  It is known that $N(n)$, the maximum number of mutually orthogonal latin
squares of order $n$, satisfies the lower bound $N(n) \ge n^{1/14.8}$ for large
$n$. For $h\ge 2$, relatively little is known about the quantity $N(h^n)$,
which denotes the maximum number of `HMOLS' or mutually orthogonal latin
squares having a common equipartition into $n$ holes of a fixed size $h$. We
generalize a difference matrix method that had been used previously for
explicit constructions of HMOLS. An estimate of R.M. Wilson on higher
cyclotomic numbers guarantees our construction succeeds in suitably large
finite fields. Feeding this into a generalized product construction, we are
able to establish the lower bound $N(h^n) \ge (\log n)^{1/\delta}$ for any
$\delta>2$ and all $n > n_0(h,\delta)$.
