-
A
-Design is an incidence structure More specifically, a
design is a set of points, with a collection of -subsets of points called blocks such that every -set of points lies in precisely blocks -
The projective planes
have the property that every two points lie in a unique block. They are therefore
designs -
Let
be a design and a set of points where . We have
So that
is also an design. -
A
-design exists if is an integer for all . -
denotes the total blocks of the design. -
is the replication number which denotes the number of blocks containing each point. -
-
(Godsil 5.10.1) In a
-design with , we have . -
A
-design with is called symmetric. -
(Godsil 5.10.2) The dual
of a symmetric design is a symmetric design with the same parameters -
(Godsil 5.10.3) A bipartite graph is the incidence graph of a symmetric
-design if and only if it is distance regular with diameter -
The incidence matrix of a design is the matrix
where rows are indexed by points, columns by blocks. if the -th block in the -th block. -
satisfies Where
is the all one’s matrix.
Steiner system
-
A Steiner System is one where
. -
A Steiner Triple system is a
-design with , and . -
(Godsil 5.10.4) The block intersection graph of a Steiner triple system with
is distance regular with diameter .