  We prove that for every positive integer $m$, there exist infinitely many
simple abelian varieties over $\mathbb{F}_2$ of order $m$. The method is
constructive, building on the work of Madan--Pal in the case $m=1$ to produce
an explicit sequence of Weil polynomials giving rise to abelian varieties over
$\mathbb{F}_2$ of order $m$. This sequence itself depends on the choice of a
suitable generalized binary representation of $m$; by making careful choices of
this representation, we can ensure that the the resulting sequence of
polynomials have 2-adic Newton polygons which guarantee the existence of
suitable irreducible factors.
