MDS ARRAY CODES TOLERATING MULTIPLE-DISK FAILURES

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The rst step is to nd the parity of the diagonals. This parity is given by exclusive-OR of the bits of the two parity columns. If this is 0, then the diagonals have even parity; otherwise, they have odd parity. This is not dif cult to see, and hence the reader is encouraged to undertake the proof. In the array we can see that the exclusive-OR of the bits in the two redundant columns is 1. Therefore the diagonals have odd parity, that is, Q 1. Next, the algorithm starts a recursion to retrieve the missing bits ci; 0 and ci; 2 , 0 i 3. The diagonal entries (3, 1), (2, 2), (1, 3), (0, 4) intersect column 2 in entry (2, 2) only. From Q 1, we conclude that c2; 2 0. So we retrieve bit (2, 0), using the horizontal parity, which is always even. We obtain c2; 0 0. Then we consider the diagonal going through entry (2, 0), which consists of the entries (2, 0), (1, 1), (0, 2), (3, 4), (2, 6). The only bit missing is in entry (0, 2), and we can conclude that c0; 2 0. Again, using the horizontal parity, we nd that c0; 0 0. Now, using the diagonal through (0, 0), we obtain c3; 2 0, which implies, by the horizontal parity, that c3; 0 1. Using the diagonal through (3, 0), we obtain c1; 2 0, which nally implies that c1; 0 1. The nal reconstructed array is illustrated below:

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We could give a decoding that corrects one error (i.e., only one column has failed but its location is unknown). This is not the model in RAID architectures, where disk failures are catastrophic events in which pointers identify the failed disks. (The reader is encouraged to reconstruct the erroneous example array with one failed column in Exercises 14.6. Algebraic Description of EVENODD In the array of p 1 p 2 , each column in the array is a polynomial modulo Mp x xp 1 = x 1 xp 1 xp 2 x 1. It is convenient to assume that the array has an imaginary row of zeros that makes it a p p 2 array. A cyclic shift of a column in such an array can cause the bit corresponding to the last row to be nonzero. However, if this is the case, the arithmetic modulo Mp x forces to take the complement of the shifted column, restoring the zero in the last position. As was mentioned earlier, we will use Eq. (14.8) to denote a polynomial modulo Mp x . The codeword for EVENODD is de ned as follows: 9 > > > = C0 a C1 a Cp 1 a Cp a Cp 1 a ; where a is a root of Mp x ; > : p 1 p 1 X X > > > Cp a Ci a ; and Cp 1 a ai Ci a : > ;

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CODES FOR MULTIPLE / DISTRIBUTED STORAGE SYSTEMS

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Note that the parameter Q, de ned in Eq. (14.14) and included in Eq. (14.13), essentially renders Eq. (14.13) to be the sum of cyclic shifts modulo Mp x rather than ordinary cyclic shifts. The following is a parity-check matrix for EVENODD: ! 1 1 1 1 0 H : 14:16 1 a ap 1 0 1 For p 5, for example, a can be expressed by a 4 4 binary matrix de ned by M5 x x4 x3 x2 x 1: 3 2 0 0 0 1 61 0 0 17 7 6 40 1 0 15 0 0 1 1 Note that the parity symbols Cp a and Cp 1 a depend on the information symbols but not on each other. It is easy to see from the parity-check matrix shown in Eq. (14.16) that the minimum distance of EVENODD is 3, which gives alternative proof to the basic MDS property of the code. 2. X-Codes A new class of MDS array codes, called X-codes, has size n n, where n is a prime [XU99a]. The X-codes are of minimum column distance 3; that is, they can correct either two column erasures or one column error. The key novelty in X-code is its simple geometrical construction that achieves optimal complexity for the encoding or the update, whereby a change of any single information bit affects exactly two parity bits. The key idea exists in the code construction such that all parity symbols are placed in rows rather than columns. In X-codes, information symbols are placed in an array of size n 2 n. Parity symbols are constructed from the information symbols along several parity-check lines or diagonals of some slopes with addition operations. But the parity symbols are placed in two additional rows. So the coded array is of size n n, with the rst n 2 rows containing information symbols, and the last two rows containing parity symbols. Note that information symbols and parity symbols are mixed in each column of the array. Encoding Procedure Let ci; j be the symbol located at the i-th row and the j-th column. The parity symbols of X-codes are constructed according to the following calculations: 9 n 3 X > > > cn 2;i ck;ji k 2jn > > > = k 0 ; 14:17 > n 3 > X > > cn 1;i ck;ji k 2jn > > ;

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where i 0; 1; ; n 1, and jxjn x mod n. Geometrically the two parity rows are just the checksums along diagonals of slopes 1 and 1, respectively. From this construction it is easy to see that the two parity rows are obtained independently; that is, each information symbol affects exactly one parity symbol in each parity row. All parity symbols depend only on information symbols, and therefore

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