when does active S box count reduce in AES difference propagation? what is the relationship between active S-boxes and MixColumns? can you please explain it with an example?
The answer to the second part of your question:
The AES MixColumns operator ensures that the 8 bytes (4 in the input column 4 in the output column) form the codewords of an MDS code over $GF(2^8)$, which means the minimum weight of the code, which is 5, equals the number of nonzero bytes.
Any nonzer byte contributes 1 to the minimum weight, by definition of Hamming Weight over $GF(2^8)$. A nonzero symbol has weight 1, regardless of how many bits of the eight is nonzero.
So the number of active Sboxes is at least five.