Latin Squares and Related Structures

Abstract

A latin square of order n is an n × n array of cells, each filled with one of n symbols such that each row and each column contain each symbol precisely once. This thesis contributes three new results from three different topics within the study of latin squares. In doing this, we give a broad overview of three distinct areas of the study of latin squares, and the results that have been accomplished in the literature so far.The first study presents new results pertaining to transversals in latin squares. Previous work on transversals has investigated the spectrum of intersection sizes of two transversals within the back-circulant latin squares. A natural extension to this work is to investigate the spectrum of intersection sizes of more than two transversals within the back-circulant latin squares. In this thesis we will accomplish this for three and four transversals, and for all but a finite list of exceptions give a design theoretic construction that recursively builds from base designs that we found by a computational search.The second study investigates µ-way k-homogeneous latin trades. These structures have been extensively studied when µ = 2, but much less is known when µ > 2. Previous investigation had filled in a fraction of the spectrum when µ = 3. We continue this study giving new constructions and show that 3-way k-homogeneous latin trades of order m exist for all but 196 possible exceptions.The third study investigates mutually nearly orthogonal latin squares (MNOLS). These MNOLS are similar to mutually orthogonal latin squares, and can also be used in the design of experiments. Continuing from previous investigations, we enumerate the number of collections of three cyclic MNOLS for latin squares with order up to 16. This required using computational enumeration techniques and a large optimised computer search, as the search space was incredibly large. We present the number of collections of three MNOLS for latin squares with order up to 16 under a variety of equivalences, where we take the collections to be either sets or ordered lists.