Title

Universal Lower Bounds for Potential Energy of Spherical Codes

Document Type

Article

Publication Date

2-29-2016

Publication Source

Constructive Approximation

Inclusive pages

1-31

DOI

10.1007/s00365-016-9327-5

Publisher

Springer US

ISBN/ISSN

0176-4276

Peer Reviewed

yes

Abstract

We derive and investigate lower bounds for the potential energy of finite spherical point sets (spherical codes). Our bounds are optimal in the following sense – they cannot be improved by employing polynomials of the same or lower degrees in the Delsarte-Yudin method. However, improvements are sometimes possible and we provide a necessary and sufficient condition for the existence of such better bounds. All our bounds can be obtained in a unified manner that does not depend on the potential function, provided the potential is given by an absolutely monotone function of the inner product between pairs of points, and this is the reason for us to call them universal. We also establish a criterion for a given code of dimension n and cardinality N not to be LP-universally optimal, e.g. we show that two codes conjectured by Ballinger et al to be universally optimal are not LP-universally optimal.

Keywords

Minimal energy problems, Spherical potentials, Spherical codes and designs, Levenshtein bounds, Delsarte–Goethals–Seidel bounds, Linear programming

Disciplines

Analysis | Discrete Mathematics and Combinatorics | Harmonic Analysis and Representation | Mathematics

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Link to Original Published Item

http://www.math.vanderbilt.edu/~esaff/texts/253