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.

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

Other articles related to "steiner system, steiner":

Projective Linear Group - Finite Fields - Mathieu Groups
... One begins with the projective 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 ... 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 ...
Constructions of The Mathieu Groups - Automorphism Groups of Steiner Systems
... There exists 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 ... exists up to equivalence a unique S(5,6,12) Steiner system W12, and the group M12 is its automorphism group ...
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 ...

Famous quotes containing the words system and/or steiner:

    Justice in the hands of the powerful is merely a governing system like any other. Why call it justice? Let us rather call it injustice, but of a sly effective order, based entirely on cruel knowledge of the resistance of the weak, their capacity for pain, humiliation and misery. Injustice sustained at the exact degree of necessary tension to turn the cogs of the huge machine-for- the-making-of-rich-men, without bursting the boiler.
    Georges Bernanos (1888–1948)

    We know that a man can read Goethe or Rilke in the evening, that he can play Bach and Schubert, and go to his day’s work at Auschwitz in the morning.
    —George Steiner (b. 1929)