SINGLE-BYTE ERROR CORRECTING AND SINGLE-BYTE PLUS SINGLE-BIT ERROR DETECTING CODES in .NET

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SINGLE-BYTE ERROR CORRECTING AND SINGLE-BYTE PLUS SINGLE-BIT ERROR DETECTING CODES
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fact ai GF 2l \ ak GF 2l ; (i 6 k). Letting x 0, v 0 or u 0 leads to a contradiction in the same way. Clearly, aj i must have a minimal polynomial of degree l 1. However, 2l elements 0; 1; an ; a2n ; . . . ; a 2 2 n , and no others have such minimal 3l polynomials in GF 2 ), which contradicts the fact 0 < ji jj < n. Therefore, the code has the detecting capability. Finally, we conclude that the code speci ed in the theorem is an SbEC-(Sb S)ED code. Q.E.D. In the theorem we have assumed that Hi has rank b 2l for i 1; 2; ; n; that is, any b 2l columns of Hi are linearly independent. It is apparent that we can make l each Hi have rank b in any case, since ai GF 2l f0; ai ; ai b; ai b2 ; . . . ; ai b2 2 g is a linear space over GF 2 of dimension l for i 1; 2; . . . ; n. For instance, provided b an , " Hi ai 1 b a i b ai ai ai ai ai b ai b2 ai bl 1 #
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are such matrices. Example 6.17 [HAMA97]
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The theorem yields the following parity-check matrix of an (84, 72) S4EC-(S4 S)ED code when l 2: H H1 H2 Hn ; where n 21, a is a root of a primitive polynomial g x x6 x 1 over GF 2 , and Hi ai a i n ai n a i ai 2n a i ai a i 2n ! i 1; 2; . . . ; n :
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A binary expression for this matrix can be found in Figure 6.28 (a). Deleting 2 bytes of the code yields a practical (76, 64) S4EC-(S4 S)ED code. In this case the 13-th and the 17-th bytes are deleted. Then the systematic form of this code determined by row operations is presented in Figure 6.28 (b).
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Figure 6.29 shows the check-bit lengths of the most ef cient SbEC-(Sb S)ED codes, where b 4 bits, which can be obtained with the design methods given in Subsection 6.4.2, as well as the tightest of those bounds on check-bit length mentioned in Subsection 6.4.1 and the check-bit lengths of the most ef cient SbECDbED codes known [CHEN92]. For any byte length b ! 2, design method I provides
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0000 0110 0000 0000 1011 0110 0111 1001 1110 0111 0111 1110 0110 0110 0110 0000 0000 1011 1110 1110 0111 0111 1110 0111 1011 1101 0110 0110 0000 0000 0000 0111 0111 1110 0111 1110 0000 1011 1101 0110 0110 0000 0111 0111 1110 0111 1110 0000 0000 0000 1011 1101 0110 0110 0000 1110 0111 1110 0000 0111 0110 0110 0000 1011 1101 0110 1110 0111 1110 0000 0111 0000 0110 0000 0110 0000 1011 1101 1001 1110 0000 0111 0000 1110 1101 1011 0000 0110 0000 1011 0111 0000 0111 0000 1110 1001 1011 0110 1011 0000 0110 0000 0111 0111 0000 1110 1001 0111 0000 1011 0110 1011 0000 0110 0000 0000 1110 1001 0111 0111 0110 0110 1011 0110 1011 0000 0000 1110 1001 0111 0111 0000 0000 0110 0110 1011 0110 1011 1110 1001 0111 0111 0000 0000 1011 1011 0110 0110 1011 0110 0111 0111 0111 0000 0000 1110 0110 1101 1011 0110 0110 1011 0000 0111 0000 0000 1110 0111 1011 1101 1101 1011 0110 0110 0111 0000 0000 1110 0111 0000 0110 1101 1101 1101 1011 0110 0111 0000 1110 0111 0000 0111 0110 0000 1101 1101 1101 1011 0111 1110 0111 0000 0111 0111
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