as part of my Ph.D. research. The results as presented in this paper are divided into four categories.

The first section discusses the classification of commutative semifields of order 64. This was done with a computer search that exploited the commutativity property and thus made the computational effort reasonable.

Next this paper documents my work classifying the flag-transitive planes of order 32. These planes were classified by Alan Prince in 2000. The work shown in this paper was done independently and without knowledge of Prince's work. Here we classify all of the projective planes of order 32 which have an automorphism group that contains a subgroup of order 11. This includes the flag-transitive planes of order 32.

Then we lay out a method for pairing down a search tree with the use of the conjugacy classes. This strategy can drastically reduce the search space and therefore the run time of a search. This section also describes the benefit of using bit operations and storage methods for problems over [special characters omitted]. The goal was to classify the semifield planes of order 64.

In the final section, we define the new mathematical concept of non singular n-dimensional arrays. This is an extension of a concept first brought to my attention by Tim Pentilla. Nonsingular cubical arrays can be used to describe semifields. In this paper we extend the idea to include n-dimensional arrays. We then provide constructions of nonsingular n-dimensional arrays for every n over every field.