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Bounds for spherical codes: The Levenshtein framework lifted


Authors: P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff and M. M. Stoyanova
Journal: Math. Comp. 90 (2021), 1323-1356
MSC (2020): Primary 94B65, 74G65, 52A40
DOI: https://doi.org/10.1090/mcom/3621
Published electronically: March 2, 2021
MathSciNet review: 4232226
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Abstract: Based on the Delsarte-Yudin linear programming approach, we extend Levenshtein’s framework to obtain lower bounds for the minimum $h$-energy of spherical codes of prescribed dimension and cardinality, and upper bounds on the maximal cardinality of spherical codes of prescribed dimension and minimum separation. These bounds are universal in the sense that they hold for a large class of potentials $h$ and in the sense of Levenshtein. Moreover, codes attaining the bounds are universally optimal in the sense of Cohn-Kumar. Referring to Levenshtein bounds and the energy bounds of the authors as “first level”, our results can be considered as “next level” universal bounds as they have the same general nature and imply necessary and sufficient conditions for their local and global optimality. For this purpose, we introduce the notion of Universal Lower Bound space (ULB-space), a space that satisfies certain quadrature and interpolation properties. While there are numerous cases for which our method applies, we will emphasize the model examples of $24$ points ($24$-cell) and $120$ points ($600$-cell) on $\mathbb {S}^3$. In particular, we provide a new proof that the $600$-cell is universally optimal, and in so doing, we derive optimality of the $600$-cell on a class larger than the absolutely monotone potentials considered by Cohn-Kumar.


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Additional Information

P. G. Boyvalenkov
Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G Bonchev Str., 1113 Sofia, Bulgaria; and Technical Faculty, South-Western University, Blagoevgrad, Bulgaria
MR Author ID: 331674
Email: peter@math.bas.bg

P. D. Dragnev
Affiliation: Department of Mathematical Sciences, Purdue University, Fort Wayne, Indiana 46805
MR Author ID: 623970
Email: dragnevp@pfw.edu

D. P. Hardin
Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
MR Author ID: 81245
ORCID: 0000-0003-0867-2146
Email: doug.hardin@vanderbilt.edu

E. B. Saff
Affiliation: Department of Mathematics, Center for Constructive Approximation, Vanderbilt University, Nashville, Tennessee 37240
MR Author ID: 152845
Email: edward.b.saff@vanderbilt.edu

M. M. Stoyanova
Affiliation: Faculty of Mathematics and Informatics, Sofia University, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
MR Author ID: 758227
ORCID: 0000-0002-8813-3398
Email: stoyanova@fmi.uni-sofia.bg

Keywords: Levenshtein framework, minimal energy problems, linear programming, bounds for codes
Received by editor(s): August 19, 2019
Received by editor(s) in revised form: September 15, 2020
Published electronically: March 2, 2021
Additional Notes: The research of the first and fifth authors was supported, in part, by a Bulgarian NSF contract DN02/2-2016. The research of the second author was supported, in part, by a Simons Foundation grant no. 282207. The research of the third and fourth authors was supported, in part, by the U. S. National Science Foundation under grant DMS-1516400.
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