Multiple coverings of the farthest-off points and multiple saturating sets in projective spaces
نویسندگان
چکیده
For the kind of coverings codes called multiple coverings of the farthestoff points (MCF) we define μ-density as a characteristic of quality. A concept of multiple saturating sets ((ρ, μ)-saturating sets) in projective spaces PG(N, q) is introduced. A fundamental relationship of these sets with MCF codes is showed. Lower and upper bounds for the smallest possible cardinality of (1, μ)-saturating sets are obtained. In PG(2, q), constructions of small (1, μ)-saturating sets improving the probabilistic bound are proposed. A number of results on the spectrum of sizes of minimal (1, μ)-saturating sets are obtained.
منابع مشابه
A note on multiple coverings of the farthest-off points
In this work we summarize some recent results, to be included in a forthcoming paper [1]. We define μ-density as a characteristic of quality for the kind of coverings codes called multiple coverings of the farthest-off points (MCF). A concept of multiple saturating sets ((ρ, μ)-saturating sets) in projective spaces PG(N, q) is introduced. A fundamental relationship of these sets with MCF is sho...
متن کاملFurther 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 spaces 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 par...
متن کاملMultiple 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 result...
متن کاملOn co-Farthest Points in Normed Linear Spaces
In this paper, we consider the concepts co-farthest points innormed linear spaces. At first, we define farthest points, farthest orthogonalityin normed linear spaces. Then we define co-farthest points, co-remotal sets,co-uniquely sets and co-farthest maps. We shall prove some theorems aboutco-farthest points, co-remotal sets. We obtain a necessary and coecient conditions...
متن کاملRemotality and proximinality in normed linear spaces
In this paper, we consider the concepts farthest points and nearest points in normed linear spaces, We obtain a necessary and coecient conditions for proximinal, Chebyshev, remotal and uniquely remotal subsets in normed linear spaces. Also, we consider -remotality, -proximinality, coproximinality and co-remotality.
متن کامل