The maximal partial spreads of PG(3,4) were recently classified by Leonard Soicher. Each such partial spread (with r lines, say) yields a translation net of order 16 and degree r and hence a set of r-2 mutually orthogonal Latin squares of order 16. We determine which of these nets are transversal-free. In particular, we obtain sets of t mutually orthogonal Latin squares of order 16. We determine which of these nets are transversal-free. In particular, we obtain sets of t MAXMOLS(16) for two previously unknown cases, namely for t=9 and t=10.

We also use maximal partial spreads in PG(3,4) \ PG(3,2) with r lines, where r is in {6,7}, to construct transversal-free translation nets of order 16 and degree r+3 and hence maximal sets of r+1 mutually orthogonal Latin squares of order 16. Thus we obtain sets of t MAXMOLS(16) for two further open cases, namely for t=7 and t=8.

We also present an infinite class of 2²ⁿ⁺¹-1 MAXMOLS(2²ⁿ⁺²), n not equal to 1.