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Upper bounds for energies of spherical codes of given cardinality and separation

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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.

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Notes

  1. Similar results for codes in Hamming spaces are obtained in [10, 12, 14].

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Acknowledgements

The authors thank the anonymous referees for helpful suggestions and comments.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G Bonchev Str., 1113, Sofia, Bulgaria

    P. G. Boyvalenkov

  2. Technical Faculty, South-Western University, Blagoevgrad, Bulgaria

    P. G. Boyvalenkov

  3. Department of Mathematical Sciences, Purdue University Fort Wayne, Fort Wayne, IN, 46805, USA

    P. D. Dragnev

  4. Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, TN, 37240, USA

    D. P. Hardin & E. B. Saff

  5. Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164, Sofia, Bulgaria

    M. M. Stoyanova

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  1. P. G. Boyvalenkov
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  2. P. D. Dragnev
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  3. D. P. Hardin
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  4. E. B. Saff
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  5. M. M. Stoyanova
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Correspondence to P. G. Boyvalenkov.

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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”.

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Boyvalenkov, P.G., Dragnev, P.D., Hardin, D.P. et al. Upper bounds for energies of spherical codes of given cardinality and separation. Des. Codes Cryptogr. 88, 1811–1826 (2020). https://doi.org/10.1007/s10623-020-00733-y

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  • DOI: https://doi.org/10.1007/s10623-020-00733-y

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