Bounds on sets with few distances

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered:•we improve the Ray-Chaudhuri–Wilson bound of the size of uniform intersecting families of subsets;•we refine the bound of Delsarte–Goethals–Seidel on the maximum size of spherical sets with few distances;•we prove a new bound on codes with few distances in the Hamming space, improving an earlier result of Delsarte. We also find the size of maximal binary codes and maximal constant-weight codes of small length with 2 and 3 distances.