A census of genus 6 curves over ${\mathbb{F}}_2$

We compile a complete list of isomorphism class representatives of curves of genus 6 over ${\mathbb{F}}_2$. We use explicit descriptions of canonical curves in each stratum of the Brill–Noether stratification of the moduli space ${\mathcal{M}}_6$, due to Mukai in the generic case. Our computed value of $\mathrm{\#}{\mathcal{M}}_6({\mathbb{F}}_2)$ agrees with the Lefschetz trace formula as recently computed by Bergstrom–Canning–Petersen–Schmitt.