Block Design - Generalization: t-designs

Generalization: t-designs

Given any positive integer t, a t-design B is a class of k-element subsets of X, called blocks, such that every point x in X appears in exactly r blocks, and every t-element subset T appears in exactly λ blocks. The numbers v (the number of elements of X), b (the number of blocks), k, r, λ, and t are the parameters of the design. The design may be called a t-(v,k,λ)-design. Again, these four numbers determine b and r and the four numbers themselves cannot be chosen arbitrarily. The equations are

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

Theorem: Any t-(v,k,λ)-design is also an s-(v,k,λ_s)-design for any s with 1 ≤ s ≤ t. (Note that the "lambda value" changes as above and depends on s.)

A consequence of this theorem is that every t-design with t ≥ 2 is also a 2-design.

There are no known examples of non-trivial t-(v,k,1)-designs with .

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