Cryptography Reference
In-Depth Information
extrinsic information about the bits. We can thus decode the product codes in
a turbo manner.
Let therefore
r
=(
r
1
, ..., r
n
)
be the received word after encoding and trans-
mission on a Gaussian channel. The Chase-Pyndiah algorithm with
t
places is
decomposed as follows:
•
Step 1: Select the
t
places
P
k
in the frame containing the least reliable
symbols in the frame (i.e. the
t
places
j
for which the
r
j
values are the
smallest in absolute value).
•
Step 2: Generate the vector of the hard decisions
h
0
=(
h
01
, ..., h
on
)
such
that
h
0
j
=1
if
r
j
>
0
and 0 otherwise. Generate the vectors
h
1
, ...,
h
2
t
−
1
such that
h
ij
=
h
0
j
if
j/
∈{
P
k
}
and
h
iP
k
=
h
0
P
k
⊕
Num
(
i, k
)
where
Num
(
i, k
)
is the
k
-th bit in the binary writing of
i
.
•
Step 3: Decode the words
h
0
, ..., h
2
t
−
1
with the hard decoder of the linear
code. We thus obtain the concurrent words
c
0
, ..., c
2
t
−
1
.
•
Step 4: Calculate the metrics of the concurrent words
M
i
=
1
≤j≤n
r
j
(1
−
2
c
ij
)
•
Step 5: Determine the index
pp
such that
M
pp
=min
{
M
i
}
Codeword
c
pp
is then the most probable codeword.
•
Step 6: For each bit
j
in the frame, calculate the reliability
F
j
=1
/
4(min
{
M
i
,c
ij
=
c
pp,j
}−
M
pp
)
If there are no concurrent words for which the
j
-th bit is different from
c
pp,j
, then the reliability
F
j
is at a fixed value
β
.
•
Step 7: Calculate the extrinsic value for each bit
j
,
E
j
=(2
×
c
pp,j
−
1)
×
F
j
−
r
j
The extrinsic values are then exchanged between row decoders and column
decoders in an iterative process. The value
β
, as well as the values of the feed-
backs are here more sensitive than in the case of decoding convolutional turbo
codes. Inadequate values can greatly degrade the error correction capability.
However, it is possible to determine them incrementally. First, for the first iter-
ation we search to see which values give the best performance (for example by
dichotomy). Then, these values being fixed, we perform a similar search for the
second iteration, and so forth.
