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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.
Received: 27.11.2017
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
Linking options:
https://www.mathnet.ru/eng/im8739https://doi.org/10.1070/IM8739 https://www.mathnet.ru/eng/im/v83/i3/p133
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Abstract page: | 396 | Russian version PDF: | 36 | English version PDF: | 19 | References: | 39 | First page: | 30 |
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