CODES FOR HIGH-SPEED MEMORIES IV: SPOTTY BYTE ERROR CONTROL CODES in .NET

Produce QR Code in .NET CODES FOR HIGH-SPEED MEMORIES IV: SPOTTY BYTE ERROR CONTROL CODES
CODES FOR HIGH-SPEED MEMORIES IV: SPOTTY BYTE ERROR CONTROL CODES
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7.4 SINGLE SPOTTY BYTE ERROR CORRECTING AND DOUBLE SPOTTY BYTE ERROR DETECTING (St=b EC-Dt=b ED) CODES Here we study the St=b EC-Dt=b ED codes that correct single t=b-errors and detect double t=b-errors. An St=b EC-Dt=b ED code is primarily inspired by the architecture of ReedSolomon SbEC-DbED code, denoted as RS SbEC-DbED code. Since the RS SbEC-DbED code has a strong error control function of single-byte error correction and double-byte error detection, this requires check-bit length equal to three times the length of a byte. In particular, in computer and communication systems that are prone only to a few transient bit errors in a byte, this code function becomes unnecessary. 7.4.1 Code Conditions and Bounds
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Theorem 7.8 Let Hi denote an r b binary submatrix for 0 i n 1. The null space of H H0 H1 H2 H3 Hn 1 is an St=b EC-Dt=b ED code, if and only if: 1. 2. 3. 4. E1 E2 Hi T 6 0 for E1 6 E2 , E1 Hi T 6 E2 Hj T for i 6 j; E1 Hi T 6 E2 E3 Hj T for i 6 j, E2 6 E3 ; E1 Hi T E2 Hj T 6 E3 Hk T for i 6 j 6 k 6 i, i; j; k n 1, 8E1 ; E2 ; E3 2 Et=b , Et=b E 2 GF 2b j1 w E t .
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Proof Conditions 1 and 2 con rm that all single t=b-error patterns within a b-bit byte generate unique nonzero syndromes. Hence these satisfy the conditions of an St=b EC code. On the other hand, condition 3 together with condition 4 con rm that the syndrome generated by a single t=b-error is different from that generated by a double t=b-error. This asserts that double t=b-errors are detectable, and hence the code that satis es these conQ.E.D. ditions is an St=b EC-Dt=b ED code. Theorem 7.9 A linear binary St=b EC-Dt=b ED code requires at least 3t check bits.
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Proof According to condition 3, at least 3t binary columns of H (t columns each corresponding to the three t=b-errors) are linearly independent. Therefore a linear binary Q.E.D. St=b EC-Dt=b ED code requires at least 3t check bits. Theorem 7.10 An N; N R St=b EC-Dt=b ED code exists only if   X  t   t N X b N b t 1 2 1! 2 1 : i j b i 1 b j 1
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The proof of this theorem will be given in a generalized form in Subsection 7.5.1. 7.4.2 Design for St=b EC-Dt=b ED Codes and St=b EC-Dt=b ED-SbED Codes
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Let H0 h00 h01 h0b 1 be a q b binary matrix (q b) whose at least min 3t; b columns are linearly independent. Here h00 ; h01 ; ; h0b 1 , are binary column vectors of
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DOUBLE SPOTTY BYTE ERROR DETECTING (St=b EC-Dt=b ED) CODES
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GF 2q . If min 3t; b b, H0 can be any b b nonsingular matrix, including the b b identity matrix. On the other hand, if min 3t; b 3t < b, we consider H0 to be a paritycheck matrix of a linear binary b; b q code with minimum distance at least 3t 1, (i.e., it is a parity-check matrix of a binary t-error correcting and 2t-error detecting code). Similarly let H00 h00 h00 h00 be an r b matrix with at least t columns that are 0 1 b 1 linearly independent. Here h00 ; h00 ; ; h00 , are binary column vectors of GF 2r . If 0 1 b 1 t b, the matrix H00 can be any b b nonsingular matrix, including the b b identity matrix. If t < b, we consider H00 to be a parity-check matrix of a linear binary b; b r code with the minimum distance being at least t 1, which is to say, it is a parity-check matrix of a binary t-error detecting code. We now use the H0 and H00 matrices de ned above in the following theorem to design the St=b EC-Dt=b ED code. Theorem 7.11 Let g be a primitive element of GF 2 R q =2 , where R ! b 2r. Let HR be an R b 2 R q =2 binary submatrix given by 2 H0 H0 g1 H00 g2 H00 H0 gi H00 H0 g2
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