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

808 ERROR CORRECTING CODES
Figure 14.1
Encoder for
linear block
codes.
--+--------------u' Cz
--+---+----------0 C 3
Commutator
Data input
1-------------U C 5
Linear Codes
A block code is a linear block code if for every pair of codewords Ca and c1, from the
block code,
is also a codeword. For this reason, linear codes must have an all-zero codeword 000 • • • 00.
For linear codes, the minimum distance equals the minimum weight.
Decoding
Let us consider some codeword properties that could be used for the purpose of decoding.
From Eq. (14.8) and the fact that the modulo-2 sum of any sequence with itself is zero, we get
d · P fJJ Cp = ld Cp] [ p ] = 0
,___, Im
(14.9)
where Im is the identity matrix of order m x m (m = n - k). Thus,
cH 7 = 0
(14.!0a)
where
(14.lOb)
and its transpose
H = [P 7
Im]
(14.lOc)
is called the parity check matrix. Every codeword must satisfy Eq. (14. lOa). This is our clue
to decoding. Consider the received word r. Because of possible errors caused by channel noise,
r in general differs from the transmitted codeword c,
r=cEBe