Let $A_2(n,d)$ denote the maximum number of words in a binary code (not necessarily linear) with length $n$ and distance $d$. The value of this function is not known in general, though there are tables for small values of $n$ and $d$, e.g. http://www.win.tue.nl/~aeb/codes/binary-1.html.
Suppose $n = 2^k$ and $d = 2^j$ (with $j < k$) are both powers of $2$. Is there a known formula for $A_2(2^k,2^j)$? It seems like a strong enough assumption that a closed formula should exist.
I am also interested if there are other exact results that apply when $n$ and $d$ can be large.