Cryptography Reference
In-Depth Information
A
w
(23,12)
A
w
(24,12)
Weight
0
1
1
7
253
0
8
506
759
11
1288
0
12
1288
2576
15
506
0
16
253
759
23
1
0
24
0
1
Table 4.6 -
A
w
forextendedGolayandGolaycodes.
For Golay and extended Golay codes, the
A
w
quantities are given in Table 4.6.
With a high signal to noise ratio, error probability
P
e,
word
is well approxi-
mated by the first term of the series:
2
A
d
min
erfc
Rd
min
E
b
1
E
b
N
0
P
e,
word
=
if
>>
1
(4.30)
N
0
The same goes for error probability
P
e,
bit
on the information symbols.
P
e,
bit
=
d
min
n
E
b
N
0
P
e,
word
if
>>
1
(4.31)
In the absence of coding, the error probability on the binary symbols is equal
to:
2
erfc
E
b
p
=
1
N
0
As seen in Section 1.5, comparing the two expressions of the binary error proba-
bility with and without coding, we observe that the signal to noise ratio
E
b
/N
0
is multiplied by
Rd
min
in the presence of coding. If this multiplying coecient is
higher than 1, the coding acts as an amplifier of the signal to noise ratio whose
asymptotic gain is approximated by
G
a
=10log(
Rd
min
)
(dB)
To illustrate these bounds, let us again take the example of the (15,7) BCH
code transmitted on a Gaussian channel with 4-PSK modulation. In Figure 4.6,
we show the evolution of the binary error probability and word error probability
obtained by simulation from the sub-optimal Chase algorithm (4 non-reliable
positions). We also show the first two terms of the sums appearing in the
bounds given by (4.28) and (4.29). As a reference, we have also plotted the
binary error probability curve of a 4-PSK modulation without coding.
