Prove that a code is d-error-correcting if and only if it is (2d)-error detecting.

The Fano code is 2 error-detecting, and it is 1-error-correcting, so i was thinking of using this as the basis step of induction..

assume: d error correcting  \implies 2d error detecting
prove: d+1 error correcting \implies 2(d+1) error detecting


d + 1 error correcting \implies 2d error detecting + 2 error detecting. (induction assumption)

= 2(d+1)-error detecting.


if this works then.... i can just use the same method to

Prove: that 2d error detecting  \implies d error correcting