Upper bound on size of minimal binary coverage code
So, according to Theorem 4.3 of this monograph by Réné Struik, if we consider $q$-ary binary codes of length $n$, such that $n \to \infty$ with $r/n \to p \in [0,(q-1)/q]$, then the minimal code size has the following asymptotic limit
$$
(1/n)\log_q N_{n,q}(r) \to 1-H_q(p),
$$
where $H_q(p)$ is the $q$-ary entropy of $p$. This answers my specific problem which corresponds to $q=2$.