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,...

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.

The Newton radius of a code is the largest weight of a uniquely correctable error. The covering radius is the largest distance between a vector and the code. Two relations between the Newton radius and the covering radius are given.

We report the first demonstration of flexible white phosphor-converted light emitting diodes (LEDs) based on p-n junction core/shell nitride nanowires. GaN nanowires containing seven radial In0.2Ga0.8N/GaN quantum wells were grown by metal-organic chemical vapor deposition on a sapphire substrate by a catalyst-free approach. To fabricate the flexible LED, the nanowires are embedded into a phosp...

The shortest possible length of a q-ary linear code of covering radius R and codimension r is called the length function and is denoted by q(r, R). Constructions of codes with covering radius 3 are here developed, which improve best known upper bounds on q(r, 3). General constructions are given and upper bounds on q(r, 3) for q = 3, 4, 5, 7 and r ≤ 24 are tabulated.

The covering radius problem in any dimension is not known to be solvable in nondeterministic polynomial time, but when in dimension two, we give a deterministic polynomial time algorithm by computing a reduced basis using Gauss’ algorithm in this paper.