# Definition:Balanced Incomplete Block Design

## Definition

A Balanced Incomplete Block Design or BIBD with parameters $v, b, r, k, \lambda$ is a block design such that:

$v$ is the number of points in the design
$b$ is the number of blocks
$k$ is the size of each block
$r$ is the number of blocks any point can be in
$\lambda$ is the number of times any two points can occur in the same block

and has the following properties:

Each block is of size $k$
All of the $\dbinom v 2$ pairs occur together in exactly $\lambda$ blocks.

A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example:

$\map {\operatorname {BIBD} } {v, k, \lambda}$
$\tuple {v, k, \lambda}$-$\operatorname{BIBD}$

## Properties

For every $\map {\operatorname {BIBD} } {v, k, \lambda}$ the following are true:

$b k = r v$
$\lambda \paren {v - 1} = r \paren {k - 1}$
$\tuple {v, k, \lambda} b = \dfrac {\dbinom v 2} {\dbinom k 2} \lambda = \dfrac {v \paren {v - 1} \lambda} {k \paren {k - 1} }$
$k < v$
$r > \lambda$

Note: All of the above are integers.

See Necessary Condition for Existence of BIBD for proofs of the above.