A construction for an infinite family of perfect mixed codes with covering radius 2 is presented. These are the first known nontrivial perfect mixed codes with covering radius greater than 1. Based on mixed codes, constructions for binary covering codes that lead to a considerable improvement of upper bounds on the sizes of covering codes are presented. These codes and some other codes can be obtained by the blockwise direct sum construction. Two infinite families of codes are of special interest. They are quasi-perfect, nonlinear, union of their disjoint translates covers the space, and their density as covering codes is remarkably low.>

It is proved that codes of length n, covering radius R, and minimum Hamming distance 2R-1 are normal if R does not divide n. Constructions for abnormal codes with covering radius R and minimum Hamming distance at least R-1 are given.>