CODES FOR DISTRIBUTED STORAGE SYSTEMS in .NET

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The following theorem gives an extension code to the one in Theorem 14.1. Theorem 14.9 If the incidence matrix P P0 P1 satis es the following conditions, the matrix H P j I is the parity-check matrix of the triple erasure correcting codes that satis es the direct decoding: 1. Every weight-3 column in P is distinct. 2. Every pair of two weight-3 columns, namely hi and hj , in each of P0 and P1 has 1 s at most in one same row position, that is, w hi ^ hj 1, where x ^ y means logical AND of binary vectors of x and y. This theorem can be easily proved such that any three weight-3 columns in P satisfy the relation (14.22). The following shows how to design P0 and P1 based on the BIBD codes and the additive-3 codes. Extended BIBD Codes Theorem 14.10 The following matrices P0 and P1 show the incidence matrices in BIBD codes that satisfy Theorem 14.9: 2 6 P0 4 I 0 2 Ps 0 6 0 P1 4 I Ps 0 0 Ps 0 I 0 I Ps 0 I 0 Ps 0 I Ps 0 0 Psq 1 I 0 Psq 1 0 I 0 Psq 1 I 0 I Psq 1 I 0 Psq 1 I 3 I 7; 5 I I 3
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Here Psi , 0 i q 1, is an s s binary matrix, where s 2q 1, shown in Theorem 14.3, in which the rst column vector of s-th degree has two 1 s at the (i 1)-th and the (2q i)-th positions, and the remaining s 1 distinct vectors are generated by the rst column vector cyclic shifted in downward by s 1 times. The information length of the code de ned by H P0 P1 j I is k r r 2 =3, where r  3 (mod 6).
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An example of Ps0 with q 1, and s 3 is shown below: 0 Ps0 4 1 1 2 3 1 1 0 1 5: 1 0
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From this example we have the incidence matrix P0 with r k0 , where r 9, and k0 12, the same as considered in Example 14.6. The maximum numbers of column vectors in P0 and P1 are shown as k0 r r 1 =6 and k1 r r 3 =6, where r  3 (mod 6), respectively. Therefore the information length . of the code de ned by H P j I P0 . P1 j I can be expressed by . k k0 k1 r r 2 ; 3
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where r  3 (mod 6). Since Psi has all weight-2 columns, the matrix P satis es two conditions of Theorem 14.9. Therefore H P0 P1 j I is a parity-check matrix of the triple erasure correcting codes that satis es the direct decoding. Example 14.9 [OHDE05b]
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The following shows the parity-check matrix of the (30, 21) codes that satis es Theorem 14.9 with parameters of q 1, and r 9:
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011 000 100 100 011 000 100 100000000 101 000 010 010 101 000 010 010000000 110 000 001 001 110 000 001 001000000
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Extended Additive-3 Codes The extended additive-3 code with larger code length than that of the existing additive code, called an additive-3a code, is de ned by the following lemmas and theorems. Lemma 14.3 The number of solutions of an integer x that satis es 3x  a (mod r), where 0 x r 1, r is an integer (! 3), and a is also an integer (0 a r 1), is denoted as ba and expressed as follows: 8 <3 ba 1 : 0 for a multiple of 3; and r multiple of 3; for a any integer; and; r integer other than multiple of 3; for a integer other than multiple of 3; and r multiple of 3:
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The reader is encouraged to prove this. This lemma gives a generalized form for a, that is, Lemma 14.1 is a special case of a 1 of this lemma.
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