In this chapter we introduce designs. Our ultimate goal is to study
symmetric designs and their relationship to difference sets. Along the
way we also introduce more general designs. Concepts of existence
and equivalence that appear here will be mirrored in our study of
Design theory is an area of combinatorics that was originally stud-
ied for its connections to statistics and the design of experiments. This
study has found use in other areas of mathematics including geometry,
coding theory, finite group theory, and difference sets. So the study
of designs is a good place to start our exploration of the connections
among these different algebraic and combinatorial structures.
2.1. Incidence structures
We start with the general notion of an incidence structure.
Definition. An incidence structure is an ordered triple (P, B, I) where
P is a set of points,
B is a set of blocks,
I ⊆ P × B is an incidence relation between P and B.
If (p, B) is in I, we say that p and B are incident.