B = 4(2 b -4+1 1) b -3 (b > 4, p = 2 1) b=4 in .NET

Generation qr-codes in .NET B = 4(2 b -4+1 1) b -3 (b > 4, p = 2 1) b=4
B = 4(2 b -4+1 1) b -3 (b > 4, p = 2 1) b=4
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Figure 9.3 Check-bit lengths compared with information-bit lengths of the SEC-S4=p 4 EL type I codes.
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Source : [FUJI94]. 1994 IEEE.
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Preliminaries De nition 9.2 Let an odd-weight column square matrix be a nonsingular b b matrix whose columns are odd weight. Let an even-weight column square matrix be a b b matrix whose columns are b copies of an even-weight vector (including the zero vector). & Because there are 2b 1 even-weight column vectors having dimension b, there exist even-weight column square matrices. 2 Here, we show an example design method of nonsingular odd-weight column b b square matrices.
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De nition 9.3 The matrix Mb shown below is de ned as the matrix having b rows and 2b 1 odd-weight columns: Mb a0 a1 a2 a2b 1 1 2 3 p0 p1 p2 p2b 1 1 6 7 6 j j j j 7 6 7 : 6 b 1 7 4 0 a0 a a2 2 5 j j j j
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j In Mb , a is a root of the b 1 -th degree binary primitive polynomial g x , ai is a j coef cient vector of xi mod g x , and pi 2 f0; 1g is a bit determined to make the column vector ai, i 0; 1; ; 2b 1 1, be odd weight. & Lemma 9.1 The following shows a nonsingular odd-weight column square matrix generated from any consecutive b column vectors in the matrix M b : Ai ai0 where ij  i j mod 2b 1 , and 0 j ai1 aib 1 b b ;
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From this lemma we obtain 2b 1 nonsingular odd-weight column square matrices. Design of the Parity-Check Matrices Let A and B be the sets of the odd-weight column square matrices and the even-weight column square matrices, respectively. The following lemma provides the basic idea of the code design method. Lemma 9.2 Consider two different vectors each having degree r (i.e., Q and Q0 ) and each being constructed of elements from the even-weight column square matrices and the odd-weight column square matrices (i.e., Qj ; Q0j 2 A [ B for j 0; 1; ; r 1). In each vector there exists at least one matrix included in A. 2 3 2 0 3 Q0 Q0 6 Q1 7 6 Q01 7 6 7 6 7 Q 6 . 7; Q0 6 . 7: 4 . 5 4 . 5 . . Qr 1 Q0r 1
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Assume that the i-th elements in Q and Q0 (i.e., Qi and Q0i ), respectively, are different square matrices. In other words, Qi is an odd-weight column square matrix and Q0i is an even-weight column square matrix, and vice versa. Then any summation of binary column vectors in Q produces a different result than that given by any summation of binary column vectors in Q0 . Proof Let V and V0 , each having r b-tuples, be the results of the summation of column vectors in Q and Q0 , respectively: 2 6 6 V 6 4 v0 v1 . . . vr 1 3 7 7 7; 5 2 6 6 V0 6 4 v00 v01 . . . v0r 1 3 7 7 7: 5
Without loss of generality, Qi and Q0i can be regarded as the odd-weight column square matrix and the even-weight column square matrix, respectively. Assume vi v0i . Then V is the vector resulting from the summation of an even number of columns in Q, and V0 is that resulting from the summation of odd number of columns in Q0 . There exists an odd-weight column matrix in Q0 , say Q0l, in the l-th row for l 6 i. So v0l is an odd-weight b-tuple and vl has even weight. Therefore, V 6 V0 . Q.E.D. Below we provide an example expressed as a submatrix Hi corresponding to the i-th block. In other words, if we use the matrix from B in the rst row, then we write in this place, and so on: 2 6 Hi 6 4 3 7 7: 5
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In order to distinguish single-bit errors from single-byte errors, excluding single-bit errors, every submatrix has at least one b b identity matrix, which is included in A. In the submatrix Hi , for example, this can be expressed as follows:
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where Ai 2 A; Bi 2 B, 0