  In this paper we prove a function field version of a theorem by Rudnick and
Soundararajan about lower bounds for moments of quadratic Dirichlet
$L$-functions. We establish lower bounds for the moments of quadratic Dirichlet
$L$--functions associated to hyperelliptic curves of genus $g$ over a fixed
finite field $\mathbb{F}_{q}$ in the large genus $g$ limit.
