A block design, in the balanced incomplete sense (BIBD), is an incidence structure consisting of a set 
of 
points and an indexed family of
subsets of
called blocks.
Each block contains 
points, each point
lies in 
blocks, and every pair of distinct points
lies in exactly 
blocks. The parameters are positive
integers with 
(Dinitz and Stinson 1992, p. 1).
The blocks need not form a set partition of 
,
since they can overlap. For example, the Fano plane
has 
, 
, and 
, so each point belongs to
three blocks.
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