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

Daniele Bartoli; Alexander A. Davydov; Massimo Giulietti; Stefano Marcugini; Fernanda Pambianco

doi:10.3934/amc.2015.9.63

Article Contents

2015, Volume 9, Issue 1: 63-85. Doi: 10.3934/amc.2015.9.63

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

1.
Department of Mathematics and Informatics, Perugia University, Perugia, 06123
2.
Institute for Information Transmission Problems (Kharkevich institute), Russian Academy of Sciences, GSP-4, Moscow, 127994
3.
Department of Mathematics and Computer Science, University of Perugia, Perugia, 06123

Received: May 2014

Published: February 2015

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Abstract

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.

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Mathematics Subject Classification: Primary: 51E21, 51E22; Secondary: 94B05.

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