Perfect codes on graphs

Author

Cull, Paul

Author_Institution

Dept. of Comput. Sci., Oregon State Univ., Corvallis, OR, USA

fYear

1997

fDate

29 Jun-4 Jul 1997

Firstpage

452

Abstract

The set of codewords for a standard error-correcting code can be viewed as a subset of the vertices of a hypercube. Two vertices are adjacent in a hypercube exactly when their Hamming distance is 1. Such a code is a perfect 1-error-correcting code if no two codewords are adjacent, and if every non-codeword is adjacent to exactly one codeword. Since a perfect code can be described using only vertices and adjacency, the definition applies to general graphs rather than only to hypercubes. How does one decide if a graph can support a perfect 1-error-correcting code? The obvious way to show that such a code exists is to display the code. On the other hand, it seems difficult to show that a graph does not support such a code. We show that this intuition is right by showing that to determine if a graph has a perfect 1-error-correcting code is an NP-complete problem

Keywords

computational complexity; decoding; error correction codes; graph theory; Hamming distance; NP-complete problem; adjacency; codewords; decoding; error-correcting code; hypercube; perfect 1-error-correcting code; vertices; Code standards; Computer science; Decoding; Displays; Error correction codes; Hamming distance; Hypercubes; Law; NP-complete problem; Poles and towers;

fLanguage

English

Publisher

ieee

Conference_Titel

Information Theory. 1997. Proceedings., 1997 IEEE International Symposium on

Conference_Location

Ulm

Print_ISBN

0-7803-3956-8

Type

conf

DOI

10.1109/ISIT.1997.613389

Filename

613389