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as;l i 2 GF 2 ; s; l b 1; 0 i
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Equation (3.1) leads to the following relation for 0
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: mod-2 sum:
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CODE DESIGN TECHNIQUES FOR MATRIX CODES
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This shows that the modulo-2 sum of the corresponding diagonal elements in the binary square matrices hi; j s is equal to 1 of the corresponding element in the identity matrix I. Thus the binary represented set al; l 0 ; al; l 1 ; . . . ; al; l r 1 has an odd number of l s, meaning it is odd weight. On the other hand, the following relation holds for 0 m 6 l b 1:
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This shows that the modulo-2 sum of the corresponding nondiagonal elements in the binary square matrices hi; j s is equal to 0 of the corresponding element in the identity matrix I. Thus the binary represented set am;l 0 ; am;l 1 ; . . . ; am;l r 1 , where m 6 l, has an even number of l s, meaning it is even weight. The foregoing equations result in the relation
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This shows that the l-th column vector of the binary H matrix is of odd weight. These relations hold for any integer l and j. Hence every column vector of the binary H matrix that satis es Eq. (3.1) is of odd weight. Q.E.D. The odd-weight-column code gives a good discrimination of even number and odd number of errors. Therefore it has better multiple error detection capability than the non odd-weight-column code, as will be shown in later chapters.
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EVEN-WEIGHT-ROW CODES Let the binary H matrix of code C be expressed by r row vectors as 2 6 6 H 6 4 P0 P1 . . . Pr 1 3 7 7 7: 5
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De nition 3.5
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If every nonzero row vector Pi hi;0 . . . hi;n 1 of H satis es Eq. (3.2), then C is called an even-weight-row code:
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for rows i 0; 1; . . . ; r 1;
3:2
Where hi; j : j-th element in the i-th row vector 2 GF 2b , 0: zero element in GF 2b , P : summation in GF 2b .
EVEN-WEIGHT-ROW CODES
It can be easily proved that the code satisfying Eq. (3.2) has all even-weight rows in the binary form of H. This code has an important characteristic given below. De nition 3.6 For every binary codeword W of code C, if its bitwise complement W is also in C, then C is called a self-complementing code. & Theorem 3.7 Proof Even-weight-row code C is a self-complementing code.
Let the codeword W of code C be expressed as W D j P d0 d1 . . . dk 1 p0 p1 . . . pr 1 ;
where D d0 d1 . . . dk 1 is an information part and P p0 p1 . . . pr 1 is a check part. In addition let the H matrix be expressed as H H0 ; I , where H0 is an r k binary matrix for information part of H, and I is an r r binary identity matrix for the check part of H. Clearly, H0 has r odd-weight-row vectors, h00 ; h01 ; . . . ; h0r 1 . Therefore each check bit can be obtained by the following relation: pi d0 d1 . . . dk 1 h0i T ; 0 i r 1:
Let the bitwise complement of W be W d0 d1 . . . dk 1 c0 . . . cr 1 D j P . For D d0 d1 . . . dk 1 , the following relation always holds because every row vector h0i is of odd weight: d0 d1 . . . dk 1 h0i pi ;