Epsilon-entropy of an Ellipsoid in a Hamming Space
Problems of Information Transmission
New bounds on binary identifying codes
Discrete Applied Mathematics
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 4
On the thinnest coverings of spheres and ellipsoids with balls in hamming and euclidean spaces
General Theory of Information Transfer and Combinatorics
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The following two problems are dealt with: P1) finding the smallest rate,R, of a binary code of lengthnadmitting a prescribed covering radiusrho n; P2) discovering whether a majority of codes with any rate larger thanRadmits the given covering radius. For the class of unrestricted (nonlinear) codes a solution to both problems is obtained by an elementary averaging argument. The solution to P1 isR = 1 - H(rho) + O(n^{-1} log n)and the answer to P2 is positive. As for the more interesting class of linear codes, Goblick's extension method shows that the solution to P1 is the same as in the unrestricted case; in contrast, P2 seems to remain an open question. A simple derivation of Goblick's result is presented, and a discussion is made of the positive conjecture concerning P2 for linear codes.