# 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
and
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 plane 2-(7,7,3,3,1) S(2,3,7) Two interleaved Fano planes 2-(7,14,6,3,2) The complement of the Fano plane 2-(7,7,4,4,2) The Paley biplane 2-(11,11,5,5,2) M12 5-(12,132,66,6,1) S(5,6,12) M11 4-(11,66,30,5,1) S(4,5,11) M10 3-(11,30,12,4,1) S(3,4,10) M9 2-(11,12,4,3,1) S(2,3,9) M24 5-(24,759,253,8,1) S(5,8,24) M23 4-(23,253,77,7,1) S(4,7,23) M22 3-(22,77,21,6,1) S(3,6,22) M21 2-(21,21,5,5,1) S(2,5,21)