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: Block Design
Famous quotes containing the words steiner and/or systems:
“Words that are saturated with lies or atrocity, do not easily resume life.”
—George Steiner (b. 1929)
“The only people who treasure systems are those whom the whole truth evades, who want to catch it by the tail. A system is just like truth’s tail, but the truth is like a lizard. It will leave the tail in your hand and escape; it knows that it will soon grow another tail.”
—Ivan Sergeevich Turgenev (1818–1883)