This paper deals with an extension of perfect codes to fractional (or weighted) coverings. We shall derive a Lloyd theorem --- a strong necessary condition of existence---and start a classification of these perfect coverings according to their diameter. We illustrate by pointing to list decoding.
Cohen, G. D.; Litsyn, S. N.; and Mattson, H. F. Jr, "Binary Perfect Weighted Coverings (PWC) I. The Linear Case" (1991). Electrical Engineering and Computer Science - Technical Reports. 100.