Jacobians in isogeny classes of supersingular abelian threefolds in characteristic 2

We determine the isogeny classes of supersingular abelian threefolds over Fn2 containing the Jacobian of a genus 3 curve. In particular, we prove that for even n>6 there always exist a maximal and a minimal curves of genus 3 over Fn2. The methods provide an explicit construction of supersingular curves of genus 3 with Jacobian in a prescribed isogeny class.