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

14.3 Linear Block Codes 807
codeword can be expressed as
C =dG
= d[h P]
= [d dP]
= [d C p
]
(14.7)
where the check digits, also known as the checksum bits or parity bits, are
C p
=dP
(14.8)
Thus, knowing the data digits, we can calculate the check digits from Eq. (14.8) and consequently
the codeword c p· The weight of the codeword c is the number of ls in the codeword.
The Hamming distance between two codewords c a and c b is the number of elements by which
they differ, or
d(c a , cb) = weight of (c a EB cz,)
Example 14. l For a (6, 3) code, the generator matrix G is
0 0 1 0 1
G = 0 1 0 0 1 1
0 0 1 1 1 0
'-,--' '-,--'
h
p
For all eight possible data words, find the corresponding codewords, and verify that this code
is a single-error correcting code.
Table 14.2 shows the eight data words and the corresponding codewords formed from
C =dG.
TABLE 14.2
Data Word d
Codeword c
111 111000
110 110110
101 101011
100 100101
011 011101
010 010011
001 001110
000 000000
Note that the distance between any two codewords is at least 3. Hence, the code can
correct at least one error. The possible encoder for this code shown in Fig. 14.1 uses a
three-digit shift register and three modulo-2 adders.