Cryptography Reference
In-Depth Information
dr dr dr dr
dr
dr
d
r
0
0
1
1
2
2
3
3
i
i
k
2
k
2
k
1
k
1
Figure 5.25 - Puncturing a systematic code to obtain a rate
2
/
3
.
In the case of systematic codes, it is generally the redundancy that is punc-
tured. Figure 5.26 shows the trellis of code
[1
,
(1 +
D
2
+
D
3
)
/
(1 +
D
+
D
3
)]
resulting from a puncturing operation according to the mask of Figure 5.25. The
"X"s mark the bits that are not transmitted and that therefore cannot be used
for the decoding.
00
0X
00
0X
00
(000)
11
1X
1X
0X
11
11
00
(001)
11
1X
01
10
0X
1X
01
10
(010)
1X
10
(011)
1X
0X
01
10
1X
10
(100)
0X
01
01
0X
0X
1X
01
10
(101)
11
1X
0X
11
00
(110)
00
0X
1X
00
(111)
11
d
d
r
i
r
i
d
i
=0
=1
d
i
Figure 5.26 - Trellis diagram of the punctured recursive code for a rate
2
/
3
.
The most widely used decoding technique involves taking the decoder of
the original code and inserting
neutral
values in the place of the punctured
elements. The neutral values are values representing information that is
a priori
not known. In the usual case of a transmission using antipodal signalling (+1
for the logical '1', -1 for the logical '0'), the null value (analogue 0) is taken as
the neutral value.
The introduction of puncturing increases the coding rate but, of course,
decreases its correction capability. Thus, in the example of Figure 5.26, the free
distance of the code is reduced from 6 to 4 (an associated RTZ sequence is shown
in the figure). Likewise Figure 5.27, in which we present the error rate curves
of code
[1
,
(1 +
D
2
+
D
3
)
/
(1 +
D
+
D
3
)]
for rates 1/2, 2/3, 3/4 and 6/7, shows
a decrease in error correction capability with the increase in coding rate.
The choice of puncturing mask obviously influences the performance of the
code. It is thus possible to favour one part of the frame, transporting sensitive
data, by slightly puncturing it to the detriment of another part that is more
highly punctured. A regular mask is, however, often chosen as it is simple to
implement.
