Bounds on codes with few distances

We prove a new bound on the size of codes with few distances in the Hamming space, improving an earlier result of P. Delsarte. We also improve the Ray-Chaudhuri-Wilson bound of the size of uniform intersecting families of subsets (constant-weight codes) and the bound of Delsarte-Goethals-Seidel on the maximum size of spherical codes with few distances. Finally, we find the size of maximal binary codes and maximal constant-weight codes of small length with 2,3, and 4 distances.