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

Abstract

AbstractIn 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.KeywordsEuclidean SpaceUniform BoundThin CoveringArbitrary RadiusBinary Linear CodeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.