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On the Thinnest Coverings of Spheres and Ellipsoids with Balls in Hamming and Euclidean Spaces

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General Theory of Information Transfer and Combinatorics

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4123))

Abstract

In this paper, we present some new results on the thinnest coverings that can be obtained in Hamming or Euclidean spaces if spheres and ellipsoids are covered with balls of some radius ε. In particular, we tighten the bounds currently known for the ε-entropy of Hamming spheres of an arbitrary radius r. New bounds for the ε-entropy of Hamming balls are also derived. If both parameters ε and r are linear in dimension n, then the upper bounds exceed the lower ones by an additive term of order logn. We also present the uniform bounds valid for all values of ε and r.

In the second part of the paper, new sufficient conditions are obtained, which allow one to verify the validity of the asymptotic formula for the size of an ellipsoid in a Hamming space. Finally, we survey recent results concerning coverings of ellipsoids in Hamming and Euclidean spaces.

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Authors
  1. I. Dumer
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  2. M. S. Pinsker
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  3. V. V. Prelov
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Editor information

Editors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, 100131, 33501, Bielefeld, Germany

    Rudolf Ahlswede

  2. Fakultät für Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615, Bielefeld, Germany

    Lars Bäumer , Ning Cai , Harout Aydinian  & Christian Deppe , ,  & 

  3. Russian Academy of Sciences Institute of Problems of Information Transmission, Bol’shoi Karetnyi per. 19, 101447, Moscow, Russia

    Vladimir Blinovsky

  4. Universität Bielefeld, Fakultät für Mathematik, Universitätsstr. 25, 33615, Bielefeld, Germany

    Haik Mashurian

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© 2006 Springer-Verlag Berlin Heidelberg

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Dumer, I., Pinsker, M.S., Prelov, V.V. (2006). On the Thinnest Coverings of Spheres and Ellipsoids with Balls in Hamming and Euclidean Spaces. In: Ahlswede, R., et al. General Theory of Information Transfer and Combinatorics. Lecture Notes in Computer Science, vol 4123. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11889342_57

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  • DOI: https://doi.org/10.1007/11889342_57

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-46244-6

  • Online ISBN: 978-3-540-46245-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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