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Asymptotic bounds for spherical codes

Manin, Yu. I. and Marcolli, M. (2019) Asymptotic bounds for spherical codes. Izvestiya: Mathematics, 83 (3). pp. 540-564. ISSN 1064-5632. http://resolver.caltech.edu/CaltechAUTHORS:20190711-104459099

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Abstract

The set of all error-correcting codes C over a fixed finite alphabet F of cardinality q determines the set of code points in the unit square [0,1]^2 with coordinates (R(C), δ(C)):= (relative transmission rate, relative minimal distance). The central problem of the theory of such codes consists in maximising simultaneously the transmission rate of the code and the relative minimum Hamming distance between two different code words. The classical approach to this problem explored in vast literature consists in inventing explicit constructions of "good codes" and comparing new classes of codes with earlier ones. A less classical approach studies the geometry of the whole set of code points (R, δ) (with q fixed), at first independently of its computability properties, and only afterwards turning to problems of computability, analogies with statistical physics, and so on. The main purpose of this article consists in extending this latter strategy to the domain of spherical codes.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1070/im8739DOIArticle
https://arxiv.org/abs/1801.01552arXivDiscussion Paper
Additional Information:© 2019 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd. Received 27 November 2017. The second author is supported by NSF grant DMS-1707882 and NSERC grant RGPIN-2018-04937.
Funders:
Funding AgencyGrant Number
NSFDMS-1707882
Natural Sciences and Engineering Research Council of Canada (NSERC)RGPIN-2018-04937
Subject Keywords:error-correcting codes, spherical codes, asymptotic bounds
Classification Code:AMS 2010 Mathematics Subject Classification. 94B60, 94B65
Record Number:CaltechAUTHORS:20190711-104459099
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20190711-104459099
Official Citation:Yu. I. Manin and M. Marcolli 2019 Izv. Math. 83 540
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97055
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:11 Jul 2019 23:07
Last Modified:11 Jul 2019 23:07

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