Triples of Orthogonal Latin and Youden Rectangles For Small Orders

TL;DR

This paper provides a comprehensive enumeration and classification of triples of mutually orthogonal Latin and Youden rectangles for small orders, revealing new structures and symmetries, including the first enumeration of Youden rectangle triples.

Contribution

It presents the first complete enumeration of non-isotopic triples of orthogonal Latin rectangles and Youden rectangles for small orders, along with analysis of their automorphism groups and construction methods.

Findings

Enumerated triples for orders up to 7

First enumeration of Youden rectangle triples

Identified symmetrical triples of 4x8 rectangles

Abstract

We have performed a complete enumeration of non-isotopic triples of mutually orthogonal $k\times n$ Latin rectangles for $k\leq n \leq 7$. Here we will present a census of such triples, classified by various properties, including the order of the autotopism group of the triple. As part of this we have also achieved the first enumeration of pairwise orthogonal triples of Youden rectangles. We have also studied orthogonal triples of $k \times 8$ rectangles which are formed by extending mutually orthogonal triples with non-trivial autotopisms one row at a time, and requiring that the autotopism group is non-trivial in each step. This class includes a triple coming from the projective plane of order 8. Here we find a remarkably symmetrical pair of triples of $4 \times 8$ rectangles, formed by juxtaposing two selected copies of complete sets of MOLS of order 4.