Enumeration of Sets of Mutually Orthogonal Latin Rectangles

We study sets of mutually orthogonal Latin rectangles (MOLR), and a natural variation of the concept of self-orthogonal Latin squares which is applicable on larger sets of mutually orthogonal Latin squares and MOLR, namely that each Latin rectangle in a set of MOLR is isotopic to each other rectangle in the set. We call such a set of MOLR co-isotopic. In the course of doing this, we perform a complete enumeration of sets of t mutually orthogonal k x n Latin rectangles for k <= n <= 7, for all t < n up to isotopism, and up to paratopism. Additionally, for larger n we enumerate co-isotopic sets of MOLR, as well as sets of MOLR where the autotopism group acts transitively on the rectangles, and we call such sets of MOLR transitive. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are co-isotopic and/or transitive in each step of the construction process. We use the prefix stepwise to refer to sets of MOLR with this property at each step of their construction. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular hypergraphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.