Figure 7.18 Shortened (285, 256) m-spotty D3=8 EC code. in .NET

Creator qr barcode in .NET Figure 7.18 Shortened (285, 256) m-spotty D3=8 EC code.
Figure 7.18 Shortened (285, 256) m-spotty D3=8 EC code.
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CODES FOR HIGH-SPEED MEMORIES IV: SPOTTY BYTE ERROR CONTROL CODES
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Decoding The m-spotty byte error control code given by Theorem 7.24 is decoded here. Let v, c, and E be the received word, codeword, and error vector, respectively. Then the syndrome S is calculated as follows: S S0 S1 S2 . . . Sd 2 v HT c E HT E HT ; where S0 2 GF 2q is an q-bit binary row vector and Si 2 GF 2r is an r-bit binary row vector for i 1; 2; . . . ; d 2. If p d d 1 =2e m-spotty byte errors, E0 ; E1 ; . . . ; Ep 1 2 Et=b have occurred in the i0 -th, i1 -th, . . ., ip 1 -th byte, respectively, then the syndrome S is given by Eq. (7.24): 3T S0 7 6 6 S1 7 7 6 7 6 S 6 S2 7 6 . 7 6 . 7 4 . 5 Sd 2 2 2 6 6 6 6 6 6 6 6 4
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E0 H0 T E1 H0 T Ep 1 H0 T
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gi0 E0 H00 T gi1 E1 H00 T gip 1 Ep 1 H00 T g2i0 E0 H00 T g2i1 E1 H00 T g2ip 1 Ep 1 H00 T . . . T d 2 i0 00 d 2 i1 g E0 H g E1 H00 T g d 2 ip 1 Ep 1 H00 T
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P T We let p 1 Ex be expressed as E . The relation S0 E H0 leads to E because H0 is a x 0 parity-check matrix of b d 1 =2c t=b-error correcting and d d 1 =2e t=b-error T T detecting code. Multiplying this by H00 from the right gives E H00 . We also let 00 T 00 T 00 T r E0 H , E1 H , . . ., Ep 1 H 2 GF 2 be e0 , e1 , . . ., ep 1 , respectively, and then the syndrome S0 is given by Eq. (7.25): 2 6 6 6 S0 6 6 4 S00 S1 S2 . . . Sd 2 3T 2 e0 e1 ep 1 gi0 e0 gi1 e1 gip 1 ep 1 g2i0 e0 g2i1 e1 g2ip 1 ep 1 . . . g d 2 i0 e0 g d 2 i1 e1 g d 2 ip 1 ep 1 3T 7 7 7 7 7 5 7:25
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This syndrome is identical to that of the RS code with a minimum Hamming distance d over GF 2r . The error patterns of GF 2r and error locations are determined next by using a decoding algorithm of the RS code such as the Berlekamp-Massey algorithm mentioned in Subsections 2.3.5 and 2.3.6. b In the nal step of the decoding, the error patterns Ex 2 GF 2b , where x 0; 1; . . . ; p 1, are transformed from the corresponding error patterns ex 2 GF 2r b b to Ex by one-to-one mapping for x 0; 1; . . . ; p 1. In this case at most one of Ex s may
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A GENERAL CLASS OF SPOTTY BYTE ERROR CONTROL CODES
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40 m-Spotty Dt / 8 EC Code 35 D8EC Code 4,070 2,024 1,002
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t = 4~8
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4,006 2,825 1,993 t=2 K = 492 1,403 3,618 987 2,553 25 1,798 693 485 1,268 336 891 233 625 162 438 20 K = 112 306 t=3 75 212 147 33 51 m-Spotty D t / 8 EC code bound t = 2 K = 69 101 15 46 10 32
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Figure 7.19 Comparison of information-bit lengths and check-bit lengths of the m-spotty Dt=8 EC codes.
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