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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 M12 and M24, 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 = m2.

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(2,35), 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.