Abstract
One of the outstanding problems in the study of Latin squares is that of improving the known lower bounds for N(n), the maximum number of Latin squares of order n in a mutually orthogonal set. After describing the methods of construction which attain the best known lower bounds for N(n), n < 32 and showing how most of these are interrelated we provide a general method of construction for sets of mutually orthogonal Latin squares (m.o.l.s.) from left neofields. We then give detailed information about the structure of all isomorphically distinct left neofields of order less than ten and about the m.o.l.s. which they produce, and summarised information for orders up to fourteen. We further show that many of the previously known constructions of m.o.l.s. effectively employ the construction which we describe, in particular the recent constructions of three m.o.l.s. of order fourteen and four of order twenty. In the course of this investigation it is noted that the number of complete mappings of both of the non-cyclic abelian groups of order eight is the same. Furthermore, it is found that both of the non-abelian groups of order eight possess the same number of complete and near complete mappings. We explain and justify why this is the case. In our study of left neofields we discuss the properties of sequenceability and R-sequenceability of groups. At the end of the thesis, we discuss a related question, raised by R. L. Graham, as to which groups are r-set-sequenceable. This is solved for abelian groups except that, for r = n - 1, the question is reduced to that of asking which abelian groups are R-sequenceable.