Notation for Balanced Block Designs

A balanced block design is a structure comprising a set of v elements, formed into b blocks, with each element being in r blocks, each block containing k elements, and each set of t elements occuring together in λ blocks.

The letters used above, and listed below, are the ones conventionally used.

b is the number of blocks.
k is the number of elements in each block.
λ is the number of blocks containing any specified set of elements of size t.
r is the number of blocks containing a specific element.
t is the size of the sets of elements such that each such set occur in λ blocks.
v is the number of vertices, elements, points, or "varieties".

We find
bk = vr
r Πt–1i=1(k-i) = λ Πt–1i=1(v-i)
which for t = 2 simplifies to
r(k-1) = λ(v-1)

A block design may be designated as t–( v, b, r, k, λ ); or, as b and r can be calculated from the other parameters, as t-( v, k, λ ). Where t is omitted, it is assumed to be 2.

If  λ  is 1, the block design is a Steiner system, and can be designated S(t, k, v).

If the b blocks of a Steiner system can be partitioned into r sets, with each such set being a partition of the set of v elements, then the Steiner system can also be called a Kirkman system, and described as parallelisable. The partitions are known as spreads.

An expression t–( v, b, r, k, λ ) does not necessarily designate a unique structure. There may be no such structure, or one, or many non-isomorphic such structures.

Some examples

The Fano plane2-(7,7,3,3,1)S(2,3,7)
Two interleaved Fano planes2-(7,14,6,3,2)
The complement of the Fano plane2-(7,7,4,4,2)
The Paley biplane2-(11,11,5,5,2)

Introduction to balanced mixed tournament plans
Explanation of list of tournament plans
List of tournament plans
Notation used for block designs
Some infinite sets of block designs
Further reading on block designs