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E *= 0
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Figure 8.1 Error E in an error vector E and the corresponding submatrix Hi in a parity-check matrix H.
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Source: [FUJI02]. 2002 IEICE Japan.
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PARALLEL DECODING BURST ERROR CONTROL CODES
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is the inverse matrix of Ai , we have " A 1 i Ai " Hy i By i # Hi Bi # I:
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Subsequently the following holds: Hy Hi IL L ; i By Hi O R L L ; i Hy Bi OL R L ; i By Bi I R L R L : i Therefore the following two equations hold for the syndrome given by Eq. 8:1 : S Hy E Hi T Hy E Hy Hi T E I E; i i i S By E Hi T By E By Hi T 0: i i i Example 8.1 [FUJI02]
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Consider the error pattern generation of the (22, 13) 3-bit burst error correcting Fire code shown below. Note that the 9 22 parity-check matrix H includes a 9 7 submatrix H5 representing binary columns starting from i 5. The error vector E represents a 3-bit burst error starting from the 9-th bit of the word.
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Error E Error vector E * = 0000000001010000000000 ,
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Parity-check matrix H =
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1 0 0 0 0 0 1 0 0
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0 1 0 0 0 0 0 1 0
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0 0 0 1 0 0 1 1 0
0 0 0 0 1 0 0 1 1
0 0 0 0 0 1 1 1 1
1 0 0 0 0 0 1 0 1
0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1
0 0 0 0 1 0 1 1 0
0 0 0 0 0 1 0 1 1
1 0 0 0 0 0 1 1 1
0 1 0 0 0 0 1 0 1
0 0 1 0 0 0 1 0 0
0 0 0 1 0 0 0 1 0
0 0 0 0 1 0 0 0 1
0 0 0 0 0 1 1 1 0
1 0 0 0 0 0 0 1 1
0 1 0 0 0 0 1 1 1
0 0 1 0 0 0 1 0 1
0 0 0 1 0 0 1 0 0
R = 9.
The syndrome is therefore written as follows: S E HT 000101010 : 5
Submatrix H5
PARALLEL DECODING BURST / BYTE ERROR CONTROL CODES
An example of B5 is given below. Matrix B5 can be appended to H5 to obtain a 9 9 nonsingular matrix A5 .
0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 , 0 0 1
B5 =
R L=2
Nonsingular matrix A 5 = H 5 B5
0 0 0 0 0 1 1 1 1
1 0 0 0 0 0 1 0 1
0 1 0 0 0 0 1 0 0
0 0 1 0 0 0 0 1 0
0 0 0 1 0 0 0 0 1
0 0 0 0 1 0 1 1 0
0 0 0 0 0 1 0 1 1
0 0 0 0 0 0 0 0 1
0 0 0 0 0 1 0 0 1
R = 9.
are then obtained as follows:
H5
Inverse matrix A 1 5
H5 B5
1 1 0 0 0 0 1 1 0
1 0 1 0 0 0 1 0 0
0 0 0 1 0 0 1 0 1
0 0 0 0 1 0 0 1 0
1 0 0 0 0 1 0 0 1
0 0 0 0 0 0 0 1 1
1 0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0 1
0 0 0 0 0 0 0 1 0