On spherical codes with inner products in a prescribed interval
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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.
KeywordsSpherical codes Linear programming Bounds for codes H-energy of a code
Mathematics Subject Classification94B65 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.
- 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.Koelink E., de los Ríos A.M., Román P.: Matrix-valued Gegenbauer-type polynomials. Constr. Approx. 46(3), 459–487 (2017).Google Scholar
- 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
- 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