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 Izv. RAN. Ser. Mat., 2019, Volume 83, Issue 3, Pages 133–157 (Mi izv8739)

Asymptotic bounds for spherical codes

Yu. I. Manina, M. Marcollib

a Max–Planck–Institute für Mathematik, Bonn, Germany
b California Institute of Technology, Pasadena, USA

Abstract: The set of all error-correcting codes $C$ over a fixed finite alphabet $\mathbf{F}$ of cardinality $q$ determines the set of code points in the unit square $[0,1]^2$ with coordinates $(R(C), \delta (C))$:= (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 $(R,\delta)$ (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.

Keywords: error-correcting codes, spherical codes, asymptotic bounds.

 Funding Agency Grant Number National Science Foundation DMS-1707882 Natural Sciences and Engineering Research Council of Canada (NSERC) RGPIN-2018-04937 The second author is supported by NSF grant DMS-1707882 and NSERC grant RGPIN-2018-04937.

DOI: https://doi.org/10.4213/im8739

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English version:
Izvestiya: Mathematics, 2019, 83:3, 540–564

Bibliographic databases:

UDC: 519.725+514.174.2
MSC: 94B60, 94B65

Citation: Yu. I. Manin, M. Marcolli, “Asymptotic bounds for spherical codes”, Izv. RAN. Ser. Mat., 83:3 (2019), 133–157; Izv. Math., 83:3 (2019), 540–564

Citation in format AMSBIB
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