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Proof

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$ _{t}({\bf c})$ and S$ _{t}({\bf c}')$. We have from the definition of t-error correcting code, that the distance between c and c' is at least $ 2t+1$, yet from the previous theorem we have that both codewords are within $ t$ of v (from definition of sphere of radius $ t$). That implies that c and c' are at most $ 2t$ apart (both are the center of a sphere containing v), a contradiction.

A $ t$-error correcting code C $ \subset$ V is called perfect if any vector of V has a distance of $ t$ or less from exactly one codeword.

Let V $ =\{0,1\}^{n}$. Let C $ \subset$ V be a 1-error correcting code. Then the number of codes in C equals $ \frac{2^n}{n+1}$ if and only if C is perfect.



Bill Cherowitzo 2001-12-11