Cryptography Reference
In-Depth Information
•
The sequences that are not RTZ have an influence on the whole circle: on
average, one parity symbol out of two is modified along the block. For
typical values of
k
(a few hundred or more), the corresponding output
weight is therefore very high and these error patterns do not contribute
to the MHD of the code, as already mentioned at the end of the previous
section. Without termination or with termination using tail bits, only the
part of the block after the beginning of the non-RTZ sequence has any
effect on the parity symbols.
To these two advantages we can, of course, add the interest of having to trans-
mit no additional information about termination and therefore losing nothing
in spectral eciency.
The circular termination technique was chosen for the DVB-RCS and DVB-
RCT [7.2, 7.1] standards, for example.
7.3.2 The permutation function
Whether we call it interleaving or permutation, the technique that involves dis-
persing the data over time has always been very useful in digital communications.
For example, we use it profitably to reduce the effects of more or less long at-
tenuations in transmissions affected by fading and, more generally, in situations
where perturbations can alter consecutive symbols. In the case of turbo codes
too, permutation allows us to eciently combat the appearance of error packets,
on at least one of the dimensions of the composite code. But its role does not
stop there: in close relation with the properties of the constituent codes, it also
determines the minimum distance of the concatenated code.
Let us consider the turbo code presented in Figure 7.3. The worst permu-
tation that we can use is, of course, identity permutation, which minimizes the
diversity of the coding (we then have
Y
1
=
Y
2
)
. On the other hand, the best
permutation that we could imagine, but that probably does not exist [7.42],
would enable the concatenated code to be equivalent to a sequential machine of
which the number of irreducible states would be
2
k
+6
. There are indeed
k
+6
binary memorization elements in the structure:
k
for the permutation memory
and 6 for the two convolutional codes. If we could assimilate this sequential
machine to a convolutional encoder and for common values of
k
,thenumberof
corresponding states would be very large, in any case large enough to guarantee
a large minimum distance. For example, a convolutional encoder with a code
memory of 60 (
10
18
states !) shows a free distance of the order of a hundred (for
R
=1
/
2
), which is quite sucient.
Thus, from the worst to the best permutation, there is a wide choice and
we have not yet discovered any perfect permutation. Having said that, good
permutations have been defined even so in order to elaborate standardized turbo
coding schemes.
