We consider coverings of a sphere Sn r of radius r with the balls of radius one in an n-dimensional Euclidean space R. Our goal is to minimize the covering density, which defines the average number of the balls covering a point in Sn r . For a growing dimension n, we obtain the covering density at most (n lnn)/2 for any sphere Sn r and the entire space R. This new upper bound reduces two times ...

Simple random coverage models, well studied in Euclidean space, can also be defined on a general compact metric space. By analogy with the geometric and discrete coupon collector's problem cover times for finite Markov chains, one expects weak concentration bound distribution of time to hold under minimal assumptions. We give two such results, fixed-radius balls other sequentially arriving rand...

We give an explicit upper bound of the minimal number νT,n of balls of radius 1/2 which form a covering of a ball of radius T > 1/2 in R , n > 2. The asymptotic estimates of νT,n we deduce when n is large are improved further by recent results of Böröczky Jr. and Wintsche on the asymptotic estimates of the minimal number of equal balls of R n covering the sphere S. The optimality of the asympto...

The aim of this paper is to provide an active inference algorithm for anomalous behavior. As a main concept we introduce fuzzy temporal consistency covering set, and put forward a fuzzy temporal selection model based on temporal inference and covering technology. Fuzzy set is used to describe network anomaly behavior omen and character, as well as the relations between behavior omen and charact...

In this paper we construct new family of binary linear completely transitive (and, therefore, completely regular) codes. The covering radius of these codes is growing with the length of the code. In particular, for any integer ρ ≥ 2, there exist two codes in the constructed class of codes with d = 3, covering radius ρ and length ( 4 ρ 2 ) and ( 4 ρ+2 2 ) , respectively. These new completely tra...

In this paper we study properties and invariants of matrix codes endowed with the rank metric, and relate them to the covering radius. We introduce new tools for the analysis of rank-metric codes, such as puncturing and shortening constructions. We give upper bounds on the covering radius of a code by applying different combinatorial methods. The various bounds are then applied to the classes o...

The search for codes of covering radius 1 led Österg̊ard, Quistorff and Wassermann to the OQW method of associating a unique graph to each code [9]. We present results on the structure and existence of OQW-associated graphs. These are used to find an upper bound on the size of a ball of radius 1 around a code of length 3 and minimum distance 2. OQW-associated graphs and non-extendable partial La...