Cryptography Reference
In-Depth Information
This classical termination has one main drawback: the protection of the data
is not independent of their position in the frame. In particular, this can lead to
edge effects in the construction of a turbocode (see Chapter 7).
Tail-biting
A technique was introduced in the 70s and 80s [5.12] to terminate the trellis of
convolutional codes without edge effects: tail-biting. This involves making the
decoding trellis circular, that is, ensuring that the departure and the final states
of the encoder are identical. This state is then called the circulation state. This
technique is trivial for non-recursive codes as the circulation state is merely the
last
ν
bits of the sequence to encode. As for RSC codes, tail-biting requires
operations that are described in the following. The trellis of such a code, called
circular recursive systematic codes (CRSC), is shown in Figure 5.24.
(000)
(001)
(010)
(011)
(100)
(101)
(110)
(111)
Figure 5.24 - Trellis of a CRSC code with 8 states.
It is possible to establish a relation between state
s
i
+1
of the encoder at an
instant
i
+1
,itsstate
s
i
and the input data
d
i
at the previous time:
s
i
+1
=
As
i
+
Bd
i
(5.15)
where
A
is the state matrix and
B
the input matrix. In the case of the recur-
sive systematic code of generator polynomials [1, (1+
D
2
+
D
3
)
/( 1+
D
+
D
3
)
]
mentioned above, these matrices are
⎡
⎤
⎡
⎤
101
100
010
1
0
0
⎣
⎦
and
B
=
⎣
⎦
.
A
=
