Hyperelliptic curves over F₂ of every 2-rank without extra automorphisms

We prove that for any pair of integers 0 ≤ r ≤ g such that g ≥ 3 or r > 0, there exists a (hyper)elliptic curve C over F₂ 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, λ) over F₂ of dimension g and 2-rank r such that Aut(A, λ) = {±1}.