Graphs on $\{0,1\}^n$ based on fixed Hamming distance

In general, this is an open problem. In the special case where $n$ is divisible by $4$ and $k=n/2$, the clique number is believed to be $n$ but this is equivalent to the Hadamard matrix conjecture. I think that the chromatic number is also unknown in this special case.

Not quite what you're looking for, but appears to be the closest thing in the literature: the clique number for the related case $d_H \le k$ is addressed in Sharifiyazdi's dissertation The Clique Number of Generalized Hamming Graphs; references therein also discuss the chromatic number.