  This paper presents an algorithm for generating all imaginary and unusual
discriminants up to a fixed degree bound that define a quadratic function field
of positive 3-rank. Our method makes use of function field adaptations of a
method due to Belabas for finding quadratic number fields of high 3-rank and of
a refined function field version of a theorem due to Hasse. We provide
numerical data for discriminant degree up to 11 over the finite fields
$\mathbb{F}_{5}, \mathbb{F}_{7}, \mathbb{F}_{11}$ and $\mathbb{F}_{13}$. A
special feature of our technique is that it produces quadratic function fields
of minimal genus for any given 3-rank. Taking advantage of certain
$\mathbb{F}_{q}(t)$-automorphisms in conjunction with Horner's rule for
evaluating polynomials significantly speeds up our algorithm in the imaginary
case; this improvement is unique to function fields and does not apply to
number field tabulation. These automorphisms also account for certain
divisibility properties in the number of fields found with positive 3-rank. Our
numerical data mostly agrees with the predicted heuristics of
Friedman-Washington and partial results on the distribution of such values due
to Ellenberg-Venkatesh-Westerland for quadratic function fields over the finite
field $\finfldq{q}$ where $q \equiv -1 \pmod{3}$. The corresponding data for $q
\equiv 1 \pmod{3}$ does not agree closely with the previously mentioned
heuristics and results, but does agree more closely with some recent number
field conjectures of Malle and some work in progress on proving such
conjectures for function fields due to Garton.
