SINGLE SPOTTY BYTE ERROR CORRECTING St=b EC) CODES in .NET

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SINGLE SPOTTY BYTE ERROR CORRECTING St=b EC) CODES
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extended cyclic Hamming code with code length 16 bits, meaning a (16, 11) SEC-DED code, can be considered as H00 matrix. Then we have 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 H 0 1 2 3 4 5 6 7 8 9 10 11 12 13 0 b b b b b b b b b b b b b b b14
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g0 g18 g5 g29 g10 g11 g8 g25 g26 g27 g22 g23 g14 g15 g16 g17 ; where b is a primitive element of GF 24 corresponding to primitive polynomial p x x4 x 1 and g is a primitive element of GF 25 corresponding to primitive polynomial p x x5 x2 1. Clearly, H0 is a parity-check matrix of a binary 3-bit error correcting code and H00 is a parity-check matrix of a binary 3-bit error detecting code, as required by Theorem 7.2. The parity-check matrix of the resulting 517; 501 S3=16 EC code is given by the following matrix: H0 0 00 gH Here gi H00 gi gi 18 gi 5 gi 29 gi 10 gi 11 gi 8 gi 25 gi 26 gi 27 gi 22 gi 23 gi 14 gi 15 gi 16 gi 17 for 0 i 14. Notice that in this case the homomorphism f : GF 25 ! GF 25 is simply given by f x x for any x 2 GF 25 . A shortened S3=16 EC code with 256 information bits (i.e., an 272; 256 S3=16 EC code) can be obtained by simply choosing the rst 272 binary columns from the original 517; 501 S3=16 EC code. Furthermore this shortened code can be made systematic by row operations. Figure 7.1 shows the binary parity-check matrix of this shortened 272; 256 S3=16 EC code in systematic form. This code requires only one RAM chip (with 16-bit I/O data) for check bits. Our second example is an S3=8 EC code that corrects up to three random bit errors corrupting a single 8-bit byte. This code is suitable for memory systems that use chips with 8-bit I/O data. Figure 7.2 shows an example of an S3=8 EC code, that is, the binary paritycheck matrix of a (132, 121) S3=8 EC code. In this case we use the parity-check matrix of an 8; 1 3-error correcting code as the H0 matrix, and that of an 8; 4 3-error detecting code as the H00 matrix. These H0 and H00 matrices are given below in binary form: 3 10000001 6 01000001 7 7 6 6 00100001 7 7 6 H0 6 00010001 7; 7 6 6 00001001 7 7 6 4 00000101 5 00000011 2 3 11111111 6 01001011 7 7 H00 6 4 00101110 5: 00010111 2 H0 1 00 gH H0 2 00 g H H0 30 00 g H H0 O ! O : I5
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0110010100001110 1110000111011001 0111011110001001 0000001101011110 0010011100010011 1101110101000011 0110000001101011 1111011101010000 0101111010100110 1101000010101100 1001110110110100 1010100001100110 0010011010110100 0110101111000000 1110110010100011 1001010000111010 1100110101111000 1100111011001010 0100101111000100 1011100001110110 0100110011010111 1010010111011001 0001000101011110 1110010000001110 1111111101111111 0011010110100010 0110001001001011 0001101011100001 1101000011100100 1000011101100001 1101100101000001 1111011001010001 0001110110010100 1001000011101100 0011001101011110 0100111000100110 1110110100001110 0010001010111100 1000010101100101 1100001010110010 0011110011001101 0000100111101111 0110101111100100 1111111111101111 1100101000010101 1010100001110010 1011001010000101 0100010011010110 0100101000011101 0101101011110001 1011010000111011 0101010011010111 0010111010111100 1010110001110110 0010110100010011 1000111111001010 0100110111111000 0111000101110101 0111100010111010 1100001110100010 1111111111110111 1111011001010100 0001001101011100 1010000111011000 1001001101011110 0011100010011010 0011101011110001 0110000100110101 0101111110001001 0011000111100010 1111110111111111 0110001101101011 1101100111000011 1111011000010000 0101111000000110 1011101100111000 1001010000110011 0011110001001001 1101000011101110 1001110110010101 1101011110001001 1000101000011101 1010011101100101 0000101011110001 1100101110110010 1000101011001010 1010101001110110 0100110001111000 0111000110110101 0111100011011010 0001001101111110 0011110001011101 0011010111101010 0101111000100010 1010000111011011 1110010100001111 0001010000111011 1010110010100001 1100010100001110 1000110110010100 0011110111100010 0011100001001101 1100000110110010 0000100010101111 0110101101000100 1011001011000111 0100010011110111 0110001001111011 1011101100100000 0100110101111100 0001101011110011 1100101000011100 0101100101000011 0010001001101011 1010111011001010 0010010111100010 0111100100110101 1001100110010100 1101001011101100 1110110110100001 1001010010111011 1000011100100101 0010011010011100 0111100010001010 0110101111001100 1100101000011001 1100001110110000 0011110001001100 1011010111100010 1011111111111111 0000111100010011 1111010100001110 1000011011001010 0101101000100110 1111010001010000 1101000111101100 1001110100010100 1110110011100001 1001010000011011 0100110101101000 0011110001000101 0100010011010011 1000011101100111 0010011010111101 1001101011110001 0000010011010111 1100110010100001 1010001010000111 1000111101100101 0010101100010011 1011100100101000 1010100101110110 1000111001001010 0100110100111000 0111000100010101 0010011010101100 1110010100000110 1101100101000111 0111100010011000 0001001101011111 1100010011010111 1111001010000111 1011110110010100 0111101111000100 1001110000111011 1011111100101000 0101010110001001 0010111000010011 1010100011110110 1000111010001010 0100110101011000 1000011101110101 1010000111010001 0110001001101111 0111000100110111 0111100010011011
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1111000100110101 0000110101111000 1001001010000111 1001111011001010 0001001011110001 1110000100001110 1010001111011001 1111111011111111 0011010101100010 0110001000101011 1101100101100011 1100101000001101 1001110110011100 0010011010111000 1111011001010010 0101111000100111
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0010100001110110 0110111100010011 0001110001001101 1010101100101000 0110001111000100 0111110010011010 0111001100110101 0001001001011110 1010000101011001 1110010101001110 1111111111011111 1011001010010111 1111011001011000 1110110010100101 0011010111100000 0110001001101010
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0010000111011001 0101001101011110 1110101000011101 0110100010011010 1011001100101000 0110011001101011 0011011111100010 1101100001000011 1111011011010000 0101111001100110 1101000011001100 0010111100000011 1000011101101101 0000100110101011 1001110110010110 1110110010100000
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