Maximal partial spreads and transversal-free translation nets

Abstract

In this paper, we are concerned with three topics in finite geometry which have generated much interest in the literature: maximal partial spreads (or t-spreads), translation nets, and maximal sets of mutually orthogonal Latin squares. We obtain large maximal sets of MOLS for infinitely many new parameter pairs by constructing appropriate transversal-free translation nets which in turn belong to suitable maximal partial spreads. To this purpose, we first obtain some improved bounds on translation nets and partial t-spreads and then give a detailed study of the possible extensions of a translation net of small or critical deficiency. In order to apply these results, we also construct maximal partial t-spreads with previously unknown parameters. Given any prime power q = p a , where p is a prime ⩾5, we show the existence of a transversal-free translation net of order q 2 (and hence a maximal set of MOLS of order q 2) for each of the deficiencies d = q – 1, q, and q + 1. Finally, using results of Evans, we also obtain interesting examples of rather small transversal-free translation nets; in particular, we determine all transversal-free translation nets of order p 2 and degree p + 1 for a prime p with abelian translation group.