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Figure 6.5 Example of grouping matrix A. Source: [KANE83]. 1983 IECE Japan.
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The 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 :
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Note 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. Since the product of A and H, which is F, is the concatenation of identity matrices, the bit having 1 in the vector Fi shows that the
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