In this chapter we discuss some basic properties of combinatorial designs and their relationship to codes. In Section 6.5, we showed how duadic codes can lead to projective planes. Projective planes are a special case of t-designs, also called block designs, which are the main focus of this chapter. As with duadic codes and projective planes, most designs we study arise as the supports of codewords of a given weight in a code.

t-designs

A t-(v, k, λ) design, or briefly a t-design, is a pair (P, B) where P is a set of v elements, called points, and B is a collection of distinct subsets of P of size k, called blocks, such that every subset of points of size t is contained in precisely λ blocks. (Sometimes one considers t-designs in which the collection of blocks is a multiset, that is, blocks may be repeated. In such a case, a t-design without repeated blocks is called simple. We will generally only consider simple t-designs and hence, unless otherwise stated, the expression “t-design” will mean “simple t-design.”) The number of blocks in B is denoted by b, and, as we will see shortly, is determined by the parameters t, v, k, and λ.