Block Design
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)
|
A BIBD is therefore commonly written as simply (
, 
, 
), since 
and 
are given in terms
of 
, 
, and 
by
 |  |  |
(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)
|
This matrix satisfies the equation
 |
(6)
|
where 
is a 
identity
matrix and 
is the 
unit
matrix (Dinitz and Stinson 1992).
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) |
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.
Weisstein, Eric W. "Block Design." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BlockDesign.html
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