Title:

On pointweighted designs.

A pointweighted structure is an incidence structure with each point assigned an
element of some set W C Z+ as a 'weight'. A pointweighted structure with
no repeated blocks and the property that the sum of the weights of the points
incident with anyone block is a constant k is called a pointweighted design. A
t  (v, k, Aj W) pointweighted design is such a structure with the sum of the
weights of all the points equal to v and the property that every set of t distinct
points is incident with exactly A blocks. This thesis introduces and examines this
generalisation of block designs.
The first chapter introduces incidence structures and designs.
Chapter 2 introduces and defines pointweighted designs. Three constructions of
families of t  (v, k, Aj W) pointweighted designs are given.
Associated with any pointweighted design is the incidence structure on which it
is based  the 'underlying' incidence structure (u.i.s.). It is shown in Chapter 3
that any automorphism of the u.i.s. of a t  (v, k, Aj W) pointweighted design
with more than one block and t > 1 preserves weights in the pointweighted design.
The u.i.s. of such a pointweighted design is shown to be a block design if
and only if every point is assigned the same weight. A necessary and sufficient
condition is obtained for the assignment of weights in any pointweighted design
to be essentially uniquely determined by the u.i.s.
Chapter 4 considers t{v, k, Aj W) pointweighted designs in which all of the points
apart from a 'special' point have the same weight. It is shown that when v > k
the weight of the special point is an integer multiple of the weight assigned to all
the other points. A class of these pointweighted designs is demonstrated to be
equivalent to a class of groupdivisible designs with specific parameters.
The final chapter uses the procedure of pointcomplementing incidence structures
to construct pointweighted designs. Trivial pointweighted designs are defined
and a necessary and sufficient condition for the existence of a member of a certain
class of these is obtained. A correspondence between this class of pointweighted
designs and certain trivial block designs is given using pointcomplementing.
