From this, the inequality (9.2) can be deduced. in .NET

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From this, the inequality (9.2) can be deduced.
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SINGLE-BIT ERROR CORRECTING AND SINGLE-BLOCK ERROR LOCATING (SEC-Sb=p b EL) CODES
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Design for SEC-Sb=p b EL Codes
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1. Codes Designed by Tensor Product Codes I In general, we can design the error locating codes by means of the tensor product of two codes, one being an error correcting code and the other an error detecting code. The codes designed by this method are called here type I codes. This method can be applied to the design of SEC-Sb=p b EL codes by using the single-bit error correcting and single b-bit byte error detecting code, or the SEC-SbED code presented in Section 6.1, and a single b-bit byte error correcting code, or an SbEC code presented in Section 5.1. Theorem 9.7 code: The code described by the following matrix H is an SEC Sb=p b EL H H0b0  H00 b H01 H00 0 00 0 H1 H00 H0 H b b H1 H0
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H0 N=B 1  H00 b H0 N=B 1 H00 b H N=B 1 ;
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where  represents tensor product, B p b, N is the code length (in bits) of the SECSb=p b EL code, H0b0 is the parity-check matrix of the Sb0 EC code, H00 is the parity-check b matrix of the B; B b0 SEC-SbED codes, and H0i is the submatrix of H0b0 corresponding to the i-th byte. 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. Because the binary columns of H are distinct, condition 2 of Theorem 9.4 is satis ed. The syndrome resulting from any single-byte error in the i-th block is different from that in the j-th block for i 6 j because each column in Hi is determined by the product of H0i and H00 . Hence condition 3 is satis ed. In b general, every H0i includes b0 b0 identity matrix, meaning Ib0 , and therefore every Hi has H00 as a column element. This implies that the syndrome resulting from any b single-bit error is different from that resulting from any single-byte error excluding single-bit errors. Based on this and on condition 3, condition 4 in Theorem 9.4 is satis ed. From Theorem 9.4 it follows that the code described by H is an SEC-Sb=p b EL code. Q.E.D. Example 9.1 [FUJI94]
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For b 4 and b0 5 the S5EC code with r 2 described by the matrix H0b0 is
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Hb
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where T5 is a primitive element in GF 25 , and O5 and I5 are the zero element and identity element in GF 25 , respectively. Let H00 be the parity-check matrix of the b 12; 7 SEC-S4ED code having b0 5 check bits. With these two codes the 396; 386
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SEC-S4=3 4 EL code obtained is shown in the following matrix H:
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Hb I5 O5 O5 I5 O Hb
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The code length (in bits) of the SEC-Sb=p b EL codes, de ned by Theorem 9.7, can be expressed as follows. In this case the maximal codes shown in Subsection 5.1.4 are used to determine the length of the Sb0 EC codes. N b 2
b0 b 1
 0 2R 1 2b 2c 1 0 1 b 2b c b 1 1 1 2b0 1
9:3
In this equation, R b0 r c, 0 c < b0 , is the check-bit length of the SEC-Sb=p b EL codes. Figure 9.3 shows the relations between the information-bit lengths and the check-bit lengths of the SEC-Sb=p b EL codes for b 4 bits. In this case, B shows the maximum block length in bits determined by the value of b0 > b . 2. Codes Designed by Odd / Even-Weight Column Square Matrices Codes II Here the SEC-Sb=p b EL codes designed by another method are presented and called type II codes.
18 17 16 31,852
B = 124 (b = 8)
Check-bit length R
15 14 13 12 11 10 9 256 512
B = 60 (b = 7)
6,382 7,278 7,726 3,183 3,631 1,584 1,808 785
12,669 14,573 15,469
25,356 29,164 30,956
B = 28 (b = 6)
K = 386