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# Steiner systems

A *Steiner system* S(*t*,*k*,*v*)
is a *t*-(*v*,*k*,1) design, that is,
a collection of *k*-subsets (called *blocks*) of a *v*-set
such that each *t*-tuple of elements of this *v*-set
is contained in a unique block.
By taking the *derived design* (taking all blocks that contain
a fixed element, and then discarding that element), one finds a
S(*t*–1,*k*–1,*v*–1) from a S(*t*,*k*,*v*).

No Steiner systems are known with *t* larger than 5.
The Witt designs S(5,6,12) and S(5,8,24) belong to the Mathieu groups
M_{12} and M_{24}, respectively.
The blocks of S(5,8,24) are called *octads*.
The Steiner systems S(5,8,24), S(4,7,23), S(3,6,22), S(2,5,21)
and S(5,6,12), S(4,5,11), S(3,4,10), S(2,3,9) are unique.
A handful of other Steiner 5-designs is known, see below.
The only known Steiner 4-designs are the derived systems from
Steiner 5-designs.

The *block graph* of a Steiner 2-design S(2,*m*,*n*)
is the graph with the blocks as vertices, where two blocks are adjacent
when they have nonempty intersection.
Such a graph is strongly regular, with parameters
*v* = *n*(*n* – 1)/*m*(*m* – 1),
*k* = *m*(*n* – *m*)/(*m* – 1),
λ = (*m* – 1)^{2} + (*n* – 1)/(*m*–1) – 2,
μ = *m*^{2}.

More generally, the block graph of a quasi-symmetric design
is strongly regular.

When *m* = 3, these designs are called
*Steiner triple systems*.

## Steiner 5-designs

Steiner systems S(5,6,*v*) are known for
*v* = 12, 24, 36, 48, 72, 84, 108, 132, 168, 244.
Also Steiner systems S(5,7,28) are known.
There is a unique S(5,8,24).
## References

A. Betten, R. Laue & A. Wassermann,
*A Steiner 5-design on 36 points*,
Designs, Codes and Cryptography **17** (1999) 181-186.
A. Betten, R. Laue, S. Molodtsov & A. Wassermann,
*Steiner systems with automorphism groups PSL(2,71), PSL(2,83)
and PΣL(2,3*^{5}),
J. Geometry **67** (2000) 35-41.

R.H.F. Denniston,
*Some new 5-designs*,
Bull. London Math. Soc. **8** (1976) 263-267.

M.J. Grannell, T.S. Griggs,
*A Steiner system S(5,6,108)*,
Discr. Math. **125** (1994) 183-186.

M.J. Grannell, T.S. Griggs & R.A. Mathon,
*Some Steiner 5-designs with 108 and 132 points*,
J. Comb. Designs **1** (1993) 213238.

W.H. Mills,
*A new 5-design*,
Ars Combinatoria **6** (1978) 193-195.

E. Witt,
*Die 5-fach transitiven Gruppen von Mathieu*,
Abh. Math. Sem. Univ. Hamburg **12** (1938) 256264.