|
Covering code - Fuhz Articles Covering code article, this Fuhz page will hopefully provide the answers to the who, what where and why on the Covering code topic. At the bottom of the page we often provide links to external documents relating to Covering code which may also help your research. Every effort is made to ensure the content on this page is as accurate and error free as possible, however whenever researching information that requires the utmost accuracy such as a term paper it is always best to cross reference facts with numerous sources. |
In coding theory, a covering code is an object satisfying a certain mathematical property: A code of length n over Q is an R-covering code if for every word of Qn there is a codeword such that their Hamming distance is
.
Contents |
Let
,
,
be integers. A code
over an alphabet Q of size |Q| = q is called q-ary R-covering code of length n if for every word
there is a codeword
such that the Hamming distance
. In other words, the spheres (or balls or rook-domains) of radius R with respect to the Hamming metric around the codewords of C have to exhaust the finite metric space Qn. 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.
C = {0134,0223,1402,1431,1444,2123,2234,3002,3310,4010,4341} is a 5-ary 2-covering code of length 4.1
The determination of the minimal size Kq(n,R) of a q-ary R-covering code of length n is a very hard problem. In many cases, only lower and upper bounds are known with a large gap between them. Every construction of a covering code gives an upper bound on Kq(n, R). Lower bounds include the sphere covering bound and Rodemich’s bounds
and
.2 The covering problem is closely related to the packing problem in Qn, i.e. the determination of the maximal size of a q-ary e-error correcting code of length n.
The standard work3 on covering codes lists the following applications.
Article from Wikipedia. All text is available under the terms of the GNU Free Documentation License - Our Privacy Policy - Thanks Auto Blog Commenter