Steiner Systems
A Steiner system (named after Jakob Steiner) is 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 the general 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.
Projective planes and affine planes are examples of Steiner systems under the current definition while only the Fano plane (projective plane of order 2) would have been a Steiner system under the older definition.
Read more about this topic: Symmetric Design
Famous quotes containing the words steiner and/or systems:
“To shoot a man because one disagrees with his interpretation of Darwin or Hegel is a sinister tribute to the supremacy of ideas in human affairs—but a tribute nevertheless.”
—George Steiner (b. 1929)
“We have done scant justice to the reasonableness of cannibalism. There are in fact so many and such excellent motives possible to it that mankind has never been able to fit all of them into one universal scheme, and has accordingly contrived various diverse and contradictory systems the better to display its virtues.”
—Ruth Benedict (1887–1948)