MODIFIED HAMMING SEC-DED CODES in .NET

Development QR Code JIS X 0510 in .NET MODIFIED HAMMING SEC-DED CODES
MODIFIED HAMMING SEC-DED CODES
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the binary representation of i: 1 60 G 6 40 0 2 0 1 0 0 0 0 1 0 0 0 0 1 3 1 17 7: 15 1
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The maximum code length (in bits) of the code is n 5 2r 4 for r ! 5. Example 4.2 (40, 33) SEC-DED 2 00000 00000 00000 6 00000 00000 11111 6 6 00000 11111 00000 6 6 H 6 10001 10001 10001 6 6 01001 01001 01001 6 4 00101 00101 00101 00011 00011 00011 code [DAVY91] 00000 11111 11111 00000 11111 00000 10001 01001 00101 00011 10001 01001 00101 00011 11111 00000 11111 10001 01001 00101 00011 11111 11111 00000 10001 01001 00101 00011 3 11111 11111 7 7 11111 7 7 7 10001 7: 7 01001 7 7 00101 5 00011
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The shortening algorithm of the code can obtain the excellent code with any code parameters. Shortening Algorithm columns of Hr are deleted ! bg ; g15 The matrix Hr is shortened by j columns, j in the following order: ! ! ! ! ! ! ! bg bg bg bg bd bs bH ; ; ; ; ; ; ; g8 g4 g2 g1 g15 g8 g4 8, where the
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where gv is a column of matrix G corresponding to the binary representation of v, and columns bg , bd , bs , bH are distinct. Let r 7, j 1, and g 7. Then the parity-check matrix H of the (39, 32) code is the one with the last column omitted. Take r 8, j 8, g 15, d 14, s 13, and H 12. The parity-check matrix of the (72, 64) code has the following form:
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2 6 6 00000 6 6 00000 6 6 6 00000 6 6 10001 6 6 6 01001 6 6 4 00101 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11 11 111 1 1111 00000 00000 00000 11111 11111 11111 11111 00000 00000 00000 00000 11 11 111 1 1111 00000 11111 11111 00000 00000 11111 11111 00000 00000 11111 11111 00 00 000 0 1111 11111 00000 11111 00000 11111 00000 11111 00000 11111 00000 11111 00 00 111 1 0000 10001 10001 10001 10001 10001 10001 10001 10001 10001 10001 10001 10 01 100 1 1000 01001 01001 01001 01001 01001 01001 01001 01001 01001 01001 01001 01 01 010 1 0100 00101 00101 00101 00101 00101 00101 00101 00101 00101 00101 00101 00 01 001 1 0010 "
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Note : the shortened place i:e:; deleted columns from the original matrix is indicated by the upwardpointing arrow " :
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CODES FOR HIGH-SPEED MEMORIES I: BIT ERROR CONTROL CODES
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Although the (72, 64) SEC-DED code is not the minimum-weight & equal-weight-row code de ned in Section 3.1, it is the best code so far obtained on error detection capabilities of triple and quadruple errors, P3 55:37% and P4 99:35%, respectively. 4.1.3 Double-Bit Error Correction Using SEC-DED Codes
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High-density memory chips create new reliability problems. Good examples are the soft errors caused by a-particles, and neutrons induced by cosmic rays in high-density RAM chips [NOOR80, SAIH82, OGOR96]. These soft errors may line up with existing hard errors, giving rise to multiple errors that are not correctable with SEC-DED codes. To solve these problems, extended reliability techniques have been proposed for largecapacity memory systems with SEC-DED facilities [KANE84b, AICH84]. Some of them are the read-retry technique in which the soft errors disappear during the repeated read cycles, the sparing technique, which replaces a defective component with a spare without requiring manual intervention, and the mask error correction technique, which requires some additional operations for detection and correction of hard errors. The mask error correction by retry method is illustrated in the following example. This idea is based on the ADR (alternate data retry) mentioned in Subsection 1.3.2. Example 4.3 The following sequence of operations allows for mask error correction of hard-plus-soft errors with using the self-complementing SEC-DED code where self-complementing code has been de ned by De nition 3.6: 1 0 0 h # 1 0 1 0 0 1 1 1 a # 0 1 1 0 1 1 1 0 1 1 Correct data Faults h : hard error position; a : soft error position 1 Read from memory : Uncorrectable error 0 Complement; written 0 Read from memory 1 Recomplement : Correctable error 1 Correct data rewritten 1 Read on refetch : Correctable error
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From the foregoing Read-Invert-Write-Read-Invert procedure we can correct double errors by using only the SEC-DED code. The soft errors will be masked and disappear during the subsequent read operations. In recent one-transistor-type dynamic RAMs, however, the stored data are destroyed in every read operation, and therefore a rewrite operation is always performed. That is, the erroneous datum caused by hard-plus-soft errors is readout from the DRAMs and then this erroneous datum is rewritten, which means the errors are retained in memory even in the read operation. The read-retry operation cannot then recover the erroneous data caused by soft errors in recent DRAMs. Therefore the read-retry operation even in cooperation with the SEC-DED code cannot tolerate the above hard-plus-soft errors. Another effective method for correcting multiple errors is to apply an error location technique to distance-4 codes (i.e., erasure correction technique). With the error