Combinatorial design

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Combinatorial design theory is the part of combinatorial mathematics that deals with the existence and construction of systems of finite sets whose intersections have specified numerical properties.

For instance, a balanced incomplete block design (usually called for short a block design) is a collection B of b subsets (called blocks) of a finite set X of v elements, such that any element of X is contained in the same number r of blocks, every block has the same number k of elements, and any two blocks have the same number λ of common elements. For example, if λ = 1, we have a projective plane: X is the point set of the plane and the blocks are the lines.

A spherical design is a finite set X of points in a (d−1)-dimensional sphere such that, for some integer t, the average value on X of every polynomial

f(x₁, ..., x_d)

of total degree at most t is equal to the average value of f on the whole sphere, i.e., the integral of f divided by the area of the sphere.

Combinatorial design theory is applied to the design of experiments.