Hyperelliptic curves over $\mathbb{F} _2$ of every $2$-rank without extra automorphisms

We prove that for any pair of integers $0\leq r\leq g$ such that $g\geq 3$ or $r>0$, there exists a (hyper)elliptic curve $C$ over $\mathbb{F} _2$ of genus $g$ and $2$-rank $r$ whose automorphism group consists of only identity and the (hyper)elliptic involution. As an application, we prove the existence of principally polarized abelian varieties $(A,\lambda)$ over $\mathbb{F} _2$ of dimension $g$ and $2$-rank $r$ such that $\operatorname{Aut}(A,\lambda)=\{\pm 1\}$.