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 () who found -designs, for all . H. Suzuki () 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.