Abstract Algebra Theory and Applications - Computer Science ...

7Algebraic Coding TheoryCoding theory is an application of algebra that has become increasinglyimportant over the last several decades. When we transmit data, we areconcerned about sending a message over a channel that could be affectedby “noise.” We wish to be able to encode and decode the information in amanner that will allow the detection, and possibly the correction, of errorscaused by noise. This situation arises in many areas of communications,including radio, telephone, television, computer communications, and evencompact disc player technology. Probability, combinatorics, group theory,linear algebra, and polynomial rings over finite fields all play important rolesin coding theory.7.1 Error-Detecting and Correcting CodesLet us examine a simple model of a communications system for transmittingand receiving coded messages (Figure 7.1). Uncoded messages may be composedof letters or characters, but typically they consist of binary m-tuples.These messages are encoded into codewords, consisting of binary n-tuples,by a device called an encoder. The message is transmitted and then decoded.We will consider the occurrence of errors during transmission. Anerror occurs if there is a change in one or more bits in the codeword. Adecoding scheme is a method that either converts an arbitrarily receivedn-tuple into a meaningful decoded message or gives an error message forthat n-tuple. If the received message is a codeword (one of the special n-tuples allowed to be transmitted), then the decoded message must be theunique message that was encoded into the codeword. For received noncodewords,the decoding scheme will give an error indication, or, if we are moreclever, will actually try to correct the error and reconstruct the original mes-108