n0 l 4 2l 1 n1 l 2; which expresses the relation (10.8). Q.E.D. in .NET

Integrating QR Code JIS X 0510 in .NET n0 l 4 2l 1 n1 l 2; which expresses the relation (10.8). Q.E.D.
n0 l 4 2l 1 n1 l 2; which expresses the relation (10.8). Q.E.D.
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Design for the (Bl EC)n 0 -(SEC)n1 UEP Code We will use here a matrix P that converts the error pattern E with length n0 n1 , including l-bit burst errors in X0 and single-bit errors in X1 in the output word, into an error pattern E PT with length np , including uniform l-bit burst errors. That is, we will use a matrix P that is an np n0 n1 conversion matrix: matrix P converts l-bit burst errors in X0 into the same l-bit burst errors in the former part with length n0 of the output word, and also converts single-bit errors in X1 into l-bit burst errors in the latter part with length np n0 of the output word. These converted errors in the output word with length np can be corrected by applying an np ; np r l-bit burst error correcting code. In order to prevent l-bit burst errors in X1 from being mistaken as correctable errors in X0 , the matrix P converts the burst errors in X1 to the l-bit burst errors, that is, correctable errors, in the latter part of the output word. Theorem 10.15 Let HBEC be a parity-check matrix of an np ; np r l-bit burst error correcting code. Then the null space of H HBEC P is an (n0 n1 , n0 n1 r) (Bl EC)n0 -(SEC)n1 UEP code, where
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I n0
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P = O
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H HA H B H H A B .. .
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HA H H B A H B
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3 1 1 HA 4 Q 5 , 1 1 l q ! Q , HB 1 1 l 1 q Q : l 2 q matrix having distinct q binary columns where l 1 q 2l 2 , HG : l 1 q0 matrix having distinct q0 binary columns where the element hi; j in HG equals zero for i > j and l 1 q0 2l 1 1. Proof The syndrome caused by error E is expressed as E HT E HBEC P T E PT HT , BEC where E is an error vector with length n0 n1 and output of E PT is called a converted error of E. From the structure of the matrix P, any error that occurs in X0 is always converted to the original error E in E PT , because there exists an identity submatrix In0 in P. That is, the l-bit burst errors occurring in X0 give the converted errors including the input l-bit burst error e0 in the former part of E PT :
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l ! . E PT 0 0 e0 0 0 . 0 0 : . ! ! np n0 n0
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These converted errors can be corrected by the l-bit burst error correcting code expressed by HBEC. Hence the code satis es conditions 1 and 2 of Theorem 10.13 for the error set EB0 . When l-bit burst errors corrupt both X0 and X1 , that is
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then the former part of the error, e0 , with length w ( < l) bits is converted to the original error e0 and the remaining part of the error, e1 , with a length of l w bits in X1 is converted to an error ey with a length of at most l w bits because of the property that hi; j 0 for i > j in the matrix H . That is, the converted error is an l-bit burst error, shown as
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l .! . ey 0 0 : E P 0 0 e0 . ! ! np n0 n0 T
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Especially in this case, the former part of the converted errors can be properly corrected, but the remaining part is not guaranteed to be properly corrected in X1 . Hence the code satis es conditions 1, 3, and 4 of Theorem 10.13 for the error set EB . Nonzero column vectors in P are all distinct, and therefore every single-bit error E is always converted to different pattern of E PT . Single-bit errors in X0 are always converted to single-bit errors, and the errors in X1 are converted to l-bit burst errors because of the matrix structure of H , HA , and HB , which must have l or fewer nonzero rows. Because these converted errors can be properly corrected, the code satis es conditions 1 and 5 of Theorem 10.13 for the error set Eb . When l-bit burst errors occur in X1 , the latter part of E PT with length np n0 can include a zero pattern or a nonzero pattern of e having at most l 1 s: . l ! E PT 0 0 . 0 0 e 0 0 : . ! ! n0 np n0 It is apparent that the syndrome caused by l-bit burst errors in X1 is different from the syndrome due to errors in X0 , and the syndrome due to errors spanned over X0 and X1 . Hence the code satis es conditions 6 and 7 of Theorem 10.13. Consequently the H matrix indicated in the theorem satis es the conditions of Theorem 10.13, so the code is a (Bl EC)n0 -(SEC)n1 UEP code. Q.E.D. Example 10.5 [NAMB02]: (55, 44) (B4 EC)8 -(SEC)47 UEP Code
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The parity-check matrix of a (16, 5) 4-bit burst error correcting Fire code is shown below: 1 60 6 60 6 60 6 60 6 60 6 60 6 61 6 60 6 40 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 0 0 1 1 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 0 1 0 1 1 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 ! 2 3 0 17 7 07 7 07 7 07 7 0 7 r 11: 7 07 7 17 7 07 7 05 0
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