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

Abstract

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 log n. 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.

New Non-Asymptotic Bounds on Numbers of Codewords for the Fixed-Length Lossy Compression

“…Thus, for possible source sequences, this distribution uniformly weights all reproduction sequences in the partial sphere A k w (x n ). Dumer et al [18] used this kind of distribution to derive an upper bound to the minimum number of balls of radius r covering a ball of radius s(> r), and they showed the bound is asymptotically tight.…”

“…Remark 9. Since the RHS of the above inequality is almost same as [18,RHS of (14)], it will be asymptotically characterized by exp(nh(p) − nh(D) + o(n)). Thus, this bound may be asymptotically optimal.…”