Error correction

Error correction recovers a transmitted or stored message despite specified corruption. A decoder uses the redundancy of an error-correcting code to select the original codeword, or reports failure if the available information does not determine one reliably.

For the repetition code $0\mapsto 000$, $1\mapsto 111$, a received $010$ is corrected to $000$ by choosing the closest word in Hamming distance. This guarantees recovery after at most one bit flip, but two flips can produce a wrong choice. If corrupted positions are known, they are erasure, which are easier to repair: a code of minimum distance $d$ can recover from any $d-1$ erasures.