Existence of Hamming code
The Hamming bound is an upper bound on the size of codes. It's not a tight bound in general, though in some specific cases it is achievable. Codes achieving the Hamming bound are called perfect codes.

When the alphabet size is a prime power, all perfect codes are known: apart from some trivial cases (repetition codes, codes with one codeword, and codes containing all possible codewords), they share the same parameters as a Hamming code or a Golay code.

What this means is that if your alphabet size is a prime power and you have parameters satisfying the Hamming bound, then a code with these parameters actually exists only in the cases listed above. In any other case, the bound isn't tight.