Graphs Associated with Codes of Covering Radius 1 and Minimum Distance 2

Graphs Associated with Codes of Covering Radius 1 and Minimum Distance 2

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...

On Codes with Given Minimum Distance and Covering Radius

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...

New Results on Codes with Covering Radius 1 and Minimum Distance 2

On Mixed Codes with Covering Radius 1 and Minimum Distance 2

Let R, S and T be finite sets with |R| = r, |S| = s and |T | = t. A code C ⊂ R×S ×T with covering radius 1 and minimum distance 2 is closely connected to a certain generalized partial Latin rectangle. We present various constructions of such codes and some lower bounds on their minimal cardinality K(r, s, t; 2). These bounds turn out to be best possible in many instances. Focussing on the speci...

Asymptotic Minimum Covering Radius of Block Codes

In this paper, we restudy the covering radius of block codes from an information theoretic point of view by ignoring the combinatorial formulation of the problem. In the new setting, the formula of the statistically defined minimum covering radius, for which the probability mass of uncovered space by M spheres can be made arbitrarily small, is reduced to a minimization of a statistically define...