Linear programming bounds for covering radius of spherical designs

TL;DR

This paper develops polynomial-based linear programming methods to establish new bounds on the covering radius of spherical designs, improving previous bounds and providing a comprehensive analytical framework.

Contribution

It introduces improved lower bounds and new upper bounds for the covering radius of spherical designs using polynomial techniques and linear programming.

Findings

Enhanced lower bounds for covering radius compared to previous results

New upper bounds derived from geometric and linear programming methods

Applicable to spherical designs of various dimensions and strengths

Abstract

We apply polynomial techniques (linear programming) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower bounds due to Fazekas and Levenshtein and propose new upper bounds. Our approach to the lower bounds involves certain signed measures whose corresponding series of orthogonal polynomials are positive definite up to a certain (appropriate) degree. Upper bounds are based on a geometric observation and more or less standard linear programming techniques.