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Figure 4.1 Parallel decoding circuit.
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MODIFIED HAMMING SEC-DED CODES
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An example of the H matrix for the 72; 64 code is shown in Figure 4.2. From the practical perspective, it is important that the SEC-DED code minimizes the probability of miscorrection when triple or more errors occur. A miscorrection in this context refers to an erroneous decoding that results in an error-free word or in an incorrect word containing miscorrected single bit. Now, again, we consider the (72, 64) codes. Since these codes are shortened SEC-DED codes triple errors have a possibility to generate syndrome patterns outside the column patterns of the H matrix. In this case the triple errors will be correctly decoded, that is, detected. If the syndrome pattern coincides with a column of H, the decoder will mistake it for a single error and performs a miscorrection. Table 4.2 shows the probability of detected triple errors and quadruple errors of the (72, 64) SEC-DED codes. From this table we see that the code shown in Figure 4.2 gives better protection. In general, the undetectable error probabilities of triple errors and quadruple errors, denoted as PUD3 and PUD4 , respectively, for odd-weight-column SEC-DED codes depend on the number of codewords with minimum weight 4, A4 , as follows [HSIA70a]: 4 A4 PUD3 n ; 3 A4 PUD4 n 4 4:4
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A code design method with a reduced number of weight-4 codewords has been studied by [MATS87]. After the code condition of minimum weight of the H matrix is relaxed, a 72; 64 SEC-DED code with improved error detection capabilities has been obtained [AZUM75]. As the preceding analyses, odd-weight-column SEC-DED codes have some practical advantages in decoding speed, amount of encoder / decoder hardware, and a lower probability of erroneous decoding. These codes were therefore widely used in the semiconductor memory systems of the 1970s and 1980s [CHEN84]. Parallel error detection and correction circuit ICs based on the Hsiao codes were designed and sold by some semiconductor industries. 4.1.2 Davydov-Tombak Codes
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Davydov and Tombak [DAVY91] have designed an excellent SEC-DED code that appears to be more capable of detecting triple and quadruple errors than the conventional SEC-DED codes. This code has neither odd-weight-column vectors nor a minimum weight in H. The H matrix of this code has the following form: H B0 G B1 G B2 G ! BD 1 : G 4:5
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Here G consists of the following 4 5 matrix, and Bi bi ; bi ; . . . ; bi , i 0; 1; . . ., D 1, where D 2r 4 , is an r 4 5 matrix consisting of identical columns bi , where bi is
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TABLE 4.2 Probability of Error Detection for (72,64) Codes Probability of triple-error detection (%); P3 43.72 24.0 $ 43.5 Probability of quadruple-error detection (%); P4 99.19 98.90 $ 99.18
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Codes Odd-weight-column code shownin Figure 4.2 Hamming distance-4 code (non^odd-weight- column code)
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1 2 3 4 5 6 7 8: Check Byte 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 c0 c1 c2 c3 c4 c5 c6 c7 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
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Figure 4.2 (72, 64) Odd-weight-column SEC-DED code. Source: [HSIA70a]. Copyright 1970 by International Business Machines Corporation; republished by permission.
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