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.

Also see

 * Fisher's Inequality: $b \ge v$
 * Definition:Pairwise Balanced Design