MULTIPLE COVERINGS OF THE FARTHEST-OFF POINTS WITH SMALL DENSITY FROM PROJECTIVE GEOMETRY

作者：Bartoli Daniele^*; Davydov Alexander A; Giulietti Massimo; Marcugini Stefano; Pambianco Fernanda

来源：Advances in Mathematics of Communications, 2015, 9(1): 63-85.

DOI：10.3934/amc.2015.9.63

• Multiple coverings
• covering codes
• farthest-off points
• saturating sets in projective spaces
• covering density

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 it-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 it-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.

• 出版日期2015-2