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On spherical codes with inner products in a prescribed interval

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Abstract

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval \([\ell ,s]\) of \([-\,1,1)\). An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in \([\ell ,s]\) and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in \([\ell ,1)\) (when the cardinality of the code is kept fixed) is revealed and explained. Thereby, we obtain a new extension of Levenshtein bounds for such codes. The universality of our bounds is exhibited by a unified derivation and their validity for a wide range of codes and potential functions.

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Notes

  1. For recent matrix generalizations of Gegenbauer polynomials, see [13] and [16].

  2. In fact, we suspect that both conditions are equivalent.

  3. The notation \(g=H(f;h)\) is taken from [6]; it signifies that g is the Hermite interpolant to the function h at the zeros (taken with their multiplicity) of f.

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Acknowledgements

The authors thank Konstantin Delchev, Tom Hanson, and Nikola Sekulov for their independent computational work on Conjecture 4.2 for small values of n and k. P. G. Boyvalenkov and M. M. Stoyanova: the research of these authors was supported, in part, by a Bulgarian NSF Contract DN02/2-2016. P. D. Dragnev: the research of this author was supported, in part, by a Simons Foundation Grant No. 282207. D. P. Hardin and E. B. Saff: the research of these authors was supported, in part, by the U.S. National Science Foundation under Grant DMS-1516400.

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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. South-Western University, Blagoevgrad, Bulgaria

    P. G. Boyvalenkov

  3. Department of Mathematical Sciences, Purdue University, 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

Authors
  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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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

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Boyvalenkov, P.G., Dragnev, P.D., Hardin, D.P. et al. On spherical codes with inner products in a prescribed interval. Des. Codes Cryptogr. 87, 299–315 (2019). https://doi.org/10.1007/s10623-018-0524-z

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  • DOI: https://doi.org/10.1007/s10623-018-0524-z

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