On Nearly Perfect Covering Codes

Nearly perfect packing codes are those codes that meet the Johnson upper bound on the size of error-correcting codes. This bound is an improvement to the sphere-packing bound. A related bound for covering codes is known as the van Wee bound. Codes that meet this bound will be called nearly perfect covering codes. This work studies such codes with covering radius 1. It is shown that the set of these codes can be partitioned into three families, depending on the distribution of the Hamming distances between neighboring codewords. General properties of these code families are presented, including a characterization of their weight and distance distributions. Constructions of codes for each of the families are presented. Finally, extended perfect covering codes are considered. Their punctured codes yield a variety of nearly perfect covering codes.