Figure 7.12 (136, 120)S3=8 EC-D3=8 ED-S8ED code. in .NET

Development Denso QR Bar Code in .NET Figure 7.12 (136, 120)S3=8 EC-D3=8 ED-S8ED code.
Figure 7.12 (136, 120)S3=8 EC-D3=8 ED-S8ED code.
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Figure 7.13 (136, 121)S3=8 EC-D3=8 ED code.
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10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 10000001 00000000 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 01000001 00000000 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00100001 00000000 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00010001 00000000 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00001001 00000000 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000101 00000000 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000011 00000000 11111111 00010111 00101110 01001011 11101000 00111001 01100101 10100011 11010001 01011100 11000110 01110010 10001101 10011010 10110100 00000000 10000000 01001011 11101000 00111001 01100101 10100011 11010001 01011100 11000110 01110010 10001101 10011010 10110100 11111111 00010111 00101110 00000000 01000000 00101110 01001011 11101000 00111001 01100101 10100011 11010001 01011100 11000110 01110010 10001101 10011010 10110100 11111111 00010111 00000000 00100000 00010111 00101110 01001011 11101000 00111001 01100101 10100011 11010001 01011100 11000110 01110010 10001101 10011010 10110100 11111111 00000000 00010000 11111111 00101110 11101000 01100101 11010001 11000110 10001101 10110100 00010111 01001011 00111001 10100011 01011100 01110010 10011010 00000000 00001000 01001011 00111001 10100011 01011100 01110010 10011010 11111111 00101110 11101000 01100101 11010001 11000110 10001101 10110100 00010111 00000000 00000100 00101110 11101000 01100101 11010001 11000110 10001101 10110100 00010111 01001011 00111001 10100011 01011100 01110010 10011010 11111111 00000000 00000010 00010111 01001011 00111001 10100011 01011100 01110010 10011010 11111111 00101110 11101000 01100101 11010001 11000110 10001101 10110100 00000000 00000001
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DOUBLE SPOTTY BYTE ERROR DETECTING (St=b EC-Dt=b ED) CODES
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30 28 26 24 Check-bit length R 22 20 18 16 14 12 10 32
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t = 2,3 t=8 t = 5,6,7 t=4
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4,088
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St/8 EC-Dt/8 ED-S8ED code in Theorem 7.12
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2,040 1,016 504 248 505 2,840 249 1,414 703 350 173 1,017 4,089 2,041
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K = 120
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K = 121
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S3/8 EC-D3/8 ED code in Theorem 7.11
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K = 41
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Bound of S 3/8 EC-D 3/8 ED code shown in Theorem 7.10 64 128 256 512 1,024 2,048 4,096
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Information-bit length K
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Figure 7.14 Comparison of check-bit lengths and information-bit lengths of the St=8 EC-Dt=8 ED codes and the St=8 EC-Dt=8 ED-S8ED codes.
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The error detection capabilities of the S3=8 EC-D3=8 ED code are shown in Table 7.2 for three types of errors: random triple-bit errors, random quadruple-bit errors, and single 4bit byte plus single-bit errors. These errors are outside the error control capability of the S3=8 EC-D3=8 ED code. In the table byte plus bit errors means that single 4-bit byte errors and single-bit errors are occurred simultaneously. The error detection capabilities of the S3=8 EC-D3=8 ED-S8ED code are shown in Table 7.3.
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TABLE 7.2 Error Detection Capabilities of the S3=8 EC-D3=8 ED Code Error detection capability (%) Errors Triple-bit errors Quadruple-bit errors Byte plus bit errors K 64 R 15 97.57 98.19 94.25 K 128 R 17 98.34 99.07 94.03 K 256 R 19 98.76 99.53 94.07
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TABLE 7.3 Error Detection Capabilities of the S3=8 EC-D3=8 ED-S8ED Code Errordetection capability (%) Errors Triple-bit errors Quadruple-bit errors Byte plus bit errors K 64 R 16 97.47 96.45 88.95 K 128 R 18 98.22 97.96 90.14 K 256 R 20 98.73 98.65 90.36
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CODES FOR HIGH-SPEED MEMORIES IV: SPOTTY BYTE ERROR CONTROL CODES
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A GENERAL CLASS OF SPOTTY BYTE ERROR CONTROL CODES
In Section 7.1 we saw that when a small number of random bit errors collect in a byte, we have a situation called a spotty byte error. From a generalized and theoretical code design standpoint, the single spotty errors in a byte are called s-spotty byte errors, and the multiple spotty errors in a byte are called m-spotty byte errors. In this section the code design for generalized s-spotty and m-spotty byte error control codes over GF 2b is discussed [KASH04, SUZU04, 05a, 05b]. 7.5.1 A General Class of Codes for s-Spotty Byte Errors
The s-spotty byte error has been de ned as a set of t or fewer bits errors con ned to a b-bit byte. In this case the maximum number of erroneous bits in each byte does not exceed t < b . Below we present a general class of s-spotty byte error control codes [KASH04]. 1. s-Spotty Byte Error Control Codes Preliminaries De nition 7.1 An error is called an s-spotty byte error if a set of random t or fewer bits errors is con ned to a byte, meaning the maximum number of erroneous bits in a byte does not exceed t. & The necessary and suf cient conditions of the s-spotty byte error control codes are presented as follows. Theorem 7.13 Let Hi be an R b binary submatrix for 0 i n 1, and also let Et=b fE 2 GF 2b j 1 w E tg be a set of all t=b-error patterns in a b-bit byte where w E denotes the Hamming weight of b-bit vector E. The null space of H H0 H1 H2 H3 Hn 1 is a l t=b-errors correcting and m t=b-errors detecting code if and only if E1 E2 HT1 E2v 1 E2v HTv E2v 1 HTv 1 E2v w HTv w 6 0R i i i i for 2v w l m; 0 v l; 0 w l m;
where m ! l, 8E1 ; E2 ; . . . ; E2v ; E2v 1 ; . . . ; E2v w 2 Et=b , i1 ; i2 ; . . . ; iv ; iv 1 ; . . . ; iv w are distinct integers of i satisfying 0 i1 ; i2 ; ; iv ; iv 1 ; . . . ; iv w n 1, 0R is an R-bit zero vector, and T is a transpose of vector or matrix. Proof Let two sets having r l and s m s-spotty byte errors be Ei fEi1 ; Ei2 ; ; Eir g and Ej fEj1 ; Ej2 ; ; Ejs g, respectively. In each set, r s-spotty byte errors are assumed to have occurred in the different r bytes and s s-spotty byte errors in the different s bytes. For l t=b-errors correcting and m t=b-errors detecting code, the following relation should be satis ed: Ei1 HT1 Ei2 HT2 Eir HTr 6 Ej1 HT1 Ej2 HT2 Ejs HTs : i i i j j j