Units of hyperelliptic curves over $\mathbb{F}_2$

We study unit groups of rings of the form $\mathbb{F}_2[x,y]/(y^2 + gy + h)$, for $g, h \in \mathbb{F}_2[x]$ -- in particular, the question of (non)triviality of such unit groups. Up to automorphisms of $\mathbb{F}_2[x,y]$ we classify such rings into 3 distinct types. For 2 of the types we show that the unit group is always trivial, and conjecture that the unit group is always nontrivial for the 3rd type. We provide support for this conjecture both theoretically and computationally, via an algorithm that has been used to compute units in large degrees.