Abstract Algebra Theory and Applications - Computer Science ...

114 CHAPTER 7 ALGEBRAIC CODING THEORYBlock CodesIf we are to develop efficient error-detecting and error-correcting codes, wewill need more sophisticated mathematical tools. Group theory will allowfaster methods of encoding and decoding messages. A code is an (n, m)-block code if the information that is to be coded can be divided into blocksof m binary digits, each of which can be encoded into n binary digits. Morespecifically, an (n, m)-block code consists of an encoding functionand a decoding functionE : Z m 2 → Z n 2D : Z n 2 → Z m 2 .A codeword is any element in the image of E. We also require that E beone-to-one so that two information blocks will not be encoded into the samecodeword. If our code is to be error-correcting, then D must be onto.Example 5. The even-parity coding system developed to detect singleerrors in ASCII characters is an (8, 7)-block code. The encoding function isE(x 7 , x 6 , . . . , x 1 ) = (x 8 , x 7 , . . . , x 1 ),where x 8 = x 7 + x 6 + · · · + x 1 with addition in Z 2 .Let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) be binary n-tuples. TheHamming distance or distance, d(x, y), between x and y is the numberof bits in which x and y differ. The distance between two codewords is theminimum number of transmission errors required to change one codewordinto the other. The minimum distance for a code, d min , is the minimumof all distances d(x, y), where x and y are distinct codewords. The weight,w(x), of a binary codeword x is the number of 1’s in x. Clearly, w(x) =d(x, 0), where 0 = (00 · · · 0).Example 6. Let x = (10101), y = (11010), and z = (00011) be all of thecodewords in some code C. Then we have the following Hamming distances:d(x, y) = 4,d(x, z) = 3,d(y, z) = 3.