100 011 1101 100 1110 000 1111 111 In the convention of left matrix-vector multiplication, the corresponding generator matrix G is:5 (24.50) 0001111 Note that the above definitions correspond to a code expressed in a systematic form, as shown by the fact that the right 3 x 3 sub-matrix of H and the left 4 x 4 sub-matrix of G are identity matrices. The 16 block codewords Y = XG of C1= (7,4 4), which were already listed in Chapter 11, are reproduced here for convenience in Table 24.4. Consider next the dual code C2 = C1. By definition, its parity-check matrix is = GT, with corresponding generator matrix G = . From the definitions in Eqs. (24.49) In Chapter 11 we used the convention of left vector-matrix multiplication. Thus, the codewords are generated according to the product Y= XG, see Eq. (11.2) in Chapter 11. Under this convention, for an (n,k) code with m = n - k, the systematic form of the generator and parity-check matrices are G = [Ik\Pkxm] and = [(PT)mxk|Im], respectively. Thus, for the (7,4) Hamming code (m=3), G is a 4 x 7 matrix and is a 3 x 7 matrix. Quantum error correction Table 24.5 Block codewords Y of the dual code C2 = C1= (7,3) of the Hamming code C1= (7, 4), as produced from the gen- erator matrix G=AT, where H is the parity- check matrix of C1, according to Y = GX. The parity bits are shown in bold. Message word X Block code Y 0000 000 0111 001 1011 010 1100 011 1101 100 1010 101 0110110 0001 111 and (24.50), we obtain: (24.51) G=HT= (24.52) 0 0 1 As defined in Eq. (24.52), the 7 x 3 generator matrix G of the dual code C2 generates seven-bit codewords from three-bit message words, this time using the right vector- matrix multiplication, i.e., Y = GX. Since G has the systematic form G= [P4x3|I3], and since GHT HTG=0= GHT= HTG the dual code is a valid (n,k) = (7,3) code (note that it is not a Hamming code since n 2n-k-1). The block codewords Y = GX are listed in Table 24.5. Analyzing Table 24.5, we first note that, as expected, the code C2 is in systematic form, with the four parity bits appeari