Consider a (7,4) block error correcting code and the table bellow

a) With generator polynomial P(X)=X^3+X^2+1. What is the transmitted codeword for data bits 0101 ?

b) With generator polynomial P(X)=X^3+X^2+1. A destination node receives bits 1011101. Compute the syndrome with the polynomial division and determine the original transmitted data bits and the original message.

Table 8.3 A Single-Error-Correcting (7,4) Cyclic Code (b) Table of syndromes for single-bit errors (a) Table of valid codewords Data Block Codeword Error pattern E 0000001 0000010 0000100 0001000 0010000 0100000 Syndrome S 001 010 100 101 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 0001101 0010111 0011010 0100011 0101110 0110100 0111001 1000110 1001011 1010001 1011100 1100101 1101000 1110010 011 110 Table 8.3 A Single-Error-Correcting (7,4) Cyclic Code (b) Table of syndromes for single-bit errors (a) Table of valid codewords Data Block Codeword Error pattern E 0000001 0000010 0000100 0001000 0010000 0100000 Syndrome S 001 010 100 101 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 0001101 0010111 0011010 0100011 0101110 0110100 0111001 1000110 1001011 1010001 1011100 1100101 1101000 1110010 011 110