Classification of Binary Constant Weight Codes

A binary code C ⊆ F 2 n with minimum distance at least d and codewords of Hamming weight w is called an (n , d , w ) constant weight code. The maximum size of an (n , d , w ) constant weight code is denoted by A ( n , d , w ), and codes of this size are said to be optimal. In a computer-aided approach, optimal (n , d , w ) constant weight codes are here classified up to equivalence for d =4, n ≤ 12; d =6, n ≤ 14; d =8, n ≤ 17; d =10, n ≤ 20 (with one exception); d =12, n ≤ 23; d =14, n ≤ 26; d =16, n ≤ 28; and d =18, n ≤ 28. Moreover, several new upper bounds on A ( n , d , w ) are obtained, leading among other things to the exact values A (12,4,5)=80, A (15,6,7)=69, A (18,8,7)=33, A (19,8,7)=52, A (19,8,8)=78, and A (20,8,8)=130 . Since A (15,6,6)=70, this gives the first known example of parameters for which A ( n , d , w -1) > A ( n , d , w ) with w ≤ n /2. A scheme based on double counting is developed for validating the classification results.