Mixed mutually orthogonal frequency squares

Abstract

A frequency square of type F(n; λ1, λ2, λ3, . . . , λt) is an array n × n filled with symbols ai, where each ai occurs λi times in every row and column, and n = Pt i=1 λi. Two such frequency squares F1(n; λ1, λ2, λ3, . . . , λt) and F2(n; µ1, µ2, µ3, . . . , µs) are mutually orthogonal, when superimposed, each of the ts possible ordered pairs (i, j), where 1 ≤ i ≤ t and 1 ≤ j ≤ s, occurs exactly λiµj times. This thesis generalizes the classical theory of mutually orthogonal Latin squares to mixed frequency settings, where symbols may appear with different frequencies. The non-uniform frequencies lead to a wider range of combinatorial structures with new methods of construction. Then, the thesis investigates maximizing the set of mixed mutually orthogonal frequency squares (MMOFS), focusing on theoretical methods rather than computational tools. The following lemmas and theorems are new results presented in Chapter 3. In Lemma 4.1, we define mappings from two Latin squares to form two MOFS. Then in Theorem 4.7, we apply this to explore the maximum size of the sets of higher-order MMOFS by using mixed orthogonal arrays. Moreover, Section 2.2 gives an original alternative proof of an existing upper bound for sets of MOFS. Furthermore, we identify sets of MMOFS for small orders in the final chapter by using the new results and the previous theories. These new results include Corollary 4.5, Lemma 5.1, Example 22, Corollary 5.6 and Example 24.