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Computing random $r$-orthogonal Latin squares

Sergey Bereg

TL;DR

The paper develops randomized, polynomial-time algorithms to construct pairs of $r$-orthogonal Latin squares ($r$-OLS) and $r$-self-orthogonal Latin squares ($r$-SOLS) of order $n$, enabling computation for larger $n$ than brute-force methods. It introduces constructive methods (A1–A4) that start from random initial rows and complete via SDR-based matchings and cycle-switching, with assignment-based refinements to maximize or minimize the resulting orthogonality $r$. Empirical results show broad coverage of the $r$-spectrum for $n$ from 5 to 20, although values near $r=n^2$ remain hard to reach and some spectra are still unresolved. The work also investigates the expected orthogonality between random squares, reporting empirical estimates around $0.63$ of $n^2$ for $r(A,B)$ and around $0.38$–$0.395$ of $n^2$ for $r(A,A^T)$, and identifies open questions about spectra and asymptotic behavior.

Abstract

Two Latin squares of order $n$ are $r$-orthogonal if, when superimposed, there are exactly $r$ distinct ordered pairs. The spectrum of all values of $r$ for Latin squares of order $n$ is known. A Latin square $A$ of order $n$ is $r$-self-orthogonal if $A$ and its transpose are $r$-orthogonal. The spectrum of all values of $r$ is known for all orders $n\ne 14$. We develop randomized algorithms for computing pairs of $r$-orthogonal Latin squares of order $n$ and algorithms for computing $r$-self-orthogonal Latin squares of order $n$.

Computing random $r$-orthogonal Latin squares

TL;DR

The paper develops randomized, polynomial-time algorithms to construct pairs of -orthogonal Latin squares (-OLS) and -self-orthogonal Latin squares (-SOLS) of order , enabling computation for larger than brute-force methods. It introduces constructive methods (A1–A4) that start from random initial rows and complete via SDR-based matchings and cycle-switching, with assignment-based refinements to maximize or minimize the resulting orthogonality . Empirical results show broad coverage of the -spectrum for from 5 to 20, although values near remain hard to reach and some spectra are still unresolved. The work also investigates the expected orthogonality between random squares, reporting empirical estimates around of for and around – of for , and identifies open questions about spectra and asymptotic behavior.

Abstract

Two Latin squares of order are -orthogonal if, when superimposed, there are exactly distinct ordered pairs. The spectrum of all values of for Latin squares of order is known. A Latin square of order is -self-orthogonal if and its transpose are -orthogonal. The spectrum of all values of is known for all orders . We develop randomized algorithms for computing pairs of -orthogonal Latin squares of order and algorithms for computing -self-orthogonal Latin squares of order .
Paper Structure (5 sections, 6 theorems, 3 equations, 5 figures, 3 tables)

This paper contains 5 sections, 6 theorems, 3 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Completing Orthogonal Latin Rectangles
  4. Computing Self-Orthogonal Latin Rectangles
  5. Conclusions and Open Questions

Key Result

Theorem 1

For any positive integer $n\notin\{1,3\}$ there exists a Latin square of order $n$ that has no orthogonal mate.

Figures (5)

  • Figure 1: Orthogonal Latin squares using Euler's notation (one Latin square uses the first $n$ upper-case letters from the Latin alphabet and the other uses the first $n$ lower-case letters from the Greek alphabet). Orthogonal Latin squares are also known as Graeco-Latin squares or Euler squares.
  • Figure 2: A pair of $42$-orthogonal Latin squares of order 7 where all pairs $(i,j)$ except $(2,7),(3,2),(3,4),(4,5),(4,7),(6,1),(7,3)$ appear if $A$ and $B$ are superimposed.
  • Figure 3: (a) A Latin square $A$. (b) Switching a row cycle in $A$ between the third row and the fifth row. (c) Switching a symbol cycle in $A$ on symbols 2 and 3.
  • Figure 4: (a) A Latin rectangle $A$. (b) Function $f_{1,3}$ for $A$ induces a cycle (3,5) and a path (1,4,2). A Latin rectangle after switching column path (1,4,2) in $A$ is shown. (c) Switching a symbol path in $A$ on symbols 3 and 4.
  • Figure 5: Algorithm A2.

Theorems & Definitions (7)

  • Theorem 1: evans06ww06
  • Theorem 2: Zhu and Zhang zz-cs-03
  • Theorem 3: Zhang z-25-13
  • Theorem 4: M. Hall hall1945
  • Theorem 5: P. Hall hall1935
  • Theorem 6
  • proof