Block design

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For an account of that concept in intelligence testing, see block design test.

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, λ ≥ 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 λ 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 λ 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, λ)-design or a (v, b, r, k, λ)-design. The parameters are not all independent; v, k, and λ determine b and r, and not all combinations of v, k, and λ are possible. The two basic equations connecting these parameters are

bk = vr, \,
\lambda(v-1) = r(k-1). \,

A fundamental theorem, Fisher's inequality, named after Ronald Fisher, is that b ≥ 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 λ = 1), and Steiner triple systems (k = 3). The former is a relatively simple example of a symmetric design.

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[edit] Projective planes

Projective planes are a special case of block designs, where we have \scriptstyle v \,>\, 0 points and, as they are symmetric designs, \scriptstyle b \,=\, v (which is the limit case of Fisher's inequality), from the first basic equation we get

k = r, \,

and since \scriptstyle \lambda \,=\, 1 by definition, the second equation gives us

v-1 = k(k-1).\,

Now, given an integer \scriptstyle n \,\geq\, 1, called the order of the projective plane, we can put k = n + 1 and, from the displayed equation above, we have \scriptstyle v \,=\, (n+1)n \,+\, 1 \,=\, n^2 \,+\, n \,+\, 1 points in a projective plane of order n.

Since a projective plane is symmetric, we have that \scriptstyle b \,=\, v, which means that \scriptstyle b \,=\, n^2 \,+\, n \,+\, 1 also. The number b is usually called the number of lines of the projective plane.

This means, as a corollary, that in a projective plane, the number of lines and the number of points are always the same. For a projective plane, k is the number of lines and it is equal to n + 1, where n is the order of the plane. Similarly, r = n + 1 is the number of lines to which the a given point is incident.

For n = 2 we get a projective plane of order 2, also called the Fano plane, with v = 4 + 2 + 1 = 7 points and 7 lines. In the Fano plane, each line has n + 1 = 3 points and each point belongs to n + 1 = 3 lines.

[edit] Generalization: t-designs

Given any integer t ≥ 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 λ 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, 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

b_i = \lambda \left.\binom{v-i}{t-i} \right/ \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.

There are no known examples of non-trivial t-(v,k,1)-designs with \scriptstyle t >\, 5.

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

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