Negative Results for Orthogonal Triples of Latin Squares of Order $10$

Abstract

We consider whether an order-ten Latin square with an order-four Latin subsquare can belong to an orthogonal triple of Latin squares. We eliminate $20$ of $28$ possibilities for how this could occur by considering the structure of possible mates. Our technique supplements the small collection of existing tools for obtaining negative results regarding
the existence of collections of orthogonal Latin squares.