# Steiner System

In combinatorial mathematics, a Steiner system (named after Jakob Steiner) is a type of block design, specifically a t-design with λ = 1 and t ≥ 2.

A Steiner system with parameters t, k, n, written S(t,k,n), is an n-element set S together with a set of k-element subsets of S (called blocks) with the property that each t-element subset of S is contained in exactly one block. In an alternate notation for block designs, an S(t,k,n) would be a t-(n,k,1) design.

This definition is relatively modern, generalizing the classical definition of Steiner systems which in addition required that k = t + 1. An S(2,3,n) was (and still is) called a Steiner triple system, while an S(3,4,n) was called a Steiner quadruple system, and so on. With the generalization of the definition, this naming system is no longer strictly adhered to.

As of 2012, an outstanding problem in design theory is if any nontrivial Steiner systems have t ≥ 6. It is also unknown if infinitely many have t = 5.

### Other articles related to "steiner system, steiner":

Constructions of The Mathieu Groups - Automorphism Groups of Steiner Systems
... up to equivalence a unique S(5,8,24) Steiner system W24 (the Witt design) ... The group M24 is the automorphism group of this Steiner system that is, the set of permutations which map every block to some other block ... Similarly, there exists up to equivalence a unique S(5,6,12) Steiner system W12, and the group M12 is its automorphism group ...
Projective Linear Group - Finite Fields - Mathieu Groups
... plane over the field with four elements, which is a Steiner system of type S(2,5,21) – meaning that it has 21 points, each line ("block", in Steiner terminology ... One calls this Steiner system W21 ("W" for Witt), and then expands it to a larger Steiner system W24, expanding the symmetry group along the way to the projective general linear ...
Fano Plane - Steiner System
... The Fano plane, as a block design, is a Steiner triple system ... As such, it can be given the structure of a quasigroup ...

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