3. A cyclic code is one in which one can rotate an existing codeword to obtain another. Consider the generator matrix for the polynomial X+X+1. To construct a codeword from this polynomial, three check bits (highest degree is 3) is added to the left of the corresponding binary string: 1011 (see last row of the following matrix. Notice that row 3 is obtained by rotating row 4, and so forth. [1 0 1 1 0 0 0 0101100 G= 0 0 1 0 1 1 0 0 0 0 1 0 1 (i) To ensure the first four columns form an identity matrix, carry out the following operation on row 1 and 2 respectively: (i) row-1 XOR row-3 XOR row-4, (ii) row-2 XOR row-4. Determine the check matrix. Using the method to compute CRC, see Lecture-2, determine the codeword of the following bitstring: 1011. Recall that the polynomial is X8+X+1. Assume an error occur at bit 2 of the codeword of (iii). Determine the resulting remainder at the receiver. In this step, encode the bitstring 1011 using generator matrix G, and decode the resulting codeword (with a corrupted 2nd bit) using the corresponding check matrix. Does the syndrome agree with the remainder of part (iv)? (iv) (v) 3. A cyclic code is one in which one can rotate an existing codeword to obtain another. Consider the generator matrix for the polynomial X+X+1. To construct a codeword from this polynomial, three check bits (highest degree is 3) is added to the left of the corresponding binary string: 1011 (see last row of the following matrix. Notice that row 3 is obtained by rotating row 4, and so forth. [1 0 1 1 0 0 0 0101100 G= 0 0 1 0 1 1 0 0 0 0 1 0 1 (i) To ensure the first four columns form an identity matrix, carry out the following operation on row 1 and 2 respectively: (i) row-1 XOR row-3 XOR row-4, (ii) row-2 XOR row-4. Determine the check matrix. Using the method to compute CRC, see Lecture-2, determine the codeword of the following bitstring: 1011. Recall that the polynomial is X8+X+1. Assume an error occur at bit 2 of the codeword of (iii). Determine the resulting remainder at the receiver. In this step, encode the bitstring 1011 using generator matrix G, and decode the resulting codeword (with a corrupted 2nd bit) using the corresponding check matrix. Does the syndrome agree with the remainder of part (iv)? (iv) (v)