Latin Matchings and Ordered Designs OD(n−1, n, 2n−1) †

This paper revisits a combinatorial structure called the large set of ordered design ((Formula presented.)). Among others, we introduce a novel structure called Latin matching and prove that a Latin matching of order n leads to an (Formula presented.) ; thus, we obtain constructions for (Formula presented.), (Formula presented.), and (Formula presented.). Moreover, we show that constructing a Latin matching of order n is at least as hard as constructing a Steiner system (Formula presented.) ; therefore, the order of a Latin matching must be prime. We also show some applications in multiagent systems.