Up: Basic Coding Topology for
Suppose, for sake of contradiction, that we have two distinct codewords c and c' and a vector v such that v is an element of both
S. We have from the definition of t-error correcting code, that the distance between c and c' is at least , yet from the previous theorem we have that both codewords are within of v (from definition of sphere of radius ). That implies that c and c' are at most apart (both are the center of a sphere containing v), a contradiction.
A -error correcting code C V is called perfect if any vector of V has a distance of or less from exactly one codeword.
. Let C V be a 1-error correcting code. Then the number of codes in C equals
if and only if C is perfect.