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14:21
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Note that the codes also enable direct decoding. This is easily proved, and therefore the reader is encouraged to attempt the proof. Example 14.8 The following shows an example code expressed by H with r 9, k 9, and n 18: y; z; w f 1; 4; 5 ; 5; 6; 8 ; 1; 2; 7 ; 0; 2; 8 ; 4; 7; 8 ; 0; 3; 7 ; 0; 4; 6 ; 1; 3; 6 ; 2; 3; 5 g;
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In the next subsection we consider a general class of codes satisfying y z w  a (mod r) and also 3x  a (mod r), where a is an integer, 0 a r 1. This is a general class of additive-3 codes mentioned earlier.
14.2.4 Extended BIBD Codes and Additive Codes The new generalized condition for direct decoding of the triple erasure correcting codes generates an ef cient code that reduces the number of check disk subsystems [OHDE05a, 05b].
Theorem 14.8 The codes can correct three erased disk data by direct decoding if and only if any three distinct binary weight-3 column vectors, hi , hj , and hk , in the matrix P with r rows (! 4) of the codes shown by H P j I satisfy the following: w hi _ hj _ hk ! 5; where x _ y means logical OR of binary vectors x and y. Proof Let h be a resultant vector of hi _ hj _ hk , then the relation (14.22) says that h has weight larger than or equal to 5. If each of the corresponding 5 rows of these three distinct column vectors has weight larger than or equal to two, then these three column vectors have the number of 1 s larger than or equal to 10. Since each of these three column vectors has weight three, h should have at most weight 9. This means there exists at least one row of these three vectors, for example, the z-th row, 0 z r 1, having weight one. Assume that hi has 1 at the z-th row position, then the other two weight-3 columns hj and hk have at least two other row positions having patterns 01 and 10 because these two column vectors are distinct. So the three erasures can be corrected by direct decoding. Conversely, for the following three distinct weight-3 column vectors, hi , hj , and hk , for example, the resultant vector h satis es w h w hi _ hj _ hk 4. This means every row has weight two or three, which does not enable direct decoding. If there exists one row with weight one in these three column vectors in order to satisfy the direct decoding, such as hj having 1 at this row position, then the remaining two column vectors hi and hk should be same. This contradicts the notion that every column is distinct. That is, w h 6 4. Therefore w hi _ hj _ hk ! 5. 14:22