Nearly Perfect Code

Let C be an error-correcting code consisting of N codewords in which each codeword consists of n letters taken from an alphabet A of length q, and every two distinct codewords differ in at least d=2e places. Then C is said to be nearly perfect if, for every possible word w_0 of length n with letters in A, there is a codeword w in C in which at most e letters of w differ from the corresponding letters of w_0. The codeword w is unique if it differs from w_0 in fewer than e places and there is at most one other codeword differing from w_0 in e places if w differs from w_0 in e places.

A nearly perfect code C^' can be derived from a perfect code C by adding a parity check digit to the end of each codeword in C, so if C is a [n,k,d]-perfect binary linear code, then C^' is a [n+1,k,d+1]-nearly perfect binary linear code. In this way, the nearly perfect extended Golay code can be obtained from the perfect Golay code and the nearly perfect extended Hamming codes from the perfect Hamming codes.

See also

Error-Correcting Code, Golay Code, Hamming Code, Perfect Code

This entry contributed by David Terr

Explore with Wolfram|Alpha

Cite this as:

Terr, David. "Nearly Perfect Code." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications