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On the minimum number of entries in a pair of maximal orthogonal partial Latin squares

Diane M. Donovan, Mike Grannell, Emine Şule Yazıcı

TL;DR

The paper resolves the conjecture that the minimum number of filled cells in a maximal orthogonal partial Latin square pair of order $n$ satisfies $F \ge \lceil n^2/3 \rceil$, and for $n \ge 21$ this bound is tight, achieved by constructions that partition the structure into three mutually orthogonal Latin squares on disjoint symbol sets. The authors prove the bound via a lemma on maximal partial transversals and a double-counting argument on row/column/entry frequencies, showing equality forces a highly regular frequency distribution and a three-OLS decomposition, with the minimum being essentially unique up to relabeling. They also connect these combinatorial objects to $n$-ary codes of length 4 with minimum distance 3 and covering radius 2, concluding that the minimum code size matches the bound for large $n$. The work links maximality, design structure, and coding theory, and points toward generalizations to $k$-MOPLS with analogous asymptotics.

Abstract

It is shown that if $F$ denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order $n$, then $F\ge n^2/3$. This resolves a conjecture raised in an earlier paper by the current authors. It is also shown that, for $n\ge 21$, the least possible number of filled cells in a pair of maximal orthogonal partial Latin squares is $\lceil n^2/3 \rceil$, and that the structure that achieves this bound is unique up to permutations of rows, columns and entries.

On the minimum number of entries in a pair of maximal orthogonal partial Latin squares

TL;DR

The paper resolves the conjecture that the minimum number of filled cells in a maximal orthogonal partial Latin square pair of order satisfies , and for this bound is tight, achieved by constructions that partition the structure into three mutually orthogonal Latin squares on disjoint symbol sets. The authors prove the bound via a lemma on maximal partial transversals and a double-counting argument on row/column/entry frequencies, showing equality forces a highly regular frequency distribution and a three-OLS decomposition, with the minimum being essentially unique up to relabeling. They also connect these combinatorial objects to -ary codes of length 4 with minimum distance 3 and covering radius 2, concluding that the minimum code size matches the bound for large . The work links maximality, design structure, and coding theory, and points toward generalizations to -MOPLS with analogous asymptotics.

Abstract

It is shown that if denotes the number of filled cells in a superimposed pair of maximal orthogonal partial Latin squares of order , then . This resolves a conjecture raised in an earlier paper by the current authors. It is also shown that, for , the least possible number of filled cells in a pair of maximal orthogonal partial Latin squares is , and that the structure that achieves this bound is unique up to permutations of rows, columns and entries.
Paper Structure (3 sections, 5 theorems, 15 equations, 7 figures)

This paper contains 3 sections, 5 theorems, 15 equations, 7 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Concluding Remarks

Key Result

Lemma 1.1

Suppose that $n>1$ and that $M$ is an MPLS($n$) of minimum size, so that $M$ has $F=\lceil n^2/2 \rceil$ filled cells. Then $M$ can be partitioned into two Latin squares, on different point sets, with orders that sum to $n$ and are either $\lfloor n/2 \rfloor$ or $\lfloor n/2 \rfloor+1$.

Figures (7)

  • Figure 1: MPLS(6) and MPLS(7)
  • Figure 2: A partition of $M$.
  • Figure 3: An MOPLS(9) with 27 filled cells.
  • Figure 4: A partition of $D$.
  • Figure 5: A partition of $M$.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Lemma 1.1
  • proof
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • proof : Proof of Theorem \ref{['th:main']}
  • Corollary 2.2.1
  • Theorem 2.3
  • proof