We extend the results of our previous paper  to the nonlinear case: The Lloyd polynomial of the covering has at least R distinct roots among 1, ... , n, where R is the covering radius. We investigate PWC with diameter 1, finding a partial characterization. We complete an investigation begun in  on linear PMC with distance 1 and diameter 2.
Cohen, G. D.; Litsyn, S. N.; and Mattson, H. F. Jr, "On Perfect Weighted Coverings with Small Radius" (1991). Electrical Engineering and Computer Science - Technical Reports. 139.