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P where indicates modulo-2 sum and 0 is the null vector with all zero elements. When the codeword is corrupted by either a single- or double-track error, the erroneous 0 0 0 0 0 0 0 0 0 received word is denoted by WZ Z0 Z1 Z2 Z3 Z4 Z5 Z6 Z7 P0 . From the received word 0 WZ , the syndromes S1 and S2 are computed as " S1 S2

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If there is no error, S1 S2 0. Suppose that only the i-th track 0 i 8 has errors and the error pattern is denoted by an eight-digit vector E (i.e., E Zi Zi0 ). Then Eqs. (11.4) and (11.5) are rewritten as S1 E; ( S2 11:6 E T 0

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If S2 6 0, then the track position i is uniquely determined by the following fact: S2 T i S1 E: 11:8

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The erroneous tracks are often identi ed externally by external pointers obtained upon detecting loss of signal in the read ampli ers, an excessive phase shift in clock, inadmissible recording patterns, and so on. We will see that if two erroneous tracks are identi ed by the external pointers, any two-track bytes in error in the ORC are correctable. Let Ei and Ej denote the two error-pattern vectors representing errors in tracks i and j, respectively (i < j). That is, the received bytes are error free except in tracks i and j, where Zi0 Zi Ei , and Zj0 Zj Ej if 0 j 7, or P0 P Ej if j 8. Then syndromes S1 and S2 of Eqs. (11.4) and (11.5) can be represented in the following way: S1 E i Ej ; ( Ei Ti Ej Tj S2 Ei T i

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These equations uniquely determine the error patterns Ei and Ej as E i S1 Ej

Figure 11.3 H matrix for the ORC in Eq. (11 Source: [PATE74]. 1974 by International Business Machines Corporation; republished by permission. .1).

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Here we have an interesting conversion of the parity-check matrix into the informationbyte format. The alternate form allows direct computation of the check byte and syndrome. In addition the new format allows a fast implementation of the ORC without requiring a buffer for the encoding. Figure 11.3 gives the binary version of the H matrix in Eq. (11.1). In the gure ai is an element of GF 28 expressed as an 8-digit column vector ai of the binary coef cients of the polynomial xi modulo g x . First, consider the column of the H matrix corresponding to the bit Zi j of the track byte Zi for all i and j such that 0 i 7 and 0 j 7. The lower half of this column is ak , where k i j, which is the same for the column corresponding to Zj i . We call this property the orthogonal symmetry of the code. To complete the symmetry, let B0 denote the check byte C. Then Zj i Bi j for 0 i 7 and 0 j 7

Figure 11.4 shows the orthogonal symmetry and the powers of a that appear in the lower half columns of the H matrix. We can proceed to re-arrange the columns of the H matrix of Figure 11.3 to obtain another H matrix, H0 in Figure 11.5, corresponding 0 0 to a codeword WB in terms of the information bytes, written as WB B0 B1 B2 B3 B4 B5 B6 B7 P . Note that this re-arrangement does not alter the parity-checking rules. The orthogonal symmetry of the code has produced Ti in the lower half of the matrix H0 , corresponding to the information byte Bi , which is the same as that corresponding to the track byte Zi . The upper half of H0 is the conventional VRC, and it can be represented by a matrix Gi , where Gi is an 8 8 all-zero matrix, except the row i, which is all ones.

B0 B1 B2 B3 B4 B5 B6 B7 Z0 0 Z1 1 Z2 2 Z3 3 Z4 4 Z5 5 Z6 6 Z7 7 P 1 2 3 4 5 6 7 8 2 3 4 5 6 7 8 3 4 5 6 7 8 4 5 6 7 8 5 6 7 8 6 7 8 7 8 9