Cryptography Reference
In-Depth Information
A new form of concatenation, called
parallel concatenation
(PC), was intro-
duced at the beginning of the 1990s to elaborate turbo codes [6.3]. Figure 6.2
presents a PC with dimension 2, which is the classical dimension for turbo codes.
In this scheme, the message is coded twice, in its natural order and in a per-
muted order. The redundant part of the codeword is formed by concatenating
the redundant outputs of the two encoders. PCs differ from SCs in several ways,
described in the next section.
6.1
Parallel concatenation and serial concatena-
tion
Limiting ourselves to dimension 2, the PC, which associates two elementary
codes with rates
R
1
(code
C
1
)and
R
2
(code
C
2
), has a global encoding rate:
R
1
R
2
R
1
+
R
2
−
R
1
R
2
R
p
=
=
(6.1)
R
1
R
2
1
−
(1
−
R
1
)(1
−
R
2
)
This rate is higher than the global rate
R
s
of a serial concatenated code (
R
s
=
R
1
R
2
), for identical values of
R
1
and
R
2
, and the lower the encoding rates
the greater the difference. We can deduce from this that with the same error
correction capability of component codes, parallel concatenation offers a better
encoding rate, but this advantage diminishes when the rates considered tend
towards 1. When the dimension of the composite code increases, the gap between
R
p
and
R
s
also increases. For example, three component codes of rate 1/2
form a concatenated code with global rate 1/4 for parallel, and 1/8 for serial
concatenation. That is the reason why it does not seem to be useful to increase
the dimension of a serial concatenated code beyond 2, except for rates very close
to unity.
However, with SC, the redundant part of a word processed by the outer
decoder has benefited from the correction of the decoder(s) that precede(s) it.
Therefore, at first sight, the correction capability of a serial concatenated code
seems to be greater than that of a parallel concatenated code, in which the values
representing the redundant part are never corrected. In other terms, the MHD
of a serial concatenated code must normally be higher than that of a parallel
concatenated code. We therefore find ourselves faced with the dilemma given
in Chapter 1: PC performs better in the convergence zone (near the theoretical
limit) since the encoding rate is more favourable, and the SC behaves better
at low error rates thanks to a larger MHD. Encoding solutions based on the
SC of convolutional codes have been studied [6.3], which can be an interesting
alternative to classical turbo codes, when low error rates are required. Serial
convolutional concatenated codes will not, however, be described in the rest of
this topic.
