Codes with minimum distance at least d and covering radius at most d− 1 are considered. The minimal cardinality of such codes is investigated. Herewith, their connection to covering problems is applied and a new construction theorem is given. Additionally, a new lower bound for the covering problem is proved. A necessary condition on an existence problem is presented by using a multiple coverin...

The covering radius of a code is the least r such that the set of balls of radius r around codewords covers the entire ambient space. We introduce a generalization of the notion of covering radius. The m-covering radius of a code is the least radius such that the set of balls of the radius covers all m-tuples of elements in the ambient space. We investigate basic properties of m-covering radii....

The minimum number of rows in covering arrays (equivalently, surjective codes) and radius-covering arrays (equivalently, surjective codes with a radius) has been determined precisely only in special cases. In this paper, explicit constructions for numerous best known covering arrays (upper bounds) are found by a combination of combinatorial and computational methods. For radius-covering arrays,...

This paper further studies the fuzzy rough sets based on fuzzy coverings. We first present the notions of the lower and upper approximation operators based on fuzzy coverings and derive their basic properties. To facilitate the computation of fuzzy coverings for fuzzy covering rough sets, the concepts of fuzzy subcoverings, the reducible and intersectional elements, the union and intersection o...

All known results on covering radius are presented, as well as some new results. There are a number of upper and lower bounds, including asymptotic results, a few exact determinations of covering radius, some extensive relations with other aspects of coding theory through the Reed-Muller codes, and new results on the least covering radius of any linear [II, k] code. There is also a recent resul...

The m-covering radius of a code is a recent generalization of the covering radius of a code. It is the smallest t such that every m-tuple of vectors is contained in a ball of Hamming radius centered at some codeword. We derive new lower bounds for the size of the smallest code that has a given length and m-covering radius.