I think F^15_2 means you have length 15 words on the symbols {0,1}. In that case, Hamming code distance should be the number of spots that two words disagree. So if your word is

000000000000000

Then 000000000000001 has code distance 1. 000000000000011 has code distance 2.

This might be just a combinatorial argument of counting the words which disagree with some fixed word in n spots where n is your code distance.