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Enumeration of Sets of Mutually Orthogonal Latin Rectangles

Gerold Jäger, Klas Markström, Denys Shcherbak, Lars-Daniel Öhman

TL;DR

This work develops a comprehensive framework for enumerating sets of mutually orthogonal Latin rectangles (MOLR), introducing homogeneous and transitive classes and a stepwise construction strategy to manage complexity. Central to the approach are the standard hypergraph representation and a reshaping transformation that relate $k\times n\, t$-MOLR to $(t+1)\times n\,(k-1)$-MOLR, enabling both structural insight and computational enumeration up to $n\le 7$. The authors implement and validate a scalable generation pipeline, classify MOLR by isotopism/paratopism and Aut$(A)$, and connect these combinatorial objects to finite geometries such as projective and affine planes, with detailed results for larger orders via restricted subclasses. Key findings include counts and patterns for $n=4$–$7$, the discovery of notable stepwise transitive MOLR at higher orders (notably a unique stepwise transitive $6$-MOLS of order $9$), and insights into how symmetry shapes the landscape of MOLR and their geometric interpretations. These results advance understanding of the existence and structure of MOLS/MOLR sets, inform connections to finite geometries, and provide a computational foundation for exploring longstanding questions about maximum MOLS, including the case $n=10$.

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.

Enumeration of Sets of Mutually Orthogonal Latin Rectangles

TL;DR

This work develops a comprehensive framework for enumerating sets of mutually orthogonal Latin rectangles (MOLR), introducing homogeneous and transitive classes and a stepwise construction strategy to manage complexity. Central to the approach are the standard hypergraph representation and a reshaping transformation that relate -MOLR to -MOLR, enabling both structural insight and computational enumeration up to . The authors implement and validate a scalable generation pipeline, classify MOLR by isotopism/paratopism and Aut, and connect these combinatorial objects to finite geometries such as projective and affine planes, with detailed results for larger orders via restricted subclasses. Key findings include counts and patterns for –, the discovery of notable stepwise transitive MOLR at higher orders (notably a unique stepwise transitive -MOLS of order ), and insights into how symmetry shapes the landscape of MOLR and their geometric interpretations. These results advance understanding of the existence and structure of MOLS/MOLR sets, inform connections to finite geometries, and provide a computational foundation for exploring longstanding questions about maximum MOLS, including the case .

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 mutually orthogonal Latin rectangles for , for all . 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.

Paper Structure

This paper contains 21 sections, 3 theorems, 1 equation, 8 figures, 40 tables.

Table of Contents

  1. Introduction
  2. Basic notation and definitions for $t$-MOLR
  3. The standard hypergraph representation
  4. Isotopism and paratopism
  5. Generation of $\mathbf{t}$-MOLR
  6. Finite geometries
  7. Basic notions for finite geometries
  8. Finite geometries from Latin rectangles
  9. A reshaping transformation
  10. Computational results and analysis
  11. The number of $\mathbf{t}$-MOLR
  12. Larger orders
  13. Autotopism group sizes of $\mathbf{t}$-MOLR
  14. Concluding remarks
  15. Sizes of autotopism groups of MOLR for $\mathbf{n=4}$
  16. ...and 6 more sections

Key Result

Lemma 5.1

The hypergraph $G(A)=(W,E)$ is linear, $(k+1)$-uniform, and $(k+1)$-partite, with $k$ vertex classes $W_1, W_2,\ldots, W_k$ of size $n$ corresponding to the rows of $A$, and one vertex class $W_0$ of size $t+1$ corresponding to the added singletons.

Figures (8)

  • Figure 1: The unique stepwise transitive $6$-MOLS of order 9.
  • Figure 2: The $G$-complete $3$-MOLS with $n=8$
  • Figure 3: A $3$-MOLR of size $9\times 10$, whose restriction to the first 8 rows is stepwise transitive.
  • Figure 4: The stepwise transitive $8$-MOLS of size $9\times9$ with $|\mathrm{Aut}|=10368.0$, corresponding to the Galois plane.
  • Figure 5: The stepwise transitive $8$-MOLS of size $9\times9$ with $|\mathrm{Aut}|=31104.0$, corresponding to the dual of the Hall plane.
  • ...and 3 more figures

Theorems & Definitions (9)

  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Lemma 5.1
  • proof
  • Theorem 5.2
  • proof
  • Theorem 6.4
  • proof