Bounds on sets with few distances

TL;DR

This paper develops new bounds on the size of finite point sets with few distances in metric spaces and applies these to improve bounds in combinatorics, coding theory, and spherical geometry.

Contribution

It introduces a novel estimate for the size of sets with few distances and refines several existing bounds in related combinatorial and geometric problems.

Findings

Improved bound on uniform intersecting families of subsets

Refined maximum size bounds for spherical sets with few distances

New bounds on codes with few distances in Hamming space

Abstract

We derive a new estimate of the size of finite sets of points in metric spaces with few distances. The following applications are considered: (1) we improve the Ray-Chaudhuri--Wilson bound of the size of uniform intersecting families of subsets; (2) we refine the bound of Delsarte-Goethals-Seidel on the maximum size of spherical sets with few distances; (3) 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.