Lower bounds for designs in symmetric spaces

TL;DR

This paper establishes lower bounds for the size of designs in symmetric spaces using harmonic analysis, revealing that large designs must cover the space with spherical caps, generalizing linear programming bounds.

Contribution

It introduces a novel approach to bounding designs in symmetric spaces via spectral estimates and harmonic analysis, extending classical bounds to higher-rank spaces.

Findings

Lower bounds depend on spectral properties of symmetry-invariant operators.

Designs in symmetric spaces must cover the entire space with spherical caps.

Results generalize linear programming bounds to broader classes of spaces.

Abstract

A design is a finite set of points in a space on which every "simple" functions averages to its global mean. Illustrative examples of simple functions are low-degree polynomials on the Euclidean sphere or on the Hamming cube. We prove lower bounds on designs in spaces with a large group of symmetries. These spaces include globally symmetric Riemannian spaces (of any rank) and commutative association schemes with 1-transitive group of symmetries. Our bounds are, in general, implicit, relying on estimates on the spectral behavior of certain symmetry-invariant linear operators. They reduce to the first linear programming bound for designs in globally symmetric Riemannian spaces of rank 1 or in distance regular graphs. The proofs are different though, coming from viewpoint of abstract harmonic analysis in symmetric spaces. As a dividend we obtain the following geometric fact: a design is large because a union of "spherical caps" around its points "covers" the whole space.