Pairs of MOLS of order ten satisfying non-trivial relations

Abstract

A relation on a \(k\text {{-}net}(n)\) (or, equivalently, a set of \(k-2\) mutually orthogonal Latin squares of order n) is an \({\mathbb {F}}_{2}\) linear dependence within the incidence matrix of the net. Dukes and Howard (2014) showed that any \(6\text {{-}net}(10)\) satisfies at least two non-trivial relations, and classified the relations that could appear in such a net. We find that, up to equivalence, there are \(18\,526\,320\) pairs of MOLS satisfying at least one non-trivial relation. None of these pairs extend to a triple. We also rule out one other relation on a set of 3-MOLS from Dukes and Howard’s classification.