What is the independence number of hamming graph? Hamming graph H(d,q) is  Cartesian product of d complete graphs Kq. We know the independence number of direct product of d complete graphs Kq. What is the independence number of hamming graph?

Experiments suggest it might be $q^{d-1}$.
Here is data from sage:
d= 2 q= 2 alpha= 2 factor= 2  alpha - q^(d-1) 0
d= 2 q= 3 alpha= 3 factor= 3  alpha - q^(d-1) 0
d= 2 q= 4 alpha= 4 factor= 2^2  alpha - q^(d-1) 0
d= 2 q= 5 alpha= 5 factor= 5  alpha - q^(d-1) 0
d= 2 q= 6 alpha= 6 factor= 2 * 3  alpha - q^(d-1) 0
d= 3 q= 2 alpha= 4 factor= 2^2  alpha - q^(d-1) 0
d= 3 q= 3 alpha= 9 factor= 3^2  alpha - q^(d-1) 0
d= 3 q= 4 alpha= 16 factor= 2^4  alpha - q^(d-1) 0
d= 3 q= 5 alpha= 25 factor= 5^2  alpha - q^(d-1) 0
d= 3 q= 6 alpha= 36 factor= 2^2 * 3^2  alpha - q^(d-1) 0
d= 4 q= 2 alpha= 8 factor= 2^3  alpha - q^(d-1) 0
d= 4 q= 3 alpha= 27 factor= 3^3  alpha - q^(d-1) 0
d= 4 q= 4 alpha= 64 factor= 2^6  alpha - q^(d-1) 0
d= 4 q= 5 alpha= 125 factor= 5^3  alpha - q^(d-1) 0

