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 \(n\ge 6\) 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, 94B65

Keywords: Johnson space, Erdös-Ko-Rado, Delsarte inequalities