Further results on multiple coverings of the farthest-off points

Multiple coverings of the farthest-off points ((R,μ)-MCF codes) and the corresponding (ρ,μ)-saturating sets in projective spa\-ces PG(N,q) are considered. We propose some methods which allow us to obtain new small (1,μ)-saturating sets and short (2,μ)-MCF codes with μ-density either equal to 1 (optimal saturating sets and almost perfect MCF-codes) or close to 1 (roughly 1+1/cq, c≥1). In particular, we provide some algebraic constructions and bounds. Also, we classify minimal and optimal (1,μ)-saturating sets in PG(2,q), q small.