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For sake of contradiction, let's assume that there exist two distinct codewords c and c', both elements of C, and let's assume that there exists some vector v $ \epsilon$ V, such that $ d({\bf v},{\bf c})\leq t$ and $ d({\bf v},{\bf c}')\leq t$. Applying the triangle inequality, we have
$ d({\bf c},{\bf c'})\leq d({\bf c},{\bf v})+d({\bf v},{\bf c'}) \leq 2t$ (by our supposition).
But this contradicts the fact that $ d($C)$ \geq 2t+1$.

Let C be a $ t$-error correcting code. Then the spheres S$ _{t}({\bf c})$ such that c $ \epsilon$ C are mutually disjoint.

Bill Cherowitzo 2001-12-11