A new approach to the covering radius of codes

We introduce a new approach which facilitates the calculation of the covering radius of a binary linear code. It is based on determining the normalized covering radius ϱ. For codes of fixed dimension we give upper and lower bounds on ϱ that are reasonably close. As an application, an explicit formula is given for the covering radius of an arbitrary code of dimension ⩽4. This approach also sheds light on whether or not a code is normal. All codes of dimension ⩽4 are shown to be normal, and an upper bound is given for the norm of an arbitrary code. This approach also leads to an amusing generalization of the Berlekamp-Gale switching game.