Introduction

A design is a set of blocks of order on a set of vertices such that
each subset
of the set of vertices is contained in exactly blocks. More formally, let denote the set of vertices and indicate by
the set
of subsets
of the set of
vertices. Then has to fulfill the following conditions in order to be a design:
The
introduction of a analog is now obvious: A design is a set of subspaces of dimension of such
that each subspace of dimension is contained in exactly blocks. More formally, let
indicate
the set of subspaces
of Then has to fulfill the following conditions in order to be a design:
As designs are suitable selections of blocks,
they can be described with the aid of the incidence matrix the rows
of which correspond to the and the columns of which correspond to the The entries of are
defined as follows:
Hence a design is nothing but a selection of
columns of that particular matrix, or, equivalently, a 01vector which solves the
system of linear equations with this particular matrix as matrix of
coefficients:
1.1 Corollary The set of designs on is the set of selections of subspaces that can
be obtained from the 01solutions of the sytem of linear equations
The set of blocks of the design
corresponding to the solution is
There are,
of course, several trivial cases, where solutions exist, for example
is a design,
but we are looking for nontrivial designs. The first examples were
presented by S. Thomas ([18]) who found designs, for all . H. Suzuki ([17]) extended this family to families
of designs for arbitrary prime powers under the same restriction on as above. As far
as we know, no nontrivial designs have been found yet for .
Thomas and Suzuki used
geometric arguments for their constructions, but we are interested here in a general
approach that allows a systematic and complete construction
of such designs (for small parameters), and therefore it has to be implemented
on computers. The above lemma opens such an approach but we note that the
matrix is
a very big matrix, so that there is not much hope to find such solutions via
solving this big system of diophantine equations.