Computing the autotopy group of a Latin square by cycle structure

TL;DR

This paper introduces an algorithm leveraging cycle structures of Latin square rows or columns to compute its autotopy group efficiently, providing bounds on group size and demonstrating polynomial-time computation for certain Latin squares.

Contribution

The paper presents a novel cycle-structure-based algorithm for autotopy group computation and establishes polynomial bounds for specific Latin square classes.

Findings

Bound on the size of the autotopy group

Polynomial-time algorithm for Latin squares with specific cycle structures

Effective computation method for autotopy groups

Abstract

An algorithm that uses the cycle structure of the rows, or the columns, of a Latin square to compute its autotopy group is introduced. As a result, a bound for the size of the autotopy group is obtained. This bound is used to show that the computation time for the autotopy group of Latin squares that have two rows or two columns that map from one to the other by a permutation which decomposes into a bounded number of disjoint cycles, is polynomial in the order $n$.