This question arose during solving an information theory problem. Suppose $l$ is the smallest integer such that $$2^l\geq {n\choose k}$$ define $\rho=\frac{k}{n}$. How we can characterize $\rho$ as a function of $l$ and $n$? i.e. what is $\rho(l,n)$?
I am interested in both asymptotic ($n\to\infty$) and non-asymptotic cases.
My effort: I know for $n\to\infty$, $l=n[H(\rho)+\epsilon]$. So, I think I can write: $\rho=H^{-1}(\frac{l}{n}-\epsilon)$, but I am not sure if this is correct, and also what is the answer in non-asymptotic case?