B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

14.4 Cyclic Codes 817
Example 1 4.4 Construct a systematic (7, 4) cyclic code using a generator polynomial (see Example 14.3).
We use
g(x) = x 3 + x 2 + 1
Consider a data vector d = 1010,
d(x) = x 3 +x
and
Hence,
x 3 + x 2 + I
x 3 + x 2 + 1lx 6 + x 4
x 6 + x 5 + x 3
x 5 + x 4 + x 3
x 5 + x 4 + x 2
x 3 + x 2
x 3 + x 2 + I
1
<f- q(x)
<f- p(x)
Hence, from Eq. (l4.17a),
c(x) = x 3 d(x) + p(x)
= x 3 cx 3 +x) + l
= x 6 +x 4 + 1
and
C = 1010001
We could also have found the codeword c directly by using Eq. (14.18c). Thus, c(x) =
q(x)g (x) = (x 3 + x 2 + l)(x 3 + x 2 + 1) = x 6 +x 4 + 1. We construct the entire code table in
this manner (Table 14.5). This is quite a tedious procedure. There is, however, a shortcut,
by means of the code generating matrix G. We can use the earlier procedure to compute the
codewords conesponding to the data words 1000, 0100, 0010, 0001. These are 1000110,
0100011, 0010111, 0001101. Now recognize that these four codewords are the four rows
of G. This is because c = d G, and when d = 1000, d • G is the first row of G, and