Coping with Bit Errors using Error Correction Codes

Recall our main goal in designing digital communication networks: to send information reliably and efficiently between nodes. Meeting that goal requires the use of techniques to combat bit errors, which are inevitable in both commmunication channels and storage media (storage may be viewed as “communication across time”; you store something now and usually want to be able to retrieve it later). The key idea we will apply to achieve reliable communication is the addition of redun­ dancy to the transmitted data, to improve the probability that the original message can be reconstructed from the possibly corrupted data that is received. The sender has an encoder whose job is to take the message and process it to produce the coded bits that are then sent over the channel. The receiver has a decoder whose job is to take the received (coded) bits and to produce its best estimate of the message. The encoder-decoder procedures together constitute channel coding; good channel codes provide error correction capabilities that reduce the bit error rate (i.e., the probability of a bit error). With proper design, full error correction may be possible, provided only a small num­ ber of errors has occurred. Even when too many errors have occurred to permit correction, it may be possible to perform error detection. Error detection provides a way for the re­ ceiver to tell (with high probability) if the message was decoded correctly or not. Error detection usually works by the sender and receiver using a different code from the one used to correct errors; common examples include the cyclic redundancy check (CRC) or hash functions. These codes take n-bit messages and produce a compact “signature” of that mes­ sage that is much smaller than the message (e.g., the popular CRC-32 scheme produces a 32-bit signature of an arbitrarily long message). The sender computes and transmits the signature along with the message bits, usually appending it to the end of the message. The receiver, after running the decoder to correct errors, then computes the signature over its estimate of the message bits and compares that signature to its estimate of the signature bits in the received data. If the computed and estimated signatures are not equal, then the receiver considers the message to have one or more bit errors; otherwise, it assumes that the message has been received correctly. This latter assumption is probabilistic: there is some non-zero (though very small, for good signatures) probability that the estimated