Covering code: Difference between revisions

Revision as of 19:27, 24 May 2008

Revision as of 12:33, 26 May 2008

Line 1:

In [[coding theory]], a '''covering code''' is an object satisfying a certain mathematical property.

{{context}}

== Definition ==

Line 5:

Let <math>q\geq 2</math>, <math>n\geq 1</math>, <math>R\geq 0</math> be [[integers]].

A [[code]] <math>C\subseteq Q^n</math> over an [[alphabet]] ''Q'' of size |''Q''| = ''q'' is called

''q''-ary ''R-'''-covering code''' of length ''n''

if for every word <math>y\in Q^n</math> there is a [[codeword]] <math>x\in C</math>

such that the [[Hamming distance]] <math>d_H(x,y)\leq R</math>.

Line 11:

with respect to the Hamming [[metric]] around the codewords of ''C'' have to exhaust

the [[finite]] [[metric space]] <math>Q^n</math>.

The covering radius of a code ''C'' is the smallest ''R'' such that ''C'' is ''R''-covering.

Every [[perfect code]] is a covering code of minimal size.