  We present an algorithm that, for every fixed genus $g$, will enumerate all
hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic
in quasilinear time; that is, the time required for the algorithm is
$\widetilde{O}(q^{2g-1})$, where $q=\#k$. Such an algorithm already exists in
the case $g=2$, thanks to work of Mestre and Cardona and Quer, and in the case
$g=3$, thanks to work of Lercier and Ritzenthaler. Experimentally, it appears
that our new algorithm is about two orders of magnitude faster in practice than
ones based on their work.
