Some Constant Weight Codes from Primitive Permutation Groups
In recent years the detailed study of the construction of constant weight codes has been extended from length at most 28 to lengths less than 64. Andries Brouwer maintains web pages with tables of the best known constant weight codes of these lengths. In many cases the codes have more codewords than the best code in the literature, and are not particularly easy to improve. Many of the codes are constructed using a specified permutation group as automorphism group. The groups used include cyclic, quasi-cyclic, affine general linear groups etc. sometimes with fixed points. The precise rationale for the choice of groups is not clear.
In this paper the choice of groups is made systematic by the use of the classification of primitive permutation groups. Together with several improved techniques for finding a maximum clique, this has led to the construction of 39 improved constant weight codes.