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

Computing quadratic function fields with high 3-rank via cubic field tabulation