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

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ERROR CORRECTING CODES
14.3 LINEAR BLOCK CODES
A codeword consists of n digits c 1 , c2, . .. , e n , and a data word consists of k digits d1 , d 2 , ... ,
d k , Because the codeword and the data word are an n-tuple and a k-tuple, respectively, they
are n- and k-dimensional vectors. We shall use row vectors to represent these words:
c = (c1 , c2, ... , C n )
d = (d1 , d2, ... , d k )
For the general case of linear block codes, all the n digits of c are formed by linear combinations
(modulo-2 additions) of k data digits. A special case in which CJ = d1 , c2 = d2, ... , C k = d k
and the remaining digits from Ck+J to e n are linear combinations of d1 , d2, ... , d k , is known
as a systematic code. In a systematic code, the leading k digits of a codeword are the data (or
information) digits and the remaining m = n - k digits are the parity check digits, formed by
linear combinations of data digits d1, d2, ... , d k :
Ck =dk
Ck+l = h11d1 EB h12d2 EB ··· EB hlkd k
ck+2 = h21d1 EB h22d2 EB ··· EB h2kdk
(14.5a)
or
C =dG
(14.5b)
where
G=
1
0
0
0 0 0 h11 h21
1 0 0 h12 h22
0
0 0 1 hlk h2 k
hm1
hm2
0
hmk
(14.6)
h (k X k) P(k x m)
The k x n matrix G is called the generator matrix. For systematic codes, G can be partitioned
into a k x k identity matrix I k and a k x m matrix P. The elements of Pare either O or 1. The