Cryptography Reference
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d (1)
d (1)
d (2)
,
,
r i
d (2)
Mux
r i
D
D
D
Figure 5.5 - Example of a recursive systematic double-binary convolutional encoder.
5.2.2 Polynomial representation
Let us first consider a classical (non-systematic and non-recursive) binary code:
c =0 , all the coecients b j are null and m =1 . The knowledge of g ( k )
k =1 ..n
j
then suces to describe the code. The coecients g ( k )
j
j =0 ..ν
j =0 ..ν
therefore define
n generator polynomials G ( k ) in D ( Delay )algebra:
G ( k ) ( D )=
j =0 ...ν
g ( k )
j
D j
(5.1)
Let us take the case of the encoder defined in Figure 5.2. The outputs r (1)
i
and
r (2)
i
are expressed as functions of the successive data d as follows:
r (1)
i
= d i + d i− 2 + d i− 3
(5.2)
which can also be written, via the transform in D :
r (1) ( D )= G (1) ( D ) d ( D )
(5.3)
with G (1) ( D )=1+ D 2 + D 3 , the first generator polynomial of the code and
d ( D ) the transform in D of the message to be encoded. Likewise, the second
generator polynomial is G 2 ( D )=1+ D + D 3 .
These generator polynomials can also be resumed by the series of their coe-
cients, (1011) and (1101) respectively, generally denoted in octal representation,
(13) octal and (15) octal respectively. In the case of a non-recursive systematic
code , like the example in Figure 5.1, the generator polynomials are expressed
according to the same principle. In this example, the encoder has generator
polynomials G (1) ( D )=1 and G (2) ( D )=1+ D + D 3 .
To define the generator polynomials of a recursive systematic code is not
straightforward. Let us consider the example of Figure 5.3. The first generator
 
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