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 $d=2e+1$ places. Then C is said to be perfect if for every possible word w_0 of length n with letters in A, there is a unique code word w in C in which at most e letters of w differ from the corresponding letters of $w_{0}$.

It is straightforward to show that C is perfect if

$\sum _{i=0}^{e}(n;i)(q-1)^{i}=(q^{n})/N$.

If C is a binary linear code, then q==2 and N==2^k, where k is the number of generators of C, in which case C is perfect if

$\sum _{i=0}^{e}(n;i)=2^{(n-k)}$.

Hamming codes and the Golay code are examples of perfect codes.