In general, a t-error correcting (n, k) linear block code is capable of correcting
a total of 2n??“k error patterns.
Now turn our attention to the BCH codes. As discussed earlier they are extensions to the Hamming
codes in that multiple bit error correction capability exists. The most commonly used BCH code
words have a block length of n 2m 1 where m 3, 4, . . .
Let us consider an example of (6, 3) BCH codes. This BCH code has an error correction capability
of t 1, since dmin 3. The codes can be constructed in the following manner. Consider the (6,3)
generator matrix as follows, written in systematic form.
(5.14)
Recall the code word is generated as follows:
(5.15)
The encoder block diagram can be easily drawn from the previous equation and is given in Fig 5.14.
SC
Sx
# G [x1 x2 x3] # G
G C1 0 0 1 1 0
0 1 0 1 0 1
0 0 1 0 1 1 S [I3 P]
PERFORMANCE IMPROVEMENT TECHNIQUES 233
+
+
+
X1
X2
X3
C1 = X1
C2 = X2
C3 = X3
C4 = X1 + X2
C5 = X1 + X3
C6 = X2 + X3
Code Word
FIGURE 5.14 A (6,3) block code encoder details.
Alternatively we can easily construct the (7,4) code word from the following generator matrix of
the encoder (written in systematic form).
(5.16)
Similarly, we can draw the encoder used to generate the output code word as shown in Fig.
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