P0 P1 P2 P3 P4 P5 P6 P7 in .NET Draw QR-Code in .NET P0 P1 P2 P3 P4 P5 P6 P7 P0 P1 P2 P3 P4 P5 P6 P7scanning qrcode on .netUsing Barcode Control SDK for visual .net Control to generate, create, read, scan barcode image in visual .net applications.1 1 1 1 1 1 1 1 1 1 1 1Make qr bidimensional barcode with .netusing barcode generator for .net control to generate, create qr code image in .net applications.1 1 1 1read qr-codes with .netUsing Barcode reader for visual .net Control to read, scan read, scan image in visual .net applications.1 1 1 1VS .NET barcode readerfor .netUsing Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications.F0 F1 F2 1 1 F3 Deploy barcode on .netusing .net vs 2010 toencode barcode on asp.net web,windows applicationFigure 6.5 Example of grouping matrix A. Source: [KANE83]. 1983 IECE Japan. .net Vs 2010 qr barcode makerin visual c#.netgenerate, create qr code jis x 0510 none on c#.net projectsSINGLE-BYTE / BURST ERROR DETECTING SEC-DED CODES Qr Bidimensional Barcode barcode library on .netgenerate, create qr-codes none on .net projectsThe rst ve rows of H are added to obtain F0 . The sixth, seventh, and eighth rows of H are the same as F1 , F2 , and F3 , respectively. The following theorem is relevant to the design of b-grouped parity checkable codes using the grouping matrix A. Theorem 6.7 A code is b-grouped parity checkable, if and only if there exists a matrix A such that the product of A and H is identical to the concatenation of the b b identity matrices. Proof Let the concatenation of the b b identity matrix be F, and let the codeword be W. Then from F A H we have W FT W A H T W HT AT 0 ,W HT 0 :Control qr code data in vb.netto encode qr bidimensional barcode and qr bidimensional barcode data, size, image with vb barcode sdkNote that the mod-2 sum of the data included in each group has a constant value (zero) regardless of the codeword W. Since each column vector of F has weight 1, and each row vector of F has equal constant weight, the code derived from the H matrix is b-grouped parity checkable. This proves the necessity of the condition, and its suf ciency can be proven in a similar manner. Q.E.D. Figure 6.6 shows the b-grouped parity checking of the arbitrary codewords, W1 and W2 . Theorem 6.7 states that if the addition of subsets of row vectors of H as prescribed by A yields a concatenation of b b identity matrices, then the code represented by H is b-grouped parity checkable. Theorem 6.8 Assume that the H matrix of a code C has distinct columns, and the code is b-grouped parity checkable. Then C is an SEC-DED-BED code with code length in bits N b 2R b : Proof Let P0 ; P1 ; . . . ; Pr 1 , be row vectors of H. Since the vector sum of some Pi s yields Fi s, 0 i b 1, there cannot be an all-0 column vector in H. Also, by the assumption, these column vectors in H are distinct. Thus the code can correct all single-bit errors. According to Theorem 6.7, the mod-2 sum of F0 ; . . . ; Fb 1 , yields an all-1 row vector. This means that every column in H has odd weight. Therefore this code can also detect all double-bit errors. Next let us examine a vector obtained by multiplying the syndrome to the grouping matrix A. It is easy to see that the result is identical to the single-byte error pattern. This pattern can indicate whether the error is a correctable single-bit error or an uncorrectable byte error. On the other hand, since the product of A and H yields the concatenation of b b identity matrices, this byte error pattern will always indicate any b-adjacent errors, that is, burst errors. This means the code can detect single b-bit burst errors (see Exercise 2.29). Let us consider the bit length of this code. 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