Error Control Coding - Digital Communication: Principles and System Modelling - page 188

C k + 1 = p 11 d 1 ⊕ p 21 d 2 ⊕···⊕ p k 1 d k

C k + 2 = p 12 d 1 ⊕ p 22 d 2 ⊕···⊕ p k 2 d k

(9.1)

...........................
C n =

p 1, n − k d 1 ⊕ p 2, n − k d 2 ⊕···⊕

p k , n − k d k

where p i,j is 0 or 1 such that C k is 0 or 1. The above rule can be written in matrix
from giving a matrix equation:

⎡ ⎤

. p 11 p 12 ···

1000 ··· 0 p 1, n − k

⎣ ⎦

. p 21 p 22 ···

0100 ··· 0 p 2, n − k

. p 31 p 32 ···

[ C 1 C 2 ··· C n ] = [ d 1 d 2 ··· d k ] (9.2)

0010 ··· 0 p 3, n − k

.

··· ··· ··· ··· ··· ··· ··· ··· ··· ···

. p k 1 p k 2 ···

0000 ··· 1 p k , n − k

Or C = DG (9.3)

Here G is the k × n matrix on the RHS of the above equation. This matrix G is

I k

. P

called the generator matrix. The general form of G is G = . A block

k × n

code generated with k × n order generator matrix is called a ( n , k ) block code.

Example 9.1 Determine the set of code words for the (6,3) code with generator
matrix

⎡ ⎤

100011
010101
001110

⎣ ⎦

G =

We need to consider the information words (0 0 1), (0 1 0), (0 1 1), ... ,(111).

= 001 into the equation gives the code word

Substituting G and i

⎡ ⎤

⎦ = 001110

100011
010101
001110

= 001

⎣

c

= 010 gives the code word

And i

⎡ ⎤

⎦ = 010101

100011
010101
001110

= 010

⎣

c

Substituting the other information words, all the code words found and they are
shown in Table 9.1 .