Cryptography Reference
In-Depth Information
Whatever the value of cycle
C
, higher degrees of parallelism of value
pC
,can
be implemented. Indeed, any multiple of
C
, the basic cycle in the permutation,
is also a cycle in the permutation, on condition that
pC
is a divisor of
k
.That
is,
j
modulo
pC
and
Π(
j
)
modulo
pC
are periodic on the circle of length
k
,
which can then be cut into
pC
fractions of equal length. For example, a degree
64 parallelism is possible for a value of
k
equal to 2048.
However, whatever the degree of parallelism, a minimum latency is unavoid-
able: the time required for receiving a packet and putting it into the buffer
memory. While this packet is being put into memory, the decoder works on the
information contained in the previous packet. If this decoding is performed in
a time at least equal to the memorization time, then the total decoding latency
is at maximum twice this memorization time. The level of parallelism in the
decoder is adjusted according to this objective, which may be a constraint in
certain cases.
For further information about the implementation of turbo decoders, of all
the publications on this topic, [7.47] is a good resource.
7.5
-binary turbo codes
m
m
binary turbo codes are built from recursive systematic convolutional (RSC)
codes with
m
binary inputs (
m
−
≥
2
). There are at least two ways to build an
m
binary convolutional code: either from the Galois field
F
2
m
,orfromthe
Cartesian product
(
F
2
)
m
. Here, we shall only deal with the latter, which is
more convenient. Indeed, a code elaborated in
F
2
m
, with a memory
ν
,has
2
νm
possible states, whereas the number of states of the code defined in
(
F
2
)
m
,with
the same memory, can be limited to
2
ν
.
The advantages of the
m
-binary construction compared to the classical (
m
=
1
) turbo code scheme, are varied: better convergence of the iterative process,
larger minimum distances, less puncturing, lower latency, robustness towards the
sub-optimality of the decoding algorithm, in particular when the MAP algorithm
is simplified into its Max-Log-MAP version [7.23].
The case
m
=2
has already been adopted in the European standards for
the return path in digital video broadcasting via the satellite network and in
the terrestrial network [7.2, 7.1] as well as in the IEEE 802.16 standard [7.5].
Combined with the circular trellis technique, these 8-state turbo codes, called
double-binary turbo codes, offer good average performance and great flexibility
in adapting to different block sizes and different rates, whilst retaining reasonable
decoding complexity.
−
7.5.1
binary RSC encoders
Figure 7.19 presents the general structure of an
m
m
−
binary RSC encoder. It uses
a pseudo-random generator with code memory
ν
and generator matrix
G
(size
−
