Introduction

Since there is some variation between authors of the terminology employed in design theory, we begin with some definitions. An incidence structure is a triple D = (P, B, I), where:

(a)
P is a set, the elements of which are called points;

(b)
B is a set, the elements of which are called blocks;

(c)
I is an incidence relation between P and B, so that I ⊂P x B. The elements of I are called flags.

Usually, blocks will be subsets of P, so that instead of writing (p, b) ∈I, we write p ∈b. In general, repeated blocks are allowed so that different blocks may correspond to the same subset of P. If D has no repeated blocks, then we say that D is simple.

An incidence structure D is said to be uniform with blocksize k if D has at least one block and all blocks contain exactly k points. A uniform incidence structure is called trivial if each k-subset of the point set appears as a block (at least once).

Let t ≥0 be an integer. Then an incidence structure D is said to be t--balanced if there exists an integer λ ≥1 such that each t--subset of the point set is contained in exactly λ blocks of D.

A near--linear space is an incidence structure in which every block contains at least two points and any two points lie in at most one block. A linear space is a near--linear space in which any two points lie in exactly one block. It is usual, when discussing near--linear spaces, to use the term line in place of the term block.

Let v, k, t and λ be integers with v ≥k ≥t ≥0 and λ ≥1. A t--design with v points and blocksize k is an incidence structure D = (P, B, I) where:

(a)
The cardinality of P is v;

(b)
D is uniform with blocksize k;

(c)
D is simple;

(d)
For each t--subset T of P there are exactly λ blocks of B incident with all the points of T (so D is t--balanced).

Such a design is usually referred to as a t--(v, k, λ) design. The parameter λ is called the index of the design. If b denotes the cardinality of B, a t--design with v = b and t ≥2 is called a symmetric design. A t--design with λ = 1 is called a Steiner design. A design which is trivial is also called a complete design. Note that a design D must contain at least one block (i.e b > 0).

The category names for the different families of incidence structures are as follows:

Incidence structure : Inc

Near-linear space : IncNsp

Linear space : IncLsp

t--design : Dsgn

V2.28, 13 July 2023