  Recently, Mubayi and Wang showed that for $r\ge 4$ and $\ell \ge 3$, the
number of $n$-vertex $r$-graphs that do not contain any loose cycle of length
$\ell$ is at most $2^{O( n^{r-1} (\log n)^{(r-3)/(r-2)})}$. We improve this
bound to $2^{O( n^{r-1} \log \log n) }$.
