On the theory of Point Weight Designs

Abstract

A point-weight incidence structure is a structure of blocks and points where each point is associated with a positive integer weight. A point-weight design is a point-weight incidence structure where the sum of the weights of the points on a block is constant and there exist some condition that specifies the number of blocks that certain sets of points lie on. These structures share many similarities to classical designs. Chapter one provides an introduction to design theory and to some of the existing theory of point-weight designs. Chapter two develops a new type of point-weight design, termed a row-sum point-weight design, that has some of the matrix properties of classical designs. We examine the combinatorial aspects of these designs and show that a Fisher inequality holds and that this is dependent on certain combinatorial properties of the points of minimal weight. We define these points, and the designs containing them, to be either `awkward' or `difficult' depending on these properties. Chapter three extends the combinatorial analysis of row-sum point-weight designs. We examine structures that are simultaneously row-sum and point-sum point-weight designs, paying particular attention to the question of regularity. We also present several general construction techniques and specific examples of row-sum point-weight designs that are generated using these techniques. Chapter four concentrates on the properties of the automorphism groups of point-weight designs with particular emphasis on row-sum point-weight designs. We introduce the idea of a structure being ``t-homogeneous with respect to its orbital partition'' and use this to derive a formula for the number of blocks a set of points lies upon. We also discuss the properties of the orbits of subgroups of the automorphism group. In chapter five we extend the idea of a dual to point-weight incidence structures and, as an extension of this, develop the idea of an underlying dual. We also examine the properties of square point-weight designs, i.e. point-weight designs that have exactly as many points as blocks. We find that there exists a result of a similar nature to the Bruck-Chowla-Ryser theorem of symmetric designs.