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

This paper revisits a combinatorial structure called the large set of ordered design (LOD). Among others, we introduce a novel structure called Latin matching and prove that a Latin matching of order n leads to an LOD(n−1,n,2n−1); thus, we obtain constructions for LOD(1,2,3), LOD(2,3,5), and LOD(4,5,9). Moreover, we show that constructing a Latin matching of order n is at least as hard as constructing a Steiner system S(n−2,n−1,2n−2); therefore, the order of a Latin matching must be prime. We also show some applications in multiagent systems.