2 Primitive Polynomials for BCH Codes
n k t G(x) octal
7 4 1 13
15 11 1 23
15 7 2 721
15 5 3 2467
31 26 1 45
31 21 2 3551
31 16 3 107657
31 11 5 5423325
31 6 7 313365047
63 57 1 103
63 51 2 12471
63 45 3 1701317
63 39 4 166623567
127 120 1 211
127 113 2 41567
127 106 3 11554743
255 247 1 435
255 239 2 267543
255 231 3 156720665
Now, for the example of the (7,4) code where the generator polynomial was set to g(p)p3p 1,
the code word generation is given in Fig. 5.17.
236 CHAPTER FIVE
D +
2
1
g0
D +
g1
D +
gn??“k??“1
D + . . .
2
1
Message
Parity
FIGURE 5.16 General cyclic code generator.
D +
2
1
1
D D +
2
1
Message
Parity
FIGURE 5.17 Cyclic code generator for the (7,4) code.
Syndrome-Based Decoding. As discussed above, for single-error correcting codes, syndrome
decoding can be easily used. Let us define the ???parity check matrix,??? H, as satisfying the following
desirable property of orthogonality
(5.27)
This orthogonality constraint results in the following ???parity check matrix???
(5.28)
This matrix has the following two properties: First no column can be all zeros and second, all columns
must be unique. This matrix can also be rewritten as follows:
(5.29)
Let us assume the received vector, , is the sum of the transmitted code word, , and the error
pattern, .
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