Error correcting codes, coding theory, etc. is an old, serious, and deep field. Once I took a grad course in it. The course was heavily abstract algebra, espe

Error correcting codes, coding theory, etc. is an old, serious, and deep field. Once I took a grad course in it. The course was heavily abstract algebra, especially finite fields. A respected text is W.\ Wesley Peterson and E.\ J.\ Weldon, Jr., {\it Error-Correcting Codes, Second Edition.\/} Last I heard, there has been some surprisingly good progress in the field. Might look up, say, "turbo codes".