The existence of such procedures has been known and exploited in complexity theory for a while, and it is relevant to program testing, average-case complexity, probabilistically checkable proofs and private information retrieval.
Partial answers follow from work by several people, including the speaker.
More general answers would follow from the resolution of certain conjectures that seem to be of independent interest. Hopefully, such codes could be useful to the practice of fault-tolerant information transmission and storage, but this remains to be seen. He is currently an assistant professor at UC Berkeley. Luca received the STOC'97 best student paper award, and is a recipient of the Sloan Research Fellowship and of the NSF Career Award.