Cryptography Reference
In-Depth Information
("all zero" path), in the trellis. Indeed, inverting any component of
d
i
modifies
the value of
y
i
. Second, expression (7.53) indicates that the Hamming weight
of the same vector is also at least equal to 2 when we return to the reference
path. In conclusion, relations (7.52) and (7.53) together guarantee that the
free distance of the code, whose rate is
R
=
m/
(
m
+1)
, is at least equal to 4,
whatever
m
.
7.5.2
-binary turbo codes
m
Figure 7.20 -
m
-binary turbo encoder.
binary RSC enoders associ-
ated with a permutation as a function of
N
words of
m
bits (
k
=
mN
)(Fig-
ure 7.20). The blocks are encoded twice by this two-dimensional code, whose
rate is
m/
(
m
+2)
. The circular trellis principle is adopted to enable encoding
of the blocks without a termination sequence and without edge effects.
The advantages of this construction compared to classical turbo codes are
the following :
We consider a parallel concatenation of two
m
−
•
Better convergence. This advantage, observed first in [7.9], commented
in [7.16] and in a different way in [7.23], can be explained by a lower
density of errors on each of the two dimensions of the iterative process.
Take relation (7.8) that provides the upper bound of the accumulated
spatial distance for a binary code and adapt it to an
m
-binary code:
sup
S
min
=
2
k
m
(7.55)
For a coding rate
R
, the number of parity bits produced by the sequence
with accumulated length
sup
S
min
is:
n
parity
(sup
S
min
)=
1
mk
2
m
2
sup
S
min
=
1
−
R
−
R
(7.56)
R
R
