In combinatorial mathematics, a block design is a particular kind of set system, which has applications to experimental design, finite geometry, software testing, cryptography, and algebraic geometry. Many variations have been studied, including balanced incomplete block designs.^{[1]}^{[2]}
Given a finite set X (of elements called points) and integers k, r, λ ≥ 1, we define a 2design B to be a set of kelement 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 kelement 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
These conditions are not sufficient as for example a (43,7,1)design does not exist. 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 and λ = 1). The former is a relatively simple example of a symmetric design. Triple systems (k = 3) are of interest in their own right.^{[3]}
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Projective planes are a special case of block designs, where we have points and, as they are symmetric designs, (which is the limit case of Fisher's inequality), from the first basic equation we get
and since by definition, the second equation gives us
Now, given an integer , called the order of the projective plane, we can put k = n + 1 and, from the displayed equation above, we have points in a projective plane of order n.
Since a projective plane is symmetric, we have that , which means that 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.
Given any integer t ≥ 2, a tdesign B is a class of kelement subsets of X (the set of points), called blocks, such that every point x in X appears in exactly r blocks, and every telement 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 b_{i} is the number of blocks that contain any ielement set of points.
There are no known examples of nontrivial t(v,k,1)designs with .
The term block design by itself usually means a 2design.
