## Bounds for spherical codes: The Levenshtein framework lifted

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P. G. Boyvalenkov, P. D. Dragnev, D. P. Hardin, E. B. Saff and M. M. Stoyanova
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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.## References

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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
- 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.
- © Copyright 2021 American Mathematical Society
- Journal: Math. Comp.
**90**(2021), 1323-1356 - MSC (2020): Primary 94B65, 74G65, 52A40
- DOI: https://doi.org/10.1090/mcom/3621
- MathSciNet review: 4232226