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

Famous quotes containing the words steiner and/or system:

    The variables are surprisingly few.... One can whip or be whipped; one can eat excrement or quaff urine; mouth and private part can be meet in this or that commerce. After which there is the gray of morning and the sour knowledge that things have remained fairly generally the same since man first met goat and woman.
    —George Steiner (b. 1929)

    The dominant metaphor of conceptual relativism, that of differing points of view, seems to betray an underlying paradox. Different points of view make sense, but only if there is a common co-ordinate system on which to plot them; yet the existence of a common system belies the claim of dramatic incomparability.
    Donald Davidson (b. 1917)