  Let $Q(n,k)$ be the number of $n$-ary quasigroups of order $k$. We derive a
recurrent formula for Q(n,4). We prove that for all $n\geq 2$ and $k\geq 5$ the
following inequalities hold: $({k-3}/2)^{n/2}(\frac{k-1}2)^{n/2} < log_2 Q(n,k)
\leq {c_k(k-2)^{n}} $, where $c_k$ does not depend on $n$. So, the upper
asymptotic bound for $Q(n,k)$ is improved for any $k\geq 5$ and the lower bound
is improved for odd $k\geq 7$. Keywords: n-ary quasigroup, latin cube, loop,
asymptotic estimate, component, latin trade.
