Universal Lower Bounds for Potential Energy of Spherical Codes

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 someti...

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Bibliographic Details
Published in:Constructive approximation 2016-12, Vol.44 (3), p.385-415
Main Authors: Boyvalenkov, P. G., Dragnev, P. D., Hardin, D. P., Saff, E. B., Stoyanova, M. M.
Format: Article
Language:English
Subjects:
Analysis
Codes
Lower bounds
Mathematics
Mathematics and Statistics
Numerical Analysis
Potential energy
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Summary: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 we 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.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-016-9327-5