Cryptography Reference
In-Depth Information
d
6
8
10
12
14
...
ω
(
d
)
6
40
245
1446
8295
...
Table 5.1 - First terms of the spectrum of the recursive systematic code with generator
polynomials
[1
,
(1 +
D
2
+
D
3
)
D
3
)]
/
(1 +
D
+
.
noise ratio than the classical code. Moreover, we find the monomials
I
2
O
8+4
c
,
where
c
is an integer, in the transfer function of the recursive code. The infinite
number of monomials of this type is due to the existence of the cycle on a
null input sequence different from the loop on state
0
.Moreover,suchacode
does not provide any monomials of the form
IO
c
, unlike non-recursive codes.
These conclusions concur with those drawn from the study of state machines in
Section 5.2.
This notion of transfer function is therefore ecient for studying the per-
formance of a convolutional code. A derived version is moreover essential for
the classification of codes according to their performance. This is the distance
spectrum
ω
(
d
)
whose definition is as follows:
∞
(
∂T
(
O, I
)
∂I
ω
(
d
)
O
d
)
I
=1
=
(5.14)
d
=
d
f
For example, the first terms of the spectrum of the recursive systematic code,
obtained from (5.13), are presented in Table 5.1. This spectrum is essential for
estimating the performance of codes in terms of calculating their error proba-
bility, as illustrated in the vast literature on this subject [5.9].
The codes used in the above examples have a rate of
1
/
2
. By increasing the
number of redundancy bits
n
the rate becomes lower. In this case, the powers
of
O
associated with the branches of the state machines will be higher than or
equal to those of the figures above. This leads to higher transfer functions with
powers of
O
, that is, to RTZ sequences with a greater Hamming weight. The
codes with lower rates therefore have a higher error correction capability.
5.3.4 Performance
The performance of a code is defined by the decoding error probability after
transmission on a noisy channel. The previous section allows us to intuitively
compare non-recursive non-systematic, non-recursive systematic and recursive
systematic codes with the same constraint length. However, to estimate the
absolute performance of a code, we must be able to estimate the decoding error
probability as a function of the noise, or at least to limit it. The literature,
for example [5.9], thus defines many bounds that are not described here and we
will limit ourselves to comparing the three categories of convolutional codes. To
do this, a transmission on a Gaussian channel of blocks of 53 then 200 bytes
