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) |
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) |
See also Affine Plane, Design, Fano Plane, Hadamard Design, Parallel Class, Projective Plane, Resolution, Resolvable, Steiner Triple System, Symmetric Block Design, Unital
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 Eric W. Weisstein