On the theory of Point Weight Designs
by Alexander W. Dent
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.