Let 
be an error-correcting
code consisting of 
codewords,in which each codeword
consists of 
letters taken from an alphabet 
of length 
, and every two distinct codewords
differ in at least 
places. Then 
is said to be perfect if for every possible word 
of length 
with letters in 
, there is a unique code word 
in 
in which at most 
letters of 
differ from the corresponding letters of 
.
It is straightforward to show that 
is perfect if
 |
If 
is a binary linear code, then 
and 
, where 
is the number of generators of 
, in which case 
is perfect if
 |
Hamming codes and the Golay code are the only nontrivial examples of perfect codes.
