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

  • I. Dumer
  • M. S. Pinsker
  • V. V. Prelov
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4123)


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.


Euclidean Space Uniform Bound Thin Covering Arbitrary Radius Binary Linear Code 
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© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • I. Dumer
  • M. S. Pinsker
  • V. V. Prelov

There are no affiliations available

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