2b 1 1, and Ib A0 is a b b identity matrix. in .NET

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2b 1 1, and Ib A0 is a b b identity matrix.
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SINGLE-BIT ERROR CORRECTING AND SINGLE-BLOCK ERROR LOCATING (SEC-Sb=p b EL) CODES
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Theorem 9.8 The code described by the following matrix H is an SEC- Sb=p b EL code with check-bit length R br and codeword length N bits:
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H0 . . . . . .
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where and are row vectors in which odd-weight square matrices and even-weight square matrices are used, respectively, and the top in each column is the row vector including all b b identity matrices. In this matrix each submatrix Hi includes different pattern of s and s and has at least one . The code length (in bits) of the code above is expressed by the following equation: N 2r 1 B b 2r 1 2 b 1 r 1 ; where r is the number of check-bytes. Proof It is apparent that the code satis es condition 1 of Theorem 9.4 for any single-bit errors and any single-byte errors. The binary columns in H are distinct. Therefore condition 2 of Theorem 9.4 is satis ed as well. If i 6 j, then matrices Hi and Hj have different patterns of s and s. By Lemma 9.2, condition 3 of Theorem 9.4 is satis ed. Since every submatrix has at least one identity matrix, condition 4 of Theorem 9.4 is satis ed. Hence the code described by H is an SEC-Sb=p b EL code. The number of blocks is equal to that of the nonzero integers expressed by r-digit binary numbers, meaning 2r 1. The block length B is determined by the number of distinct column vectors using r matrices of the odd-weight column square matrices and the evenweight column square matrices. B is also determined by the condition that every Hi includes at least one row of identity matrices. Based on this, the maximum block length B bits can be Q.E.D. expressed as b 2b 1 r 1 b2 b 1 r 1 . Therefore Eq. (9.4) is valid. Example 9.2 [FUJI94] 9:4
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For b 4 and R 8 the 96; 88 SEC-S4=8 4 EL code is given by the following matrix H: i h H H0 H1 H2 # " " B0 I4 B1 I4 B7 I4 I4 B0 I4 B1 I4 B7 I4 A0 I4 A1 # I4 A7 :
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Figure 9.4 shows this matrix expressed in binary form.
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1000 1000 1000 1000 1000 1000 1000 1000 0100 0100 0100 0100 0100 0100 0100 0100 0010 0010 0010 0010 0010 0010 0010 0010 0001 0001 0001 0001 0001 0001 0001 0001 0000 1111 0000 0000 1111 1111 0000 1111 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 1111 1111 0000 1111 0000 1111 0000 0000 0000 1111 1111 0000 1111 1111 1000 1000 1000 1000 1000 1000 1000 1000 0100 0100 0100 0100 0100 0100 0100 0100 0010 0010 0010 0010 0010 0010 0010 0010 0001 0001 0001 0001 0001 0001 0001 0001 1000 0001 0010 0101 1011 0111 1110 1100 0100 1001 0011 0110 1101 1010 0101 1010 0010 0101 1011 0111 1110 1100 1000 0001 0001 0010 0101 1011 0111 1110 1100 1000
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0000 1111 0000 0000 1111 1111 0000 1111 0000 1111 1111 0000 0000 0000 1111 1111 0000 0000 1111 1111 0000 1111 0000 1111 0000 0000 0000 1111 1111 0000 1111 1111 1000 1000 1000 1000 1000 1000 1000 1000 0100 0100 0100 0100 0100 0100 0100 0100 0010 0010 0010 0010 0010 0010 0010 0010 0001 0001 0001 0001 0001 0001 0001 0001
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Figure 9.4 Binary form of the (96, 88) SEC-S4=8 4 EL codes. Source : [FUJI94]. 1994 IEEE.
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SINGLE-BIT ERROR CORRECTING AND SINGLE-BLOCK ERROR LOCATING (SEC-Sb=p b EL) CODES
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Expanding the Code Length The code design method of Theorem 9.8 says that the check-bit length should be a multiple of the byte length b. This condition can be relaxed by taking any check-bit length R > 2b, as shown by the following theorem. Theorem 9.9 Let the following matrix H be a parity-check matrix of an SEC-Sb=p b EL code with a code length in bits b 2r 1 2 b 1 r 1 and a check-bit length br: h H H0 H1 i Hn 1 ;
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where Hi , i 0; 1; ; n 1, is a submatrix of H. The code described by the following matrix H0 is an SEC-Sb=p b EL code with check-bit length br 1: 2 6 H0 4 H0 0 0 H0 1 1 H1 0 0 H1 1 1 0 0 1 1 Hn 1 Hn 1 3 7 5:
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If this procedure is performed c times on H, then the code length in bits of the expanded code with check-bit length R br c, 0 c < b can be expressed as follows: N 2r 1 B b 2r 1 2 b 1 r 1 c : 9:5
Figure 9.5 illustrates the relation between the information-bit length and the check-bit length of the SEC-Sb=p b EL code determined by Theorems 9.8 and 9.9 for b 4.
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