We present recent results on the computation of quadratic function fields
with high 3-rank. Using a generalization of a method of Belabas on cubic field
tabulation and a theorem of Hasse, we compute quadratic function fields with
3-rank $ \geq 1$, of imaginary or unusual discriminant $D$, for a fixed $|D| =
q^{\deg(D)}$. We present numerical data for quadratic function fields over
$\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$ with
$\deg(D) \leq 11$. Our algorithm produces quadratic function fields of minimal
genus for any given 3-rank.