A Steiner system 
is a set 
of 
points, and a set
of subsets of 
of size 
(called blocks), such that any
points of 
are in exactly
one of the blocks. The special case 
and 
corresponds to a so-called Steiner
triple system. For a projective plane, 
, 
, 
, and the blocks are simply lines.
The number 
of blocks containing a point in an 
Steiner system is independent of the point.
In fact,
 |
where 
is a binomial coefficient. The total number
of blocks 
is also determined and is given by
 |
For 
, these numbers also satisfy 
and 
by Fisher's
block design inequality.
An 
is a block design with 
when 
. The formulas for 
and 
require that 
divide 
and 
divide 
. For fixed 
, Wilson (1975) proved existence for all sufficiently
large 
satisfying both conditions. At smaller values of 
, constructions settling cases left open in the Handbook
of Combinatorial Designs (Colbourn and Dinitz 2007) include 
and 
(Banakh et al. 2026), 
and 
(Hetman 2026a), and 
and 
(Hetman 2026b). Hetman (2025) constructed 
and used a recursive construction to obtain 
.
The permutations of the points preserving the blocks of a Steiner system 
form the automorphism group
of 
. For example, consider 
the set of 9 points
in the two-dimensional vector
space over the field of three elements.
The blocks are the 12 lines of the form 
with 
, which have three elements
each. The system is an 
because any two points uniquely determine a line.
For this construction in affine geometry, the affine transformations preserving the lines
form a subgroup of the automorphism
group. For a vector space of dimension 
over a field
of 
elements, the same construction gives a Steiner system 
.
Several interesting groups arise as automorphism groups of Steiner systems. The following table lists the Mathieu
groups associated with the Witt designs. In the 
case, 
has index 2 in the
full automorphism group 
; the other listed groups
are the full automorphism groups. These groups are unique up to isomorphism,
and are not only sporadic simple
groups, but are also highly transitive.
| Mathieu group | Steiner system |
 |  |
 |  |
 |  |
 |  |
 |  |
A Steiner system is block-transitive when its automorphism group has a transitive group action
on the blocks. Wang et al. (2026) proved that no 
with 
is block-transitive. This establishes the Steiner
case of the conjecture of Cameron and Praeger (1993) that no nontrivial block-transitive
6-design exists.
and 
Exist." 18 May 2026. 
-Designs, II: Large 
." In Finite Geometries and Combinatorics (Ed.
F. de Clerck and J. W. P. Hirschfeld). Cambridge, England: Cambridge
University Press, pp. 103-120, 1993. 
, New Unitals of Order 
and Other Steiner Systems with Block Length 
." 24 Nov 2025. 
and 
." 12 Oct 2026a. 
and 
." J. Combin. Designs 34, 259-261,
2026b.