On spherical codes with inner products in a prescribed interval

We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval [ ℓ , s ] of [ - 1 , 1 ) . An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in [ ℓ , s ] and lower bou...

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Bibliographic Details
Published in:Designs, codes, and cryptography codes, and cryptography, 2019-03, Vol.87 (2-3), p.299-315
Main Authors: Boyvalenkov, P. G., Dragnev, P. D., Hardin, D. P., Saff, E. B., Stoyanova, M. M.
Format: Article
Language:English
Subjects:
Circuits
Codes
Coding and Information Theory
Computer Science
Cryptology
Data Structures and Information Theory
Discrete Mathematics in Computer Science
Information and Communication
Linear programming
Lower bounds
Potential energy
Special Issue: Coding and Cryptography
Upper bounds
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Summary:We develop a framework for obtaining linear programming bounds for spherical codes whose inner products belong to a prescribed subinterval [ ℓ , s ] of [ - 1 , 1 ) . An intricate relationship between Levenshtein-type upper bounds on cardinality of codes with inner products in [ ℓ , s ] and lower bounds on the potential energy (for absolutely monotone interactions) for codes with inner products in [ ℓ , 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.
ISSN:0925-1022
1573-7586
DOI:10.1007/s10623-018-0524-z