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C2 | 2n in .NET Develop Data Matrix barcode in .NET C2 | 2n

C2 . using visual studio .net togenerate data matrix barcodes for asp.net web,windows application Office Word 2n. ( 1)x z z . . C2. (24.46). The nal step consists of p assing each qubit in the above state through a Hadamard gate, i.e., to obtain the state H n H n .

(x + C2 ) X Y . Here, ther Visual Studio .NET data matrix barcodes e is no point in going through the detailed calculation of such an operation, because as a useful property, H and H n are self-inverse operators (H H = I, H n H n = I n ).

If we let eY = 0 in Eq. (24.44),.

Quantum error correction we obtain H n (x + C2 ) . H. C2 . 2n . ( 1)x z z . z C2 (24.47). x + C2 ,. which is precisely the same state as H n (x + C2 ) X Y , and also the H n transform of the error-free state x + C2 ! Thus, a second app .NET Data Matrix barcode lication of H n on the states de ned in either Eq. (24.

46) or Eq. (24.47) yields H n H n .

(x + C2 ) . = H n H n (x + C2 ) = . x + C2 1 x + y , . C2 . y C (24.48). which yields our initial CSS-encoded state x + C2 . Thus, the second r visual .net Data Matrix ound of correction concerning Z -errors has successfully restored the CSS codeword in its full integrity.

The next section concerning the Steane code provides an applied illustration of the CSS(C1 , C2 ) codes.. Hadamard Steane code The Hadamard Steane code, a .net framework Data Matrix 2d barcode lso sometimes called the Steane code, belongs to the CSS(C1 , C2 ) family. It has the same bit- ip and phase- ip error correction capability as the earlier-described Shor code, namely, up to one error in either or both cases, but it uses 7-qubit codewords as opposed to nine qubits in the second case.

It is based on the Hamming code C1 = C = (7, 4), which was described in 11, and its dual C2 = C . To recall, a possible parity-check matrix H for the (7, 4) Hamming code, 4 which we used in that chapter, is de ned as: 1 1 0 1 1 0 0 (24.49) H = 1 0 1 1 0 1 0 .

0 1 1 1 0 0 1. Some other possible parit y-check matrices H or the Internet are 1 H = 0 0 1 H = 0 1 0 H = 0 1. for the (7, 4) Hamming code .NET Data Matrix 2d barcode commonly used in the literature 0 1 0 1 1 1 0 1 0 0 0 1 0 0 1 0 1 1 1 1 0 1 1 0 1 0 0 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 1 , 1 1 0 , 0 1 1 . 1.

24.4 Hadamard Steane code Table 24.4 Block codewords Data Matrix for .NET Y of the Hamming code C1 = (7, 4), as pro duced from the generator matrix G = H T , according to Y = X G.

The parity bits are shown in bold. Message word X 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 Block code Y 0000 000 0001 111 0010 011 0011 100 0100 101 0101 010 0110 110 0111 001 1000 110 1001 001 1010 101 1011 010 1100 011 1101 100 1110 000 1111 111. In the convention of left m atrix-vector multiplication, the corresponding generator matrix G is:5 1 0 G= 0 0 0 1 0 0 0 0 1 0 0 0 0 1 1 1 0 1 1 0 1 1 0 1 . 1 1. (24.50). Note that the above de niti ons correspond to a code expressed in a systematic form, as shown by the fact that the right 3 3 sub-matrix of H and the left 4 4 sub-matrix of k 4 G are identity matrices. The . C1 . = 2 = 2 = 16 block codewor ds Y = X G of C1 = (7, 4), which were already listed in 11, are reproduced here for convenience in Table 24.4. Consider next the dual code C2 = C .

By de nition, its parity-check matrix is H = T T G , with corresponding generator matrix G = H . From the de nitions in Eqs. (24.

49). In 11 we used the conventi VS .NET Data Matrix 2d barcode on of left vector-matrix multiplication. Thus, the codewords are generated according to the product Y = X G, see Eq.

(11.2) in 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 .

Pk m ] and H = [(P T )m k Im ], respectively. Thus, f .net vs 2010 Data Matrix 2d barcode or the (7, 4) Hamming code (m = 3), G is a 4 7 matrix and H is a 3 7 matrix.

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