I'm defining the Hamming graph $H(d,q)$ in the usual way, so we have a set $S$ of $q$ elements, the hamming graph $H(d,q)$ has vertex set $S^{d}$ (the set of all ordered $d$-tuples of elements of $S$) and the edge set formed of all pairs of vertices which differ in exactly one element.

I'm interested in the following questions of which I've found it difficult to find answers.

- Is the exact independence number of $H(d,q)$ known?
- For any set of size $k$ less than or equal to the independence number, how many independent sets of size $k$ does the hamming graph contain?