CODES FOR MASS MEMORIES in .NET

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CODES FOR MASS MEMORIES
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Figure 11.22 Decoding method.
OPTICAL DISK MEMORY CODES
erasure correction method to the distance-3 single-symbol error correcting codes, which allows correction of up to double-symbol errors. Therefore, at the rst stage, when double-symbol errors are detected in C1 decoding (i.e., where code C1 acts as a double-symbol error detecting code), error pointers are set to all symbols included in the positive slopes. At the next stage, C2 decoding can correct these double-symbol errors indicated by the error pointers. This is because the codes C1 and C2 are distance3 codes, and therefore can correct up to double-symbol erasures. Codes for Compact Disc (CD) Digital Audio Systems The following two distance-5 RS codes over GF 28 are applied to the compact disc (CD) digital audio system [VRIE80]. The codes C1 and C2 are as follows: C1 : (32,28) double-symbol error correcting RS code. C2 : (28,24) double-symbol error correcting RS code. In this coding scheme many decoding strategies can be considered. Among these, this system adopts the following decoding method called super strategy: C1 Decoding 1. All-zero syndrome ! error free. 2. Single-symbol errors ! single-symbol error correction. 3. Double-symbol errors ! double-symbol error correction. Error location pointers are given to the C2 decoding. 4. More than triple-symbol errors ! only detection. Error location pointers are given to the C2 decoding. C2 Decoding 1. All-zero syndrome ! error free. 2. Single-symbol errors ! single-symbol error correction. 3. Double-symbol errors ! (a) Number of error pointers from the C1 decoding is less than or equal to four, and at the same time the error locations of the two error pointers are equal to those calculated from the syndrome ! double-symbol error correction. (b) Others ! only error detection. 4. More than triple-symbol errors ! only error detection. Note that when double-symbol errors are corrected in the C1 decoding, these two error location pointers are given to the C2 decoding in order to avoid miscorrection of triplesymbol errors as double-symbol errors in the C1 decoding. In case where the C2 decoder cannot correct, it lets pass through 24 data symbols uncorrected but marked only with error pointers originally given by the C1 decoder. This
CODES FOR MASS MEMORIES
way, even if the C2 decoder cannot decode, most of the symbols are nevertheless probably error free, and the uncorrected marked sample values can be reconstructed via linear interpolation [VRIE80] in digital audio systems. For the marked sample values, this can estimate and interpolate their correct values from both sides of values that have not been marked with error pointers. The CIRC scheme having super strategy is said to have a maximum fully correctable burst length up to 4,000 bits. Codes for Digital Data Storage (CD-ROM) For digital data storage systems, called compact disc ROM (CD-ROM), linear interpolation cannot be applied, and therefore another method for increasing data quality should be added. One system applies both doubly encoded RS SbEC codes and cyclic redundancy check (CRC) code to the data already encoded by CIRC. That is, the CD-ROM employs such a powerful error coding scheme that the original data are encoded by CIRC, and then the resulting data are further encoded by two RS SbEC codes. Thus the data are quadruply encoded as follows. (If CRC is included, they are quintuply encoded.) One disc has about 540-megabyte data capacity. 1. CRC code. This code is appended in order to check the miscorrection of the RS codes, or to detect uncorrectable errors. The generator polynomial of the CRC code is as follows: g x x16 x15 x2 1 x16 x2 x 1 : 2. Doubly encoded RS codes. The codes are determined by the following primitive polynomial g0 x , and a is a primitive element in GF 28 (i.e., a root of g0 x ): g0 x x8 x4 x3 x2 1: The code includes two RS codes (i.e., (26, 24) RS code C1 , and (45, 43) RS code C2 ). Both codes are distance-3 RS codes. Two check bytes of the code C1 are generated from the data included in the vertical direction. Also two check bytes of the code C2 are generated from the user data included in the negative slope that has 24 43 bytes ( 1; 032 bytes) in a two-dimensional data format. This can be seen in Figure 11.23. The H matrices of the code C1 and the code C2 are expressed as follows: H1 1 1 a24 1 a43 1 a23 1 a42 ... 1 ... a ... 1 ... a ! ; 1 ! 1 : 1 1