Enumeration of Sets of Mutually Orthogonal Latin Rectangles

TL;DR

This paper provides a comprehensive enumeration of non-isotopic sets of mutually orthogonal Latin rectangles up to size 7, exploring their properties and connections to finite geometries and regular graphs.

Contribution

It introduces the concept of homogeneous and transitive sets of MOLR and performs a complete enumeration for various parameters, including the construction process and symmetry properties.

Findings

Complete enumeration of non-isotopic MOLR sets up to size 7.

Identification of connections between MOLR sets and finite projective planes.

Discovery that most projective planes of order ≤ 9 relate to stepwise transitive MOLR.

Abstract

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 \emph{homogeneous}. In the course of doing this, we perform a complete enumeration of non-isotopic sets of $t$ mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$, for all $t < n$. Specifically, we keep track of homogeneous 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 \emph{transitive}. We build the sets of MOLR row by row, and in this process we also keep track of which of the MOLR are homogeneous and/or transitive in each step of the construction process. We use the prefix \emph{stepwise} to refer to sets of MOLR with this property. Sets of MOLR are connected to other discrete objects, notably finite geometries and certain regular graphs. Here we observe that all projective planes of order at most 9 except the Hughes plane can be constructed from a stepwise transitive MOLR.