Multiple coverings of the farthest-offpoints and multiple saturating sets in projective spaces

For the kind of coverings codes called multiple coverings of the farthest-off points (MCF) we define mu-density as a characteristic of quality. A concept of multiple saturating sets (rho, mu)-saturating sets in projective spaces PG(N,q) is introduced. A fundamental relationship of these sets with MCF codes is showed. Lower and upper bounds for the smallest possible cardinality of (1,mu)-saturating sets are obtained. In PG(2,q), constructions of small (1, mu)-saturating sets improving the probabilistic bound are proposed. A number of results on the spectrum of sizes of minimal (1, mu)-saturating sets are obtained.