# Steiner Systems

A particularly interesting special case of (v,k,t)-covering designs is Steiner systems, where each t-set is coverered exactly once.  This page describes known results about Steiner systems, and gives links to them (including ones that don’t fit in the normal covering designs tables).

The tables below give a list of known infinite families and 5-designs.  For particular parameters that are small enough you can search in the La Jolla Covering Repository.  If a Steiner system exist it will be given; if it is known not to exist the lower bound for the covering design size will indicate that, and the comment should give a reference to that result.

## Infinite Families

There are a few infinite families of Steiner systems with t=2 and 3:

NameParameters
Steiner Triple Systems(v,3,2) for v ≡ 1 or 3 (mod 6)
Steiner Quadruple Systems(v,4,3) for v ≡ 2 or 4 (mod 6)
Affine Geometries(qn,q,2), q a prime power, n ≥ 2
Spherical Geometries(qn+1,q+1,3), q a prime power, n ≥ 2
Projective Geometries(qn+...+q+1,q+1,2), q a prime power, n ≥ 2
Unitals(q3+1,q+1,2), q a prime power
Denniston Designs(2r+s+2r-2s,2r,2), 2 ≤ r < s

## Known Steiner 5-designs

SystemSizeComment
S(5,6,12)132
S(5,8,24)759Unique
S(5,6,24)7084Three nonisomorphic systems
S(5,7,28)4680
S(5,6,36)62832
S(5,6,48)285384
S(5,6,72)2331924
S(5,6,84)5145336
S(5,6,108)18578196
S(5,6,132)51553216
S(5,6,168)175036708
S(5,6,244)1152676008

## References

There is a huge body of research on Steiner systems. For a good recent survey (from which most of the information on this page came from), see Steiner Systems, Charles J. Colbourn and Rudolf Mathon, in Handbook of Combinatorial Designs, second edition, (2007) pp. 102-110.

Another good reference is Design Theory, T. Beth, D. Jungnickel and H. Lenz, second edition (1999).