Year

2000

Degree Name

Doctor of Philosophy

Department

School of Information Technology and Computer Science

Abstract

W e study critical sets of F-squares, latin squares and Youden squares. We prove that some subsets of F-squares, latin squares and Youden squares are critical sets. We consider minimal critical sets. We show that some critical sets for certain types of F-squares and a certain class of Youden squares are minimal. We study subsets of critical sets and their importance in the completion of a critical set to a full design. We study the nest, influence, power and strongbox of these subsets, and their ability to hold information about the critical set. We calculate the influence of elements and subsets of a critical set of a back circulant latin square, as well as the strongbox of a critical set of a back circulant latin square. We solve the spectrum of the type 𝐹(𝑛;1,𝑛 - 1) of F-squares and give partial spectrums for type 𝐹(𝑛;2,𝑛 - 2). Connections to finite topology and graph theory are considered. We show that certain collections of subsets of critical sets may form the basis of some topology on the critical set. The problem of ranking elements in a critical set is looked at, with some examples of possible rankings given. We look very briefly at the notion of secret sharing schemes based on critical sets, and show how an access structure m a y be formed based on the properties of the influence of a set in a critical set.

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