From this we have Nmax 2r 2b b.

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Substituting b 1 in the equation shown in Theorem 10.2 gives the maximum length of the SEC codes, which is Nmax 2r 1. Table 10.1 lists the maximum information-bit lengths of the FbECjSEC codes (i.e., Nmax r) for xed-byte X0 length b and check-bit length r. Lemma 10.1 In the FbEC j SEC codes the byte errors in X0 and single-bit errors in X1 occurring simultaneously, which are denoted as byte errors in X0 plus single-bit errors in X1 , are not miscorrected as the errors occurring in the xed-byte X0 . Proof If byte errors in X0 plus single-bit errors in X1 lead to miscorrection of xed-byte X0 in the codewords of the FbECjSEC codes, there exist such errors Ei ; Ep ; and Eq having the following relation: Ep HT Ei HT Eq HT ,

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TABLE 10.1 Bounds on Information-Bit Lengths of FbECjSEC Codes b r 3 b 1 b 2 b 3 b 4 b 5 7 22 53 116 243 4 15 46 109 236 491 5 31 94 221 476 987 6 63 190 445 956 1,979 7 127 382 893 1,916 3,963 8 255 766 1,789 3,836 7,931 10 1,023 3,070 7,165 15,356 31,739 12 4,095 12,286 28,669 61,436 126,971

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10:2

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Source: [FUJI98]. 1998 IEEE.

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where Ep ; Eq , Ep 6 Eq , are byte errors in X0 , and Ei is single-bit error in X1 . Since Ep Eq Ep0 is a byte error in X0 , the relation shown in Eq. (10.2) is equivalent to the following relation: Ei HT Ep0 HT 0: This relation contradicts condition 4 of Theorem 10.1. So the xed-byte X0 is not Q.E.D. miscorrected by the byte errors in X0 plus the single-bit errors in X1 . Design for Optimal FbECjSEC Codes Theorem 10.3 The following H matrix shows a systematic FbEC jSEC code satisfying the bounds on the code length given by Theorem 10.2: H Ib P r b b M0 Qe M3 Qe Mi Qe M Ne Qe M1 Qo M2 Qo Mj Qo ! M No ; Qo

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where I b : b b identity matrix, P r b b : r b b matrix with all 1 s, i: integer whose binary representation has even weight, j: integer whose binary representation has odd weight, Ne : maximum integer i having value no greater than 2b 1, No : maximum integer j having value no greater than 2b 1, M i M j : b 2r b 1 matrix whose binary column vector of b-bit length indicates integer i(j), Qe : r b 2r b 1 matrix whose distinct column vectors indicate integers from 1 to 2r b 1, Qo : r b 2r b 1 matrix whose distinct column vectors indicate integers from zero to 2r b 2. Proof Since nonzero column vectors in H are all distinct, the code satis es conditions 1 and 2 of Theorem 10.1 for the error set E1 , and therefore the code has an SEC function. Since matrix Ib is nonsingular, the code satis es conditions 1 and 3 for the error set E0 . Let the upper b bits of syndrome S be SF and the lower r b bits of S be Sp . For the byte errors in X0 , Sp is an all-0 or an all-1 vector. If Sp is an all-0 vector, then X0 includes an even number of bit errors and SF is of even weight. If Sp is an all-1 vector, then X0 has an odd number of bit errors and SF is of odd weight. Syndromes caused by single-bit errors occurring outside X0 are different from those caused by byte errors in X0 . This is because in the case of single-bit errors, Sp is not all-0 for SF with even weight while Sp is not all-1 for SF with odd weight. So condition 4 of Theorem 10.1 is satis ed. Hence the H matrix shown in the theorem is an FbECjSEC code.

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CODES FOR UNEQUAL ERROR CONTROL / PROTECTION (UEC / UEP)

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The maximum number of columns in H1 shown in the previous H is 2b 2r b 1 , and hence the maximum code length in bits equals Nmax b 2b 2r b 1 2r 2b b: This code length is equal to the maximum code length of the FbECjSEC code shown in Theorem 10.2. Q.E.D. We see from the proof that the code indicated in Theorem 10.3 is optimal. Example 10.1 2 Systematic (27, 22) F3ECjSEC Code 000 000 000 011 101 "" 000 111 111 011 101 111 000 111 011 101 111 111 000 011 101 000 000 111 001 010 " 000 111 000 001 010 " 111 000 000 001 010 " 3 111 111 7 7 111 7: 7 001 5 010

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100 6 010 6 H 6 001 6 4 111 111

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The place of the check bits is indicated by the upward-pointing arrow ("). Decoding Procedure Single-bit error correction can be easily performed such that if the nonzero syndrome is equal to one of the column vectors in H, the corresponding bit is inverted and then corrected as a single-bit error. If the nonzero syndrome is not equal to any column vector in H, then byte errors in X0 can be assumed to exist. Let the upper b bits of the syndrome S be SF and the lower r b bits of S be Sp . Further let calculation of SF Ib 1 be Ep , meaning Ep SF . If Ep PT r b b Sp , then Ep is a byte-error pattern, which is added to the original xed-byte information of X0 . This provides a correction of the erroneous xed-byte X0 . If Ep PT r b b 6 Sp , then we can assume that there exist multiple-bit errors in the word other than single-bit errors in X1 and xed-byte errors in X0 (i.e., uncorrectable errors) that are nally detected by the code. The FbECjSEC code does not require large decoding hardware augmentation compared to the existing SEC-DED code. For example, the decoder of the 72; 64 F7ECjSEC code requires only 11.6% hardware augmentation compared to that of the (72, 64) SEC-DED code. Evaluation The FbECjSEC codes are evaluated by their check-bit lengths and error detection capabilities. Figure 10.5 shows the relation between the information-bit lengths and the check-bit lengths of the FbECjSEC codes for the xed-byte X0 with lengths b 4; 6; 7; 8, and 10 bits. For comparison, the lengths of SEC codes are indicated in the gure as well. Note that the F4ECjSEC code has almost the same check-bit length as that of the SEC code. Figure 10.6 provides an example of (72, 64) F7ECjSEC code in a shortened version of the original (135, 127) F7ECjSEC code. In the obtained H matrix of the (135, 127)

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