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) |
-Configuration."
§8.1 in