Abstract:
The following two problems are dealt with: P1) finding the smallest rate,R, of a binary code of lengthnadmitting a prescribed covering radius\rho n; P2) discovering wheth...Show MoreMetadata
Abstract:
The following two problems are dealt with: P1) finding the smallest rate,R, of a binary code of lengthnadmitting a prescribed covering radius\rho 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.
Published in: IEEE Transactions on Information Theory ( Volume: 32, Issue: 6, November 1986)