Abstract. We give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets. Most known constructions depend on finite fields either directly or via Galois geometry. Our subject splits naturally in two parts, namely the existence problem for small and large maximal sets of MOLS, respectively; in the first case, difference matrix methods have proved to be particularly useful, whereas the second case rests almost completely on the study of maximal partial t-spreads in finite protective spaces. For this reason, we also give a short review of what is known on the existence of maximal partial t-spreads.

INTRODUCTION

In what follows, we shall give a survey on a topic in Finite Geometry which has generated considerable interest in the literature: the construction of maximal sets of mutually orthogonal Latin squares (MOLS) or, equivalently, of maximal nets. As is well-known, the existence of a (Bruck) net of order s and degree r (for short, an (s, r)-net)) is equivalent to that of a set of r–2 mutually orthogonal Latin squares of order s; it is also well-known that this correspondence respects maximality as well as the stronger property of being transversal-free.