  We show that k=w+2 mutually unbiased bases can be constructed in any square
dimension d=s^2 provided that there are w mutually orthogonal Latin squares of
order s. The construction combines the design-theoretic objects (k,s)-nets
(which can be constructed from w mutually orthogonal Latin squares of order s
and vice versa) and generalized Hadamard matrices of size s. Using known lower
bounds on the asymptotic growth of the number of mutually orthogonal Latin
squares (based on number theoretic sieving techniques), we obtain that the
number of mutually unbiased bases in dimensions d=s^2 is greater than s^1/14.8
for all s but finitely many exceptions. Furthermore, our construction gives
more mutually orthogonal bases in many non-prime-power dimensions than the
construction that reduces the problem to prime power dimensions.
