Designs based on the cycle structure of a Latin square autotopism

TL;DR

This paper investigates the cycle structures of Latin square autotopisms to classify symmetries, analyze their incidence structures, and determine parameters of related combinatorial designs for Latin squares up to order 7.

Contribution

It introduces a classification of Latin square autotopism cycle structures and characterizes the associated incidence structures as 1-designs, extending known parameters for orders up to 7.

Findings

Incidence structures form 1-$(v,k,r)$ designs

Complete parameter sets obtained for Latin squares of order up to 7

Classification of symmetries based on cycle structures

Abstract

Latin squares have been historically used in order to create statistical designs in which, starting from a small number of experiments, it can be obtained a large experimental space. In this sense, the optimization of the selection of Latin squares can be decisive. A factor to take into account is the symmetry that the experimental space must verify and which is established by the autotopism group of each Latin square. Although the size of this group is known for Latin squares of order up to 10, a classification of the different symmetries has not yet been done. In this paper, given a cycle structure of a Latin square autotopism, it is studied the regularity of the incidence structure formed by the set of autotopisms having this cycle structure and the set of Latin squares remaining stable by at least one of the previous autotopisms. Moreover, it is proven that every substructure given by the isotopism class of a Latin square is a 1-$(v,k,r)$ design. Since the corresponding parameter $k$ is known for Latin squares of order up to 7, we obtain the rest of the parameters of all these substructures and, consequently, a classification of all possible symmetries is reached for these orders.