Upper bounds for energies of spherical codes of given cardinality and separation
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Upper bounds for energies of spherical codes of given cardinality and separation P. G. Boyvalenkov1,2 M. M. Stoyanova5
· P. D. Dragnev3 · D. P. Hardin4 · E. B. Saff4 ·
Received: 4 September 2019 / Revised: 20 January 2020 / Accepted: 1 February 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We introduce a linear programming framework for obtaining upper bounds for the potential energy of spherical codes of fixed cardinality and minimum distance. Using Hermite interpolation we construct polynomials to derive corresponding bounds. These bounds are universal in the sense that they are valid for all absolutely monotone potential functions and the required interpolation nodes do not depend on the potentials. Keywords Spherical codes · Potential functions · Energy of a code · Separation Mathematics Subject Classification 74G65 · 94B65 · 52A40 · 05B30
1 Introduction Let Sn−1 denote the unit sphere in Rn and C ⊂ Sn−1 be a spherical code; i.e. a finite subset of Sn−1 . Given an (extended real-valued) function h : [−1, 1] → [0, +∞], the (unnormalized) potential energy (or h-energy) of C is given by E h (C) := h(x, y), (1) x,y∈C,x = y
where x, y denotes the usual inner product of x and y.
The research of P. G. Boyvalenkov was partially supported by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)”, financed by the Bulgarian Ministry of Education and Science. The research of P. D. Dragnev was supported, in part, by a Simons Foundation Grant No. 282207. The research of D. P. Hardin and E. B. Saff was supported, in part, by the U. S. National Science Foundation under Grant DMS-1516400. The research of M. M. Stoyanova was supported by a Bulgarian NSF contract DN2/02-2016. Research for this article was started while the authors were in residence at the Institute for Computational and Experimental Research in Mathematics in Providence, RI, during the “Point Configurations in Geometry, Physics and Computer Science” program supported by the National Science Foundation under Grant No. DMS-1439786. This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography 2019”. Extended author information available on the last page of the article
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Denote by s(C) := max{x, y : x, y ∈ C, x = y}, the maximal inner product of a spherical code C and by C(n, M, s) := {C ⊂ Sn−1 : |C| = M, s(C) = s} the family of all spherical codes on Sn−1 of given cardinality M with maximal inner product s. Note that the set C(n, M, s) can be empty, for example C(n, n + 1, s) is empty for s < −1/n and C(n, 2n, s) is empty for s < 0. Given, n, M, s, and h, we are interested in upper bounds on the quantity Gh (n, M, s) :=
sup
{E h (C)},
(2)
C∈C(n,M,s)
where we use the convention that supremum of the empty set is −∞. Hereafter, we shall consider the class AM([−1, 1]) of potentials h, which are absolutely monotone in [
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