# Definition:Balanced Incomplete Block Design

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## 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

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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.