Cryptography Reference
In-Depth Information
Figure 1.5 - Error correction capability of (8, 4, 4) and (7, 4, 3) Hamming codes on a
Gaussian channel, with hard input and soft input decoding.
rate, the
asymptotic performance
of
P
e
is approximated by:
exp
E
b
N
0
−
1
2
P
e
≈
π
E
N
0
(1.15)
To evaluate the probability
P
e,word
that the soft-input decoder of a code with
rate
R
with minimum distance
d
min
produces an erroneous codeword, in the
previous equation we replace
E
N
0
by
Rd
min
E
N
0
and we introduce a multiplicative
coe
cient denoted
N
(
d
min
)
:
exp
2
N
(
d
min
)
erfc
Rd
min
E
b
Rd
min
E
N
0
−
1
1
2
N
(
d
min
)
P
e,word
=
N
0
≈
πRd
min
E
N
0
(1.16)
The replacement of
E
b
by
RE
b
comes from (1.14) for the energy received by
symbol is
E
s
. The multiplication by
d
min
is explained by the ML decoding
rule (relation (1.11)), through which the decoder can discriminate the correct
codeword and its closest competitor codewords thanks to
d
min
distinct values.
Finally, the coecient
N
(
d
min
)
, called
multiplicity
, takes into account the num-
ber of competitor codewords that are the minimum distance away. For example,
in the case of the extended Hamming code, we have
N
(
d
min
=4)=14(seeTa-
ble 1.1).
