Multiple coverings of the farthest-off points with small density from projective geometry

In this paper we deal with the special class of covering codes consisting of multiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the μ-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main results consist in a number of upper bounds on such a length function, obtained through explicit constructions, especially for the case of covering radius R = 2. A key tool is the possibility of computing the μ-length function in terms of Projective Geometry over finite fields. In fact, linear (R,μ)-MCF codes with parameters [n, n − r, d]qR have a geometrical counterpart consisting of special subsets of n points in the projective space PG(n− r− 1, q). We introduce such objects under the name of (ρ, μ)-saturating sets and we provide a number of example and existence results. Finally, Almost Perfect MCF (APMCF) codes, that is codes for which each word at distance R from the code belongs to exactly μ spheres centered in codewords, are considered and their connections with uniformly packed codes, two-weight codes, and subgroups of Singer groups are pointed out.