Abstract Algebra Theory and Applications - Computer Science ...

7.1 ERROR-DETECTING AND CORRECTING CODES 109m-digit message❄Encodern-digit codeword❄TransmitterNoise❄Receivern-digit received word❄Decoder❄m-digit received message or errorFigure 7.1. Encoding and decoding messagessage. Our goal is to transmit error-free messages as cheaply and quickly aspossible.Example 1. One possible coding scheme would be to send a message severaltimes and to compare the received copies with one another. Suppose thatthe message to be encoded is a binary n-tuple (x 1 , x 2 , . . . , x n ). The messageis encoded into a binary 3n-tuple by simply repeating the message threetimes:(x 1 , x 2 , . . . , x n ) ↦→ (x 1 , x 2 , . . . , x n , x 1 , x 2 , . . . , x n , x 1 , x 2 , . . . , x n ).To decode the message, we choose as the ith digit the one that appears in theith place in at least two of the three transmissions. For example, if the originalmessage is (0110), then the transmitted message will be (0110 0110 0110).If there is a transmission error in the fifth digit, then the received codewordwill be (0110 1110 0110), which will be correctly decoded as (0110). 1 This1 We will adopt the convention that bits are numbered left to right in binary n-tuples.