Orthogonal sets of latin squares and class-r hypercubes generated by finite algebraic systems.

摘要

Latin squares are combinatorial objects which have applications in some various and slightly surprising settings. A latin square of order n is a square array on n symbols such that each symbol occurs once in each row and column. Two latin squares are called orthogonal when superimposing them gives each of the n 2 ordered pairs of symbols exactly once. It is well known that if q is a prime power, the squares formed from the polynomials ax + y, a ∈ Fq form q - 1 latin squares of order q which are mutually orthogonal (each pair of squares is orthogonal). In this dissertation, we explore four problems relating to latin squares and other objects with similar properties, especially focusing on constructing large mutually orthogonal sets.;We explore the extent to which sets of mutually orthogonal latin squares, hypercubes, and frequency squares can be obtained by polynomials over finite fields. We are able to rescue two classical conjectures of Euler and MacNeish which are false for general latin squares but which are true when out attention is restricted to polynomial-generated squares only.;We also introduce the theory of the finite algebraic structures called uniform cyclic neofields, and explore the construction of sets of latin squares which are "nearly orthogonal." Our main result with be to give a simple construction of large sets of such nearly orthogonal squares for all even orders n where n - 1 is prime.;We then examine a new generalization of latin squares called class-r hypercubes which feature a larger alphabet (n r rather than n symbols). We give solutions to several open problems in this area, most notably the construction of large mutually orthogonal sets for r ≥ 3.;As our last topic, we give some partial progress toward solutions about a long-standing problem on the computability of partially filled latin cubes. Although the immediate extension of the famous Evans' conjecture seems fail for latin cubes of type 1 although it is true for latin squares, we explore what weaker versions of this conjecture can be said to hold.