SINGLE-BIT ERROR CORRECTING AND SINGLE-BYTE ERROR LOCATING (SEC-Se=b EL) CODES

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errors in that byte that result in a zero syndrome. This violates condition 3 of Theorem 9.10. Conditions 3 and 4 say that syndromes produced by e 1 =2 1 e 1 =2-bit errors in a byte should be different from each other and also different from those produced by e 1 =2 or fewer bits errors in that byte. In this case it is because, otherwise, there might exist e or fewer bits errors in that byte that result in a zero syndrome. This case violates the conditions of the code. Therefore syndromes produced by e 1 =2 or fewer bits errors in a byte should differ from each other. It is apparent that syndromes produced by e 1 =2 or fewer bits errors in a byte should also be different from those in another byte. Hence the following inequality holds:

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e 1 =2 N X b : i b i 1

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2. For even e. By condition 3 of Theorem 9.10, the syndromes produced by e=2 or fewer bits errors in a byte should differ from each other. However, syndromes produced by e 2 =2-bit errors in a byte are capable of being equal to those produced by other e 2 =2-bit errors in the same byte only when there exist no erroneous bits at the same bit positions in these two e 2 =2-bit errors. There exist at most " e 2 b= 2 # 2b e 2 " #

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of the e 2 =2-bit errors in a byte that have the same syndromes, where bxc means the largest integer less than or equal to x. From conditions 3 and 4 of Theorem 9.10, any syndromes produced by e 2 =2-bit errors should be different from those produced by e=2-bit errors. Hence there exists at least the following number of distinct nonzero syndromes: 9 8 b > > > >X e=2 e 2 =2 = N< b : b > i 1 i b2b= e 2 c > > > ; : Hence we have the following inequality: 9 8 b > > > >X e=2 e 2 =2 = N< b R : 2 !1 > b > i 1 i b2b= e 2 c > > ; : Q.E.D. Theorem 9.13 Linear N; N R SEC-Se=b EL codes satisfy R ! 2e: This theorem can be easily proved, and the proof is therefore omitted.

CODES FOR ERROR LOCATION: ERROR LOCATING CODES

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Design for SEC-Se=b EL Codes

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The tensor product of the error correcting codes and the error detecting codes is also applied in the design of SEC-Se=b EL codes. That is, tensor product of an SbEC code and a single-bit error correcting and e-bit error detecting code (or SEC-eED code) produces the SEC-Se=b EL code, as shown in the following theorem. Theorem 9.14 The code described by the following matrix H is an SEC-Se=b EL code whose code length N b n bits: H H0 H00 H00 H01 0 00 H0 H H01 H00 H0 H1

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H0n 1 H00 H0n 1 H00 Hn 1 ;

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where represents the tensor product, H0 is the parity-check matrix of the Sb0 EC code whose code length is n bytes, H0i is the submatrix of H0 corresponding to the i-th byte, H00 is the parity-check matrix of the b; b b0 SEC-eED code, and Hi is the submatrix of H corresponding to the i-th byte. Proof It is apparent that the code satis es condition 1 of Theorem 9.10 for any singlebit errors and any e or fewer bits errors. Condition 2 of Theorem 9.10 is satis ed because the binary columns of H all differ. The syndromes resulting from any singlebyte errors in the i-th byte is different from those in the j-th byte, where i 6 j, because each column in Hi is determined by the product of H0i by H00. In general, every H0i includes a b0 b0 nonsingular matrix, and every Hi includes a b0 b matrix obtained by the product of the nonsingular matrix and H00 . Consequently any syndrome resulting from e or fewer errors in a byte is nonzero and therefore satis es condition 3 of Theorem 9.10. This condition also tells us that the syndromes caused by any single-bit errors are different from those caused by any single-byte errors excluding single-bit errors, and therefore condition 4 is also satis ed. From the above, the indicated matrix H satis es all conditions of Theorem 9.10, and hence the code described by H is an SEC-Se=b EL code. Q.E.D. If we apply the maximal Sb0 EC codes [HONG72] shown in Subsection 5.1.4, the maximum code length in bits of the SEC-Se=b EL codes, de ned by Theorem 9.14, can be expressed as follows: 2R 1 2b 2c 1 b00 ; 2b0 1

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9:6

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where R is the check-bit length of the SEC-Se=b EL code and b00 is the code length in bits of the SEC-eED code having c check bits where c R mod b0 . If e 2, for example, then 0 b 2b 1 and b00 2c 1 .

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