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

14.4 Cyclic Codes 821
TABLE 14.6
e
1000000
0100000
0010000
0001000
0000100
0000010
0000001
s
110
011
111
101
100
010
001
The first step is to construct the decoding table. Because n - k - 1 = 2, the syndrome
polynomial is of the second order, and there are seven possible nonzero syndromes. There
are also seven possible correctable single-error patterns because n = 7. We can use
S = e . y T
to compute the syndrome for each of the seven correctable error patterns. Note that
(Example 14.4)
0 1 1 1 0 0
1 1 0 0 1 0
l 1 1 0 0 1
]
We can now compute the syndromes based on H. For example, for e = 1000000,
S = [1000000] . y T
= 110
In a similar way, we compute the syndromes for the remaining error patterns (see
Table 14.6).
When the received word r is 1101101, we can compute s(x), either according to
Eq. (14.24) or by simply applying the matrix product
S = r . y T
= [1 101101] . y T
= 101
Hence, From Table 14.6, this gives e = 0001000, and
Because this code is systematic,
c = r EB e = 1101101 EB 0001000 = 1100101
d = 1100
In a similar way, we determine for r = 0101000, s = 110 and e = 1000000; hence
c = r EB e = 1101000, and d = 1101. For r = 0001100, s = 001 and e = 0000001;
hence c = r EB e = 0001101, and d = 0001.