Cryptography Reference
In-Depth Information
Likewise, the recursive systematic code already studied has as its transfer func-
tion:
T
(
O, I
)= 2
I
3
O
6
+(
I
6
+8
I
4
+
I
2
)
O
8
+(8
I
7
+33
I
5
+8
I
3
)
O
10
+(
I
10
+47
I
8
+ 145
I
6
+47
I
4
+
I
2
)
O
12
+(14
I
11
+ 254
I
9
+ 649
I
7
+ 254
I
5
+14
I
3
)
O
14
+
(5.13)
ยทยทยท
Comparing the transfer functions from the point of view of the monomial with
the smallest degree allows us to appreciate the error correction capability at
very high signal to noise ratio (asymptotic behaviour). Thus, the non-recursive
systematic code is weaker than its rivals since it has a lower minimum distance.
A classical code and its equivalent recursive systematic code have the same free
distance, but their monomials of minimal degree differ. The first is in
(
I
3
+
I
)
O
6
and the second in
2
I
3
O
6
. This means that with the classical code an input
sequence with weight 3 and another with weight 1 produce an RTZ sequence
with weight 6 whereas with the recursive systematic code two sequences with
weight 3 produce an RTZ sequence with weight 6. Thus, if an RTZ sequence
with minimum weight is introduced by the noise, the classical code will introduce
one or three errors, whereas its recursive systematic code will introduce three
or three other errors. In conclusion, the probability of a binary error on such
a sequence is lower with a classical code than with a recursive systematic code,
which explains that the former will be slightly better at high signal to noise ratio.
Things are generally different when the codes are punctured (see Section 5.5) in
order to have higher rates [5.13].
To compare the performance of codes with low signal to noise ratio, we
must consider all the monomials. Let us take the example of the monomial in
O
12
for the non-recursive systematic code, the classical code and the recursive
systematic code, respectively:
(7
I
9
+30
I
8
+77
I
7
+73
I
6
+42
I
5
+3
I
4
)
O
12
(8
I
12
+44
I
10
+90
I
8
+77
I
6
+22
I
4
)
O
12
(
I
10
+47
I
8
+ 145
I
6
+47
I
4
+
I
2
)
O
12
If
12
errors are introduced by the noise on the channel,
232
RTZ sequences
are "available" as errors for the first code,
241
for the second and
241
again
for the third. It is therefore (a little) less probable that an RTZ sequence will
appear if the code used is the non-recursive systematic code. Moreover, the
error expectancy per RTZ sequence of the three codes is
6
.
47
,
7
.
49
and
6
.
00
,
respectively: the recursive systematic code therefore introduces, on average,
fewer decoding errors than the classical code on RTZ sequences with
12
errors
on the frame coded. This is also true for higher degree monomials. Recursive
and non-recursive systematic codes are therefore more e
cient at low signal to
