An incidence system (, , , , ) in which a set of points is partitioned into a family of
subsets (blocks) in such a way that any two points determine blocks with points in each block, and
each point is contained in different blocks. It is also generally required that , which is where the
``incomplete'' comes from in the formal term most often encountered for block designs, Balanced Incomplete Block
Designs (BIBD). The five parameters are not independent, but satisfy the two relations

(1) |

(2) |

(3) | |||

(4) |

A BIBD is called Symmetric if (or, equivalently, ).

Writing
and
, then the Incidence Matrix of the BIBD is given by the
Matrix
defined by

(5) |

(6) |

Examples of BIBDs are given in the following table.

Block Design | (, , ) |

Affine Plane | (, , 1) |

Fano Plane | (7, 3, 1)) |

Hadamard Design | Symmetric (, , ) |

Projective Plane | Symmetric (, , 1) |

Steiner Triple System | (, 3, 1) |

Unital | (, , 1) |

**References**

Dinitz, J. H. and Stinson, D. R. ``A Brief Introduction to Design Theory.'' Ch. 1 in
*Contemporary Design Theory: A Collection of Surveys* (Ed. J. H. Dinitz and
D. R. Stinson). New York: Wiley, pp. 1-12, 1992.

Ryser, H. J. ``The
-Configuration.'' §8.1 in
*Combinatorial Mathematics.* Buffalo, NY: Math. Assoc. Amer., pp. 96-102, 1963.

© 1996-9

1999-05-26