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

Consider a Hamming (7, 4, 3) code with generator polynomial
14.4 Cyclic Codes 819
g(x) = x 3 + x + I.
x 6
Rem-- =x 2 + 1
g(x)
x 5 ?
Rem-- = x~ + x + 1
g(x)
x 4 ?
Rem-- =x- + x
g(x)
x 3
Rem-- = x + 1
g(x)
(14.20)
Therefore, the cyclic code generator matrix is
0 0 0 1 0
1 0 0 1 1
G - [ 0 0 1 0 1 1
0 0 0 0 1
Correspondingly, one form of its parity check matrix is
ff=[ 1 1 0 I 0
1 I 1 0 1
1 0 0 0
J
]
(14.22)
(14.21)
Cyclic Code Generation
One of the advantages of cyclic codes is that their encoding and decoding can be implemented
by means of such simple elements as shift registers and modulo-2 adders. A systematically
generated code is described in Eqs. (14. 17). It involves a division of x n - k d (x) by g (x) that can
be implemented by a dividing circuit consisting of a shift register with feedback connections
according to the generator polynomial* g (x) = x n - k + g1x n - k - I
+ • • • + g n - k -IX + I. The
gain gk are either O or l.An encoding circuit with n-k shift registers is shown in Fig. 14.2.An
understanding of this dividing circuit requires some background in linear sequential networks.
An explanation of its functioning can be found in Peterson and Weldon. 3 The k data digits
Figure 14.2
Encoder for
systematic cyclic
code.
Data
input
k data
digits
Switch s
"-\ o-
P2
n - k parity check digits
* It can be shown that for cyclic codes, the generator polynomial must be of this form.