A note on multiple coverings of the farthest-off points

n this work we summarize some recent results, to be included in a forthcoming paper [D. Bartoli, A. A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco, Multiple coverigns of the farthest-off points with small density from projective geometry, preprint]. We define μ-density as a characteristic of quality for the kind of coverings codes called multiple coverings of the farthest-off points (MCF). A concept of multiple saturating sets ( ( ρ , μ ) -saturating sets) in projective spaces P G ( N , q ) is introduced. A fundamental relationship of these sets with MCF is showed. Bounds for the smallest possible cardinality of ( 1 , μ ) -saturating sets are obtained. Constructions of small ( 1 , μ ) -saturating sets improving the probabilistic bound are proposed.