Designs, Codes and Cryptography

, Volume 87, Issue 2–3, pp 299–315 | Cite as

On spherical codes with inner products in a prescribed interval

  • P. G. BoyvalenkovEmail author
  • P. D. Dragnev
  • D. P. Hardin
  • E. B. Saff
  • M. M. Stoyanova
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography


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.


Spherical codes Linear programming Bounds for codes H-energy of a code 

Mathematics Subject Classification

94B65 52A40 74G65 



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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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria
  2. 2.South-Western UniversityBlagoevgradBulgaria
  3. 3.Department of Mathematical SciencesPurdue UniversityFort WayneUSA
  4. 4.Department of Mathematics, Center for Constructive ApproximationVanderbilt UniversityNashvilleUSA
  5. 5.Faculty of Mathematics and InformaticsSofia UniversitySofiaBulgaria

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