Definition:Balanced Incomplete Block Design
Definition
A Balanced Incomplete Block Design or BIBD with parameters $v, b, r, k, \lambda$ is a block design where:
- $v$ 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 $\displaystyle \binom v 2$ pairs occur together in exactly $\lambda$ blocks.
A BIBD with parameters $v, b, r, k, \lambda$ is commonly written several ways, for example:
- $\operatorname{BIBD} \left({v, k, \lambda}\right)$
- $\left ({v, k, \lambda}\right)$-$\operatorname{BIBD}$
Properties
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For any $\operatorname{BIBD} \left({v, k, \lambda}\right)$ the following are true:
- $b k = r v$
- $\lambda (v-1) = r (k-1)$
- $\displaystyle \left({v, k, \lambda}\right)b = \frac{\binom v 2}{\binom k 2} \lambda = \frac{v(v-1)\lambda} {k(k-1)}$
- $k < v$
- $r > \lambda$
Note: All of the above are integers.
See Necessary Condition for Existence of BIBD for proofs of the above.
Also see
- Fisher's Inequality: $b \ge v$.
- Pairwise Balanced Design (PBD)