Classification of Binary Constant Weight Codes

A binary code <i>C</i> ⊆ F₂<i>n</i> with minimum distance at least <i>d</i> and codewords of Hamming weight <i>w</i> is called an <i>(n</i>,<i>d</i>,<i>w</i>) constant weight code. The maximum size of an <i>(n</i>,<i>d</i>,<i>w</i>) constant weight code is denoted by <i>A</i>(<i>n</i>,<i>d</i>,<i>w</i>), and codes of this size are said to be optimal. In a computer-aided approach, optimal <i>(n</i>,<i>d</i>,<i>w</i>) constant weight codes are here classified up to equivalence for <i>d</i>=4, <i>n</i> ≤ 12; <i>d</i>=6, <i>n</i> ≤ 14; <i>d</i>=8, <i>n</i> ≤ 17; <i>d</i>=10, <i>n</i> ≤ 20 (with one exception); <i>d</i>=12, <i>n</i> ≤ 23; <i>d</i>=14, <i>n</i> ≤ 26; <i>d</i>=16, <i>n</i> ≤ 28; and <i>d</i>=18, <i>n</i> ≤ 28. Moreover, several new upper bounds on <i>A</i>(<i>n</i>,<i>d</i>,<i>w</i>) are obtained, leading among other things to the exact values <i>A</i>(12,4,5)=80, <i>A</i>(15,6,7)=69, <i>A</i>(18,8,7)=33, <i>A</i>(19,8,7)=52, <i>A</i>(19,8,8)=78, and <i>A</i>(20,8,8)=130 . Since <i>A</i>(15,6,6)=70, this gives the first known example of parameters for which <i>A</i>(<i>n</i>,<i>d</i>,<i>w</i>-1) > <i>A</i>(<i>n</i>,<i>d</i>,<i>w</i>) with <i>w</i> ≤ <i>n</i>/2. A scheme based on double counting is developed for validating the classification results.