Block design

Overview
In combinatorial mathematics, a block design (more fully, a balanced incomplete block design) is a particular kind of set system, which has long-standing applications to experimental design (an area of statistics) as well as purely combinatorial aspects.

Given a finite set X (of elements called points) and integers k, r, &lambda; &ge; 1, we define a 2-design B to be a set of k-element subsets of X, called blocks, such that the number r of blocks containing x in X is independent of x, and the number &lambda; of blocks containing given distinct points x and y in X is also independent of the choices.

Here v (the number of elements of X, called points), b (the number of blocks), k, r, and &lambda; are the parameters of the design. (Also, B may not consist of all k-element subsets of X; that is the meaning of incomplete.) The design is called a (v, k, &lambda;)-design or a (v, b, r, k, &lambda;)-design. The parameters are not all independent; v, k, and &lambda; determine b and r, and not all combinations of v, k, and &lambda; are possible. The two basic equations connecting these parameters are
 * $$ bk = vr, $$
 * $$ \lambda(v-1) = r(k-1). $$

A fundamental theorem (Fisher's inequality) is that b &ge; v in any block design. The case of equality is called a symmetric design; it has many special features.

Examples of block designs include the lines in finite projective planes (where X is the set of points of the plane and &lambda; = 1), and Steiner triple systems (k = 3). The former is a relatively simple example of a symmetric design.

Generalization: t-designs
Given any integer t &ge; 2, a t-design B is a class of k-element subsets of X (the set of points), called blocks, such that the number r of blocks that contain any point x in X is independent of x, and the number &lambda; of blocks that contain any given t-element subset T is independent of the choice of T. The numbers v (the number of elements of X), b (the number of blocks), k, r, &lambda;, and t are the parameters of the design. The design may be called a t-(v,k,&lambda;)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are
 * $$ b_i = \lambda \binom{v-i}{t-i} / \binom{k-i}{t-i} \text{ for } i = 0,1,\ldots,t, $$

where bi is the number of blocks that contain any i-element set of points.

Examples include the d-dimensional subspaces of a finite projective geometry (where t = d + 1 and &lambda; = 1).

The term block design by itself usually means a 2-design.