Universal Lower Bounds for Potential Energy of Spherical Codes

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• 作者：P. G. Boyvalenkov ; P. D. Dragnev ; D. P. Hardin ; E. B. Saff…
• 关键词：Minimal energy problems ; Spherical potentials ; Spherical codes and designs ; Levenshtein bounds ; Delsarte–Goethals–Seidel bounds ; Linear programming
• 刊名：Constructive Approximation
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：44
• 期：3
• 页码：385-415
• 全文大小：847 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Numerical Analysis
  Analysis
  
• 出版者：Springer New York
• ISSN：1432-0940
• 卷排序：44

文摘

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.