Duality between packings and coverings of the Hamming space

We investigate the packing and covering densities of linear and nonlinear binary codes, and establish a number of duality relationships between the packing and covering problems. Specifically, we prove that if almost all codes are good packings, then only a vanishing fraction of codes are good coverings, and vice versa: if almost all codes are good coverings, then at most a vanishing fraction of codes are good packings. We also show that any specific maximal binary code is either a good packing or a good covering, in a certain well-defined sense.