A generalisation of \(t\)-designs (Q1044956)

By imposing additional structure on the blocks of a design, a generalization of \(t\)-designs is introduced. It is shown to encompass latin squares, mutually orthogonal latin squares, triple systems including Steiner and Kirkman triple systems, 1-factorizations, and \(t\)-designs. This general setting is used to provide a common framework for many structural properties of these designs. Of particular interest is the extension of a Markov chain method for generating random latin squares to Steiner triple systems and 1-factorizations. The paper provides a valuable synthesis of a number of diverse research directions, and points out questions that remain open.