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Transitive Sets of Mutually Orthogonal Latin Squares

Amadou Keita, Ilya Shapiro

TL;DR

This paper examines MacNeish’s conjecture for Mutually Orthogonal Latin Squares (MOLS) within the framework of transitive autotopy actions, introducing group-packet data to encode the symmetry that relates MOLS to orthogonal arrays. It proves that MacNeish’s conjecture holds for simply transitive sets of MOLS by reducing to group-theoretic constraints and Sylow arguments, while also providing computational constructions and classifications that explore transitive and non-transitive cases. The authors develop a computational pipeline in SageMath/GAP to search for group packets, construct corresponding Latin squares, and classify them by their autotopy symmetry, generating new examples and comprehensive tables of transitivity behavior up to order 10. The work clarifies when the lower bound N(n) ≥ f(n) can be tight in the transitive regime and isolates counterexamples to Euler’s conjecture and MacNeish’s conjecture to non-simply transitive configurations, thereby refining the landscape of MOLS construction and symmetry.

Abstract

We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the conjecture holds. We provide some partial results towards the transitive case, as well as the outcome of a computer search, which introduces a new construction of MOLS. In particular, we were unable to find any transitive large (conjecture-violating) sets of MOLS in the literature.

Transitive Sets of Mutually Orthogonal Latin Squares

TL;DR

This paper examines MacNeish’s conjecture for Mutually Orthogonal Latin Squares (MOLS) within the framework of transitive autotopy actions, introducing group-packet data to encode the symmetry that relates MOLS to orthogonal arrays. It proves that MacNeish’s conjecture holds for simply transitive sets of MOLS by reducing to group-theoretic constraints and Sylow arguments, while also providing computational constructions and classifications that explore transitive and non-transitive cases. The authors develop a computational pipeline in SageMath/GAP to search for group packets, construct corresponding Latin squares, and classify them by their autotopy symmetry, generating new examples and comprehensive tables of transitivity behavior up to order 10. The work clarifies when the lower bound N(n) ≥ f(n) can be tight in the transitive regime and isolates counterexamples to Euler’s conjecture and MacNeish’s conjecture to non-simply transitive configurations, thereby refining the landscape of MOLS construction and symmetry.

Abstract

We investigate MacNeish's conjecture (known to be false in general) in the setting of what we call "transitive" Mutually Orthogonal Latin Squares (MOLS). When we restrict our attention to "simply transitive" MOLS, we find that the conjecture holds. We provide some partial results towards the transitive case, as well as the outcome of a computer search, which introduces a new construction of MOLS. In particular, we were unable to find any transitive large (conjecture-violating) sets of MOLS in the literature.
Paper Structure (14 sections, 17 theorems, 97 equations, 2 tables)

This paper contains 14 sections, 17 theorems, 97 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Sets of MOLS and orthogonal arrays
  3. Transitive MOLS and groups
  4. Some simply transitive examples
  5. MacNeish’s conjecture
  6. Latin square methods
  7. Autotopy groups of MOLS based on a group
  8. Examples of the construction
  9. Computational construction and classification pipeline
  10. Group and subgroup search
  11. Construction of Latin squares
  12. Classification of Latin squares
  13. Implementation and parameters
  14. Autotopy group data and summary tables

Key Result

Lemma 16

Let $(S,X_i,\pi_i)$ with $|X_i|=n$ be a transitive orthogonal array. Choose $s\in S$, and set $x_i=\pi_i(s)\in X_i$. Then is a group packet in $(q+2)\text{-GP}(n)$.

Theorems & Definitions (65)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • Remark 10
  • ...and 55 more