Advertisement

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
Article
Part of the following topical collections:
  1. Special Issue: Coding and Cryptography

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.

Keywords

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

Mathematics Subject Classification

94B65 52A40 74G65 

Notes

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.

References

  1. 1.
    Beckermann B., Bustamante J., Martinez-Cruz R., Quesada J.: Gaussian, Lobatto and Radau positive quadrature rules with a prescribed abscissa. Calcolo 51, 319–328 (2014).MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Borodachov S., Hardin D., Saff E.: Discrete Energy on Rectifiable Sets. Springer, New York (2018). (to appear).zbMATHGoogle Scholar
  3. 3.
    Boyvalenkov P., Dragnev P., Hardin D., Saff E., Stoyanova M.: Universal lower bounds for potential energy of spherical codes. Constr. Approx. 44, 385–415 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boyvalenkov P., Dragnev P., Hardin D., Saff E., Stoyanova M.: Energy bounds for codes and designs in Hamming spaces. Des. Codes Cryptogr. 82(1), 411–433 (2017). (arxiv:1510.03406).MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bultheel A., Cruz-Barroso R., Van Barel M.: On Gauss-type quadrature formulas with prescribed nodes anywhere on the real line. Calcolo 47, 21–48 (2010).MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Cohn H., Kumar A.: Universally optimal distribution of points on spheres. J. Am. Math. Soc. 20, 99–148 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Cohn H., Woo J.: Three point bounds for energy minimization. J. Am. Math. Soc. 25, 929–958 (2012).MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cohn H., Zhao Y.: Energy-minimizing error-correcting codes. IEEE Trans. Inf. Theory 60, 7442–7450 (2014). (arXiv:1212.1913).MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davis P.J., Rabinowitz P.: Methods of Numerical Integration, 2nd edn. Academic Press, New York (1984).zbMATHGoogle Scholar
  10. 10.
    Delsarte P., Goethals J.-M., Seidel J.J.: Spherical codes and designs. Geom. Dedic. 6, 363–388 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gasper G.: Linearization of the product of Jacobi polynomials, II. Can. J. Math. 22, 582–593 (1970).MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kabatyanskii G.A., Levenshtein V.I.: Bounds for packings on a sphere and in space. Probl. Inf. Transm. 14, 1–17 (1989).Google Scholar
  13. 13.
    Koelink E., de los Ríos A.M., Román P.: Matrix-valued Gegenbauer-type polynomials. Constr. Approx. 46(3), 459–487 (2017).Google Scholar
  14. 14.
    Levenshtein V.I.: Designs as maximum codes in polynomial metric spaces. Acta Appl. Math. 25, 1–82 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Levenshtein V.I.: Universal bounds for codes and designs. In: Pless V.S., Huffman W.C. (eds.) Handbook of Coding Theory, pp. 499–648. Elsevier, Amsterdam (1998).Google Scholar
  16. 16.
    Pacharoni I., Zurrián I.: Matrix Gegenbauer polynomials: the \(2 \times 2\) fundamental cases. Constr. Approx. 43(2), 253–271 (2016).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Szegő G.: Orthogonal Polynomials, vol. 23. American Mathematical Society, Providence (1939).zbMATHGoogle Scholar
  18. 18.
    Yudin V.A.: Minimal potential energy of a point system of charges. Discret. Mat. 4, 115–121 (1992) (in Russian). English translation: Discret. Math. Appl. 3, 75–81 (1993).Google Scholar

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

Personalised recommendations