Maximum Hamming distance when the Hamming bound is satisfied

Question

Say that we have K different codewords and each of them is of length N. Using the hamming bound theorem, one could determine the largest H possible so that any two codewords will have a larger Hamming distance than H. However since the solution is not unique, I wonder if there is any bound on how large the Hamming distance can be once the Hamming bound is satisfied. That is, does there exist a H' such that while any two codewords have larger than H Hamming distance, the largest of them can't be more than H'.