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where w ! 1 such that mw is the smallest nonzero digit in the binary form of m besides m0 . Example 8.7 [LO05]
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The worst-case TAG number of a 28-input XOR circuit can be computed as follows: First, we nd m4 m3 m2 1, m1 m0 0, and w 2. Therefore TAG28 TAG24 TAG23 TAG22 14 22 23 14 23 14 21 68: We will now use this formula to calculate the maximum TAG number of the minimumweight & equal-weight-row Hsiao s SEC-DED code. Table 8.5 shows Hsiao s code parameters and the maximum TAG number for several power of 2 information bit lengths up to 1,024. All codes listed in Table 8.5 are minimum-weight & equal-weight-row codes.
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TABLE 8.5 Code Parameters and Maximum TAG Numbers of the Minimum-Weight & Equal-Weight-Row Hsiao SEC-DED Codes n 13 22 39 72 137 266 523 1,036 k 8 16 32 64 128 256 512 1,024 r 5 6 7 8 9 10 11 12 H structurea     5 5 8= 1 3     6 6 16= 3 1     7 7 32= 1 3       8 8 8 8= 5 1 3       9 9 9 44= 5 3 1       10 10 10 136= 5 3 1       11 11 11 347= 1 3 5         12 12 12 12 12= 7 3 5 1 H Weightb 24 48 96 208 472 1,040 2,230 4,704 Average Hb 4.8 8 13.7 26 52.4 104 202.8 392 Maximum TAG 28 72 183 504 1,376 3,560 8,764 20,976
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Source: [LO05]. 2005 IEEE. a The notation j= r means that j out of all possible r combinations is used. i i
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Figure 8.25 H weights and maximum TAGs of odd-weight-column SEC-DED codes for various k. Source:
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[LO05]. 2005 IEEE.
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From this table we observe that the maximum TAG number is intimately tied to the H weight, or the total number of 1 s in H. Also the maximum TAG grows exponentially with respect to k. This can be clearly seen in Figure 8.25. EXERCISES 8.1 (a) In Example 8.2, a 2-bit burst error 0110000 is assumed to have occurred in the second frame (frame 2). Using the (22, 13) 3-bit burst error correcting and 4-bit burst error detecting Fire code indicated in this example, calculate Hy S, and 10 By S for each frame, and then explain how this error can be corrected. 10 (b) For the 5-bit burst error 010011 occurred in the frame 2, explain how this error can be detected. (c) Design the parallel decoding circuit of the (22, 13) 3-bit burst error correcting and 4-bit burst error detecting Fire code in Example 8.2, implemented by the combinational circuits. 8.2 For word length N and frame length L, prove that the number of frames m overlapped with adjacent ones by length z is expressed as $ m % N z ; L z
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where dxe represents the smallest integer greater than or equal to x. 8.3 8.4 8.5 Prove Lemma 8.2. Prove Theorem 8.3. For the (42, 33) cyclic 3-bit burst error correcting and 4-bit burst error detecting Fire code generated by g x x6 1 x3 x 1 over GF 2 , answer the following: (a) Find the binary 9 9 companion matrix T. (b) Express the parity-check matrix H of this (42, 33) code over GF 2 . (c) Using the H, write the matrices of T0 ; T7 ; T14 ; T21 , T28 , and T35 necessary for parallel decoding. (d) Design the parallel decoding circuit of this (42, 33) code.
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