Chapter 2

Designs

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

difference sets.

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.

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http://dx.doi.org/10.1090/stml/067/02