Do most binary linear codes achieve the Goblick bound on the covering radius?
IEEE Transactions on Information Theory
Sphere-packings, lattices, and groups
Sphere-packings, lattices, and groups
Epsilon-entropy of an Ellipsoid in a Hamming Space
Problems of Information Transmission
Problems of Information Transmission
On the Epsilon-entropy of One Class of Ellipsoids in a Hamming Space
Problems of Information Transmission
On coverings of ellipsoids in Euclidean spaces
IEEE Transactions on Information Theory
On the reconstruction of block-sparse signals with an optimal number of measurements
IEEE Transactions on Signal Processing
On computation of entropy of an ellipsoid in a Hamming space
Problems of Information Transmission
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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.