3.2 DUAL CODE
In general,
the polynomial x n + 1 may be factored as
x n
+ 1 = g(x)h(x) (3.12)
Where g(x)
denotes the generator polynomial for the (n , k) cyclic code and h(x)
denotes the parity polynomial that has degree k. The latter
may be used to generate the dual code.
For this
purpose, we define the reciprocal polynomial of h(x) as
x
k h(x -1) = x k (x -k + hk-1 x
-k+1+ hk-2 x - k+2 + . . . + h1 x -1 + 1)
= 1+hk-1
x1 +hk-2 x 2 + . . . + h1 x k-1
+ x k (3.13)
The
reciprocal polynomial is also a factor of x n + 1. Hence x
k h(x -1) is
the
generator
polynomial of an (n, n- k) cyclic code. This cyclic code is the dual code
to the (n , k) code generated from g(x).
Example 3.2
Let us consider the dual code to the (7, 4)
cyclic code generated in Example 3.1. This dual code is a (7, 3) cyclic
code associated with the parity polynomial
h1(x)
= (x +1) (x 3 +x+1)
= x 4 + x 3 + x 2 +1
The
reciprocal polynomial is
x 4 h1(x -1) =1+ x+ x 2 + x 4
This
polynomial generates the (7, 3) dual code given in Table 3.2
|
Information bits
|
Code words
|
|
x
2 x 1 x
0
|
x 6 x
5 x 4
x 3 x 2 x 1 x0
|
|
0
0 0
0
0 1
0
1 0
0
1 1
1
0 0
1
0 1
1
1 0
1
1 1
|
0
0 0 0
0 0 0
0 0 1
0 1
1 1
0 1 0
1 1
1 0
0 1 1 1
0 0 1
1 0 1 1
1 0 0
1 0 0 1
0 1 1
1 1
1
0 0 1
0
1 1
0 0
1 0 1
|
Table 3.2 (7, 3) Dual Code With Polynomial x4 h1(x -1)
= 1+ x + x 2 +x 4
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