Introduction
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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 0-1-vector
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 0-1-solutions
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.