Weighted coverings and packings

We introduce a generalization of the concepts of coverings and packings in Hamming space called weighted coverings and packings. We study the existence of perfect weighted codes, discuss connections between weighted coverings and packings, and present many constructions for them. Conventional packings and coverings are arrangements of Hamming spheres of a given radius in the Hamming space. We generalize these concepts by attaching weights to different layers of the Hamming sphere. If several such spheres intersect in a point of the space we define the density at that point as the sum of the weights of the corresponding layers. We study the general problem of weighted packings (coverings) for which the density at each point is at most one (resp. at least one). We can consider several known types of codes, e.g., the uniformly packed codes, list codes, multiple coverings, L-codes, in a uniform way.