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 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


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