A generalisation of [formula omitted]-designs

Abstract

This paper defines a class of designs which generalise t -designs, resolvable designs, and orthogonal arrays. For the parameters t = 2 , k = 3 and λ = 1 , the designs in the class consist of Steiner triple systems, Latin squares, and 1-factorisations of complete graphs. For other values of t and k , we obtain t -designs, Kirkman systems, large sets of Steiner triple systems, sets of mutually orthogonal Latin squares, and (with a further generalisation) resolvable 2-designs and indeed much more general partitions of designs, as well as orthogonal arrays over variable-length alphabets. The Markov chain method of Jacobson and Matthews for choosing a random Latin square extends naturally to Steiner triple systems and 1-factorisations of complete graphs, and indeed to all designs in our class with t = 2 , k = 3 , and arbitrary λ , although little is known about its convergence or even its connectedness.