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

816 ERROR CORRECTING CODES
TABLE 14.4
d
1010
1111
0001
1000
C
1110010
1001011
0001101
1101000
Similarly, codewords for other data words can be found (Table 14.4). Note the structure
of the codewords. The first k digits are not necessarily the data bits. Hence, this is not a
systematic code.
In a systematic code, the first k digits are data bits, and the last m = n - k digits are the
parity check bits. Systematic codes are a special case of general codes. Our discussion thus far
applies to general cyclic codes, of which systematic cyclic codes are a special case. We shall
now develop a method of generating systematic cyclic codes.
Systematic Cyclic Codes
We shall show that for a systematic code, the codeword polynomial c(x) corresponding to the
data polynomial d(x) is given by
c(x) = x n -k d(x) + p(x)
(14.17a)
where p(x) is the remainder from dividing x n -k d(x) by g(x),
x n -k d(x)
p(x) = Rem --­
(14.17b)
g(x)
To prove this we observe that
x n -k d(x) p(x)
= q(x) + (14.18a)
g(x)
g(x)
where q(x) is of degree k - I or less. We add p(x)/g(x) to both sides of Eq. (14.18a), and
becausef (x) + f (x) = 0 under modulo-2 operation, we have
or
x n -k d(x) + p(x)
----- =q(x)
g(x)
q(x)g(x) = x n -k d(x) + p(x)
(14.18b)
(14.18c)
Because q(x) is on the order of k - I or less, q(x)g(x) is a code polynomial. As x n -k d(x)
represents d (x) shifted to the left by n - k digits, the first k digits of this codeword are
precisely d, and the last n - k digits corresponding to p (x) must be parity check digits. This
will become clear by considering a specific example.