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.

Read more about Steiner System: Properties, History, Mathieu Groups, The Steiner System S(5, 6, 12), The Steiner System S(5, 8, 24)

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

**Steiner System**s

... 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 ...

... 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 ...

**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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