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: (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.

💡 Research Summary

The paper introduces a unified method for bounding the cardinality of finite point sets that realize only a few distinct distances in a metric space. Building on Delsarte’s linear‑programming (LP) framework, the authors develop a new “k‑distance bound” that applies to any space where the distance values are limited to at most k distinct numbers. The core of the technique is the construction of distance‑polynomial matrices whose entries are weighted by non‑negative coefficients (Lagrange multipliers). By enforcing positive semidefiniteness of the resulting matrix, a determinant inequality is obtained, which together with eigenvalue interlacing yields an explicit upper bound on the size of the set.

The general bound is then specialized to three classical settings, each of which previously had its own ad‑hoc bound:

1. Uniform intersecting families – For a t‑uniform family of subsets of an n‑element ground set that is intersecting, the classic Ray‑Chaudhuri–Wilson (RCW) bound (\binom{n-1}{t-1}) is improved to (\binom{n-1}{t-1}-\binom{n-t-1}{t-1}). The improvement follows by interpreting the family as a 2‑distance set (distances 0 and 1) in the Johnson scheme and applying the new LP inequality.

2. Spherical few‑distance sets – In the unit sphere (S^{m-1}), Delsarte‑Goethals‑Seidel (DGS) gave an upper bound (\frac{(m+1)(m+2)}{2}) for sets with at most two distances. By refining the spherical harmonic expansion and using a tighter normalization of the associated Gegenbauer polynomials, the authors reduce the bound to (\frac{(m+1)(m+2)}{2}-\frac{m(m-1)}{2}) for the two‑distance case, and more generally to (\frac{(m+1)(m+2)}{2}-\frac{k(k-1)}{2}) for k distances.

3. Hamming‑space codes with few distances – For binary codes in ({0,1}^{n}) whose pairwise Hamming distances belong to a set of size k, the classical Delsarte bound (derived from Krawtchouk polynomials) is sharpened by introducing an additional set of constraints that capture the limited distance spectrum. In the concrete cases of two and three distances the new bound exceeds Delsarte’s by roughly 8–12 % for moderate n, and exact maximal sizes are determined for all n ≤ 15 by exhaustive computer search.

The paper also provides exact extremal values for small parameters: maximal binary codes and constant‑weight codes of length up to 15 with 2 or 3 distances are listed, together with explicit constructions that attain the new bounds. These constructions include the familiar Hamming code (which remains optimal for the 2‑distance case) and several novel non‑linear configurations for the 3‑distance scenario.

Methodologically, the authors blend three ingredients: (i) a systematic choice of distance‑polynomial bases (Krawtchouk, Gegenbauer, or Johnson polynomials depending on the ambient space), (ii) a refined LP formulation that adds “distance‑spectrum” constraints, and (iii) a determinant‑based positivity argument that yields tighter eigenvalue interlacing inequalities than previously known. The combination leads to a bound that is simultaneously more general and strictly stronger than the earlier results in each of the three domains.

In the concluding section the authors discuss possible extensions. The framework naturally extends to other association schemes (e.g., Grassmann or Hamming schemes over larger alphabets) and to non‑symmetric metrics such as graph distances. They also suggest that the determinant‑based approach might be combined with semidefinite programming hierarchies to tackle open problems like the maximal size of equiangular lines or spherical designs with prescribed angle sets.

Overall, the paper delivers a significant advance in the theory of few‑distance configurations, unifying disparate combinatorial bounds under a single LP‑determinant methodology and delivering concrete improvements for intersecting families, spherical codes, and Hamming‑space codes.