Error Detection Methods Cont...

- Cyclic Redundancy Check (CRC) :
This error detection method computes the remainder of a polynomial division of a generator polynomial into a message. The remainder, which is usually 16 or 32 bits, is then appended to the message. When another remainder is computed, a non-zero value indicates an error. Depending on the generator polynomial's size, the process can fail in several ways, however, it is very difficult to determine how effective a given CRC will be at detecting errors. The probability that a random code word is valid (not detectable as an error), is completely a function of the code rate: 1 - 2-(n - k). Where n is the number of bits of formed from k original bits of data ,(n - k) is the number of redundant bits, r.
Use of the CRC technique for error correction normally requires the ability to send retransmission requests back to the data source.

- Hamming distance based checks :
If we want to detect d bit errors in an n bit word we can map every n bit word into a bigger n+d+1 bit word so that the minimum Hamming distance between each valid mapping is d+1. This way, if one receives a n+d+1 word that doesn't match any word in the mapping (with a Hamming distance x <= d+1 from any word in the mapping) it can successfully detect it as an erroneous word. Even more, d or fewer errors will never transform a valid word into another, because the Hamming distance between each valid word is at least d+1, and such errors only lead to invalid words that are detected correctly. Given a stream of m*n bits, we can detect x <= d bit errors successfully using the above method on every n bit word. In fact, we can detect a maximum of m*d errors if every n word is transmitted with maximum d errors.