Do most binary linear codes achieve the Goblick bound on the covering radius? (Corresp.)

The following two problems are dealt with: P1) finding the smallest rate, R , of a binary code of length n admitting a prescribed covering radius \rho n ; P2) discovering whether a majority of codes with any rate larger than R admits 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 is R = 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.