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Linear Programming Bounds for Covering Radius of Spherical Designs

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Abstract

We apply polynomial techniques (i.e., techniques which invole polynomials) to obtain lower and upper bounds on the covering radius of spherical designs as function of their dimension, strength, and cardinality. In terms of inner products we improve the lower bounds due to Fazekas and Levenshtein and propose new upper bounds. Our approach to the lower bounds involves certain signed measures whose corresponding series of orthogonal polynomials are positive definite up to a certain (appropriate) degree. The upper bounds are based on a geometric observation and more or less standard in the field linear programming techniques.

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Acknowledgements

All numerical examples were implemented by Maple with truncation and rounding for the lower and upper bounds, respectively. Pdf files of our Maple program for the model examples can be seen at the webpage of the second author https://store.fmi.uni-sofia.bg/fmi/algebra/mstoyanova.shtml. The authors thank to Arseniy Akopyan and Nikolai Nikolov for helpful discussions and an anonymous reviewer for useful comments which significantly improved the exposition.

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  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 8 G.Bonchev str., 1113, Sofia, Bulgaria

    Peter Boyvalenkov

  2. Faculty of Mathematics and Informatics, Sofia University, 5 James Baucher blvd, 1164, Sofia, Bulgaria

    Maya Stoyanova

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  1. Peter Boyvalenkov
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  2. Maya Stoyanova
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Correspondence to Maya Stoyanova.

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The research of the first author was partially supported by Bulgarian NSF under Project KP-06-N32/2-2019. The research of the second author was supported, in part, by the National Scientific Program “Information and Communication Technologies for a Single Digital Market in Science, Education and Security (ICTinSES)”, NIS-3317, financed by the Bulgarian Ministry of Education and Science.

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Boyvalenkov, P., Stoyanova, M. Linear Programming Bounds for Covering Radius of Spherical Designs . Results Math 76, 95 (2021). https://doi.org/10.1007/s00025-021-01400-x

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