Combinatorial designs

The last several posts have been building up to this one, moving from tangible to more abstract. The posts on finite geometry gave examples of Steiner triples and more generally Steiner systems, introduced in the previous post. Combinatorial designs take this one step further. The previous post gave the following definition: A Steiner triple system, written STS(n), is a collection of three-element subsets, called blocks, of a set X with n elements, such that every two-element subset of X belongs to exactly one block. The post pointed out that the smallest projective plane is STS(7) and the smallest affine plane is STS(9). The post then defined a general Steiner system: A Steiner system S(t, k, n) is a collection of k-element subsets, called blocks, of a set X with n elements, such that every t-element subset of X belongs to exactly one block. Moving from Steiner triples to Steiner systems we generalized the size of the blocks from 3 to k, and generalized the size of sets contained in…

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