19)
In fact, performing a third shift will result in the time reversal of . Now we can take this generator
polynomial and create the generator matrix as follows (where T is used to denote the transpose operation).
(5.20) G [g S
T1
g S
T2
g S
T0
Ink ] [PInk]
g S
0
g S
3 [ 1 0 1 1]
g S
2 [ 0 1 1 1]
g S
1 [1 1 1 0]
g S
0 [1 1 0 1]
g S
[1 1 0 1]
+
+ +
X1
X2
X3
C4 = X1
C5 = X2
C6 = X3
C7 = X4
C1 = X1 + X2 + X3
C2 = X2 + X3 + X4
Code Word
X4
C3 = X1 + X2 + X4
+
+ +
FIGURE 5.15 A (7,4) block code encoder details.
Note the columns of P can be arranged in any
order without affecting the distance property of
the code. This was a specific example for (7,4)
code, generally speaking elementary row operations
on the matrix G can produce the systematic
form (see Table 5.2) [8].
The next group of error correcting codes
to be discussed are cyclic codes. These
codes satisfy the following property: if C
is a code word of a cyclic
code, then is also a
code word. In other words, all cyclic shifts of C
are code words. These codes have wonderful
properties that enable efficient encoding/decoding
operations. Thus making it practically possible
to implement long codes with many code
words.
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