Kirkman Cubes

Publisher This chapter discusses the Kirkman cubes. A Kirkman cube KC3(v) is a 3-dimensional array on v elements such that the set of triples in the nonempty cells of each of its planes is a parallel class and the set of triples in the nonempty cells of the whole array is the set of triples of an STS(v)while the projection on any two dimensions never yields a Kirkman square (i.e. a UMGRD with k = 3, d = 2, λ = l, t = 2). A general method of constructing a KC3(v) directly is the well-known starter-adder method. The construction involves a “twisted” direct product. The chapter describes various recursive constructions for KC3 (v)s. The constructions are more general and applies to any UMGRD (k,t,λ,V). The most powerful of these constructions is PBD-closure.