Abstract Algebra Theory and Applications - Computer Science ...

110 CHAPTER 7 ALGEBRAIC CODING THEORYtriple-repetition method will automatically detect and correct all single errors,but it is slow and inefficient: to send a message consisting of n bits,2n extra bits are required, and we can only detect and correct single errors.We will see that it is possible to find an encoding scheme that will encode amessage of n bits into m bits with m much smaller than 3n.Example 2. Even parity, a commonly used coding scheme, is much moreefficient than the simple repetition scheme. The ASCII (American StandardCode for Information Interchange) coding system uses binary 8-tuples, yielding2 8 = 256 possible 8-tuples. However, only seven bits are needed sincethere are only 2 7 = 128 ASCII characters. What can or should be done withthe extra bit? Using the full eight bits, we can detect single transmissionerrors. For example, the ASCII codes for A, B, and C areA = 65 10 = 01000001 2 ,B = 66 10 = 01000010 2 ,C = 67 10 = 01000011 2 .Notice that the leftmost bit is always set to 0; that is, the 128 ASCII charactershave codes00000000 2 = 0 10 ,01111111 2 = 127 10 .The bit can be used for error checking on the other seven bits. It is set toeither 0 or 1 so that the total number of 1 bits in the representation of acharacter is even. Using even parity, the codes for A, B, and C now becomeA = 01000001 2 ,B = 01000010 2 ,C = 11000011 2 .Suppose an A is sent and a transmission error in the sixth bit is caused bynoise over the communication channel so that (01000101) is received. Weknow an error has occurred since the received word has an odd number of1’s, and we can now request that the codeword be transmitted again. Whenused for error checking, the leftmost bit is called a parity check bit.By far the most common error-detecting codes used in computers arebased on the addition of a parity bit. Typically, a computer stores informationin m-tuples called words. Common word lengths are 8, 16, and 32 bits..