A linear algebraic approach to orthogonal arrays and Latin squares

TL;DR

This paper introduces a linear algebraic framework for analyzing orthogonal arrays and Latin squares, providing new methods for generating bases and understanding their structure.

Contribution

It develops an inclusion matrix approach and offers a straightforward algorithm for Latin squares using intercalates, extending previous module space methods.

Findings

Defined an inclusion matrix for orthogonal arrays

Provided an algorithm for Latin squares basis generation

Extended the approach to broader cases

Abstract

To study orthogonal arrays and signed orthogonal arrays, Ray-Chaudhuri and Singhi (1988 and 1994) considered some module spaces. Here, using a linear algebraic approach we define an inclusion matrix and find its rank. In the special case of Latin squares we show that there is a straightforward algorithm for generating a basis for this matrix using the so-called intercalates. We also extend this last idea.