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*See block design test for an account of that concept in intelligence testing.*

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* 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 *x* and *y* in *X* is also independent of the choices. Here *v* (the number of elements of *X*), *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

**(**. The parameters are not all independent;

*v*,*b*,*r*,*k*, λ)-design*v*,

*k*, and λ determine

*b*and

*r*, and not all combinations of

*v*,

*k*, and λ are possible.

Examples 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).

Given any integer *t* ≥ 2, a *t*-design*B* is a class of *k*-element subsets of *X*, called **blocks**, such that the number *r* of blocks that contain any *x* in *X* is independent of *x* and the number λ 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.

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

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

## ReferencesEdit

- Steven H. Cullinane,
*Block Designs in Art and Mathematics*, (2004) - Eric W. Weisstein,
*Block Designs*at MathWorld.

- ru:Блок-дизайн

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