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arXiv · 1801.07334

On spherical codes with inner products in a prescribed interval

P. G. Boyvalenkov·P. D. Dragnev·D. P. Hardin·E. B. Saff·M. M. Stoyanova
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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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P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff, M. M. Stoyanova. 2018-01-22. On spherical codes with inner products in a prescribed interval. https://arxiv.org/abs/1801.07334

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