Existence of L_d(n) and ML_d(n,k)

Abstract

We are given an n x n array, ML(n,k), with integers n, d, k > 0 such that n=mk and each symbol in {0, ..., m-1} appears in each row and column of the ML(n,k) exactly k times. We aim to construct an ML_d(n,k) with the restrictions below. It is required that the array is filled so that every symbol i in {0, ..., m-1} appears exactly k times in each row and column, as before. We will add the restriction that at most one of symbol i appears in each d x d block inside of the original array. What are the possible values of n, k, and d? How do we arrange the symbols? In this dissertation, we will find the answer to these questions by finding necessary conditions for an ML_d(n,k) to exist: m > d^2, then creating a construction to produce one. We will first assess the easier case of Latin squares, where k=1, then move on to multi-Latin squares, where k >1.