MDS ARRAY CODES TOLERATING MULTIPLE-DISK FAILURES in .NET

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will be explained in the following paragraphs. In the decoding of correcting multiple erasures and single errors, the only multiplications required during encoding and decoding involve ring elements P the form xi , i 0; 1; 2; , as one of the operands. Because of of the fact that x 1 ti 0 xi xt 1 1 x p 1, the multiplication can be performed by cyclic shifts of binary vectors of length p t 1, and the implementation of any arithmetic ring operation requires only XOR operations. The following shows the p 1 n array codeword of linear array code Cp 1 C p 1; n; r C0 C1 2 c0;0 6 c1;0 6 6 . 6 . 4 . cp 2;0 C2 c0;1 c1;1 . . . cp 2;1 Cn 1 c0;2 c1;2 . . . cp 2;2 c0;n 1 c1;n 1 . . . 3 7 7 7; 7 5 14:1
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where Ci , i 0; 1; 2; ; n 1, is an i-th symbol expressed by p 1 -th degree column vector, p is a prime, and n p. For an integer a, let jajp stand for the integer b 2 f0; 1; 2; ; p 1g such that b  a mod p . The linear array code Cp 1 over F GF q , where GCD p; q 1, is de ned as a subspace of all arrays of the C p 1; n; r above that satisfy the following pr linear constraints: n 1 X cjm jzjp ; j 0; 14:2
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P where means summation modulo 2, 0 m p 2, 0 z r 1, and this has an extra all-zero row cp 1;0 cp 1;1 . . . cp 1;n 1 . Figure 14.1 shows C p 1; n; r illustrated for the case where n p 5 and r 3. By this gure it is easy to verify that the array in Figure 14.2 is a codeword of array code C 4; 5; 3 over GF 2 . The r n parity-check matrix H of this code over Rp q , which is the ring of polynomials of degree less than p 1 over F GF q with multiplication taken modulo Mp x , where Mp x xp 1 = x 1 x p 1 x p 2 x 1 is de ned by 2 3 7 7 7; 7 5 14:4 14:3
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1 1 61 a 6 H 6. . 6. . 4. . r 1 1 a where a is a root of Mp x , and r Rp q is de ned as n
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p. And also the linear code C of length n over 14:5
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(a) Horizontal line (z = 0) 0 0 1 2 3 (b) Slope 1 (z = 1) 0 0 1 2 3 (c) Slope 2 (z = 2) : m = 0, : m = 1, : m = 2, : m = 3, : not defined 1 2 3 4 1 2 3 4
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Figure 14.1 Elements of the array code C(4, 5, 3) satisfying Eq. (14.2). Source: [BLAU93]. 1993 IEEE.
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Since every r columns in H are linearly independent over Rp q , C is a linear code of length n and minimum distance r 1, and it is MDS. The lengthened code with additional 2 columns is expressed as 3 2 1 0 6 0 07 7 6 6 . . 7: . .7 14:6 H0 6 H . .7 6 5 4 0 0 0 1
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Figure 14.2 Example of a codeword of the array code C(4, 5, 3) over GF(2). Source: [BLAU93]. 1993 IEEE.