Cryptography Reference
In-Depth Information
Assuming that code
C
(
n, k
)
has
A
w
codewords of weight
w
, probability
P
e,
word
can again be written in the form:
A
w
erfc
w
E
s
N
0
n
1
2
P
e,
word
<
(4.27)
w
=
d
min
Introducing the energy
E
b
received per bit of information transmitted, proba-
bility
P
e,
word
can finally be upper bounded by:
A
w
erfc
w
RE
b
N
0
n
1
2
P
e,
word
<
(4.28)
w
=
d
min
where
R
is the coding rate.
We can also establish an upper bound of the binary error probability on the
information symbols after decoding.
n
A
w
erfc
w
RE
b
n
1
2
w
P
e,
bit
<
(4.29)
N
0
w
=
d
min
To calculate probabilities
P
e,
word
and
P
e,
bit
we must know the number
A
w
of codewords of weight
w
. For extended BCH codes the quantities
A
w
are given
in [4.1].
As an example, Table 4.5 gives the
A
w
quantities for three extended Ham-
ming codes.
n
k
d
min
A
4
A
6
A
8
A
10
A
12
A
14
A
16
8
4
4
14
-
1
-
-
-
-
16
11
4
140
448
870
448
140
-
1
32
26
4
1240
27776
330460
2011776
7063784
14721280
18796230
Table 4.5 -
A
w
for three extended Hamming codes.
For the code (32,26) the missing
A
w
quantities are obtained from the relation
A
w
=
A
n−w
for
0
n/
2
,
n/
2
even.
The
A
w
quantities for non-extended Hamming codes can be deduced from
those of extended codes by resolving the following system of equations:
(
n
+1)
A
w−
1
=
wA
extended
≤
w
≤
w
wA
w
=(
n
+1
−
w
)
A
w−
1
where
n
is the length of the words of the non-extended code.
For the Hamming code (7,4), for example, the
A
w
quantities are:
=4
A
extended
4
8
A
3
A
3
=7
4
A
4
=4
A
3
A
4
=7
=8
A
extended
8
8
A
7
A
7
=1
