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We address the maximum size of binary codes and binary constant weight codes with few distances. Previous works established a number of bounds for these quantities as well as the exact values for a range of small code lengths. As our main results, we determine the exact size of maximal binary codes with two distances for all lengths \ (n 6\) as well as the exact size of maximal binary constant weight codes with \ (2\), \ (3\), and \ (4\) distances for several values of the weight and for all but small lengths. Mathematics Subject Classifications: 52C10, 05D05, 94B65Keywords: Johnson space, Erdös-Ko-Rado, Delsarte inequalities
Barg et al. (Mon,) studied this question.