Cryptography Reference
In-Depth Information
Example 4.13
Let there be a code
C
(
n, k
)
with correction capability
t
=3
. The subset of
the codewords is made up of 8 codewords, 7 of which are elaborated from the
words
e
i
.Words
e
i
are words of length
n
whose components are null except
possibly those with indices
j
1
,
j
2
and
j
3
(see the table below).
i
e
i,j
1
e
i,j
2
e
i,j
3
1
001
2
010
3
011
4
100
5
101
6
110
7
111
•
Probability of erroneous decoding of a codeword
Let us assume that the transmitted codeword is
c
0
=(
c
01
···
c
0
j
···
c
0
n−
1
)
and let
r
0
=(
r
0
···
r
j
···
r
n−
1
)
be the received word with:
r
j
=
E
s
c
0
j
+
b
j
Codeword
c
0
will be wrongly decoded if:
n−
1
n−
1
r
j
c
0
,j
<
r
j
c
l,j
∀
c
l
=
c
0
∈
C
(
n, k
)
j
=0
j
=0
The code being linear, we can, without loss of generality, assume that the code-
word transmitted is the null word, that is,
c
0
,j
=0
for
j
=0
,
1
,
1
.
The probability of erroneous decoding
P
e,
word
of a codeword is then equal
···
,n
−
to:
⎛
⎞
n
−
1
n
−
1
⎝
⎠
P
e,
word
=Pr
r
j
c
1
,j
>
0
or
...
r
j
c
l,j
>
0
or
...
j
=0
j
=0
Probability
P
e,
word
can be upper bounded by a sum of probabilities and,
after some standard computation, it can be written in the form:
erfc
w
j
E
s
N
0
2
k
1
2
P
e,
word
≤
j
=2
where
w
j
is the Hamming weight of the
j
-th codeword.
