### Journal article

Adv. Math. Commun., 2015

D. Bartoli, A. Davydov, M. Giulietti, S. Marcugini, F. Pambianco

Adv. Math. Commun., 2015

Adv. Math. Commun., 2015

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**APA**

Bartoli, D., Davydov, A., Giulietti, M., Marcugini, S., & Pambianco, F. (2015). Multiple coverings of the farthest-off points with small density from projective geometry. Adv. Math. Commun.

**Chicago/Turabian**

Bartoli, D., A. Davydov, M. Giulietti, S. Marcugini, and F. Pambianco. “Multiple Coverings of the Farthest-off Points with Small Density from Projective Geometry.” Adv. Math. Commun. (2015).

**MLA**

Bartoli, D., et al. “Multiple Coverings of the Farthest-off Points with Small Density from Projective Geometry.” Adv. Math. Commun., 2015.

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 $\mu$-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 $\mu$-length function in terms of Projective Geometry over finite fields. In fact, linear $(R,\mu )$-MCF codes with parameters $ [n,n-r,d]_{q}R$ 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 $(\rho,\mu)$-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} $\mu $ spheres centered in codewords, are considered and their connections with uniformly packed codes, two-weight codes, and subgroups of Singer groups are pointed out.