Cryptography Reference
In-Depth Information
Linear Encoding
: Earlier we stated the ease with which one mayencode
data bysending a codeword
c
∈
C
,an[
n,k
]-code over
F
q
, via
c
→
cG
.To
q
to a
k
-
summarize, the encoding function
c
→
cG
maps the vector space
F
n
dimensional subspace, of
F
q
, namely, the code
C
. When the generating matrix
G
is in standard form [
I
k
|
M
k,n
−
k
], with
M
k,n
−
k
=(
m
i,j
), then
c
=(
c
1
,c
2
,...,c
k
)
is encoded via
c
=
cG
=(
c
1
,c
2
,...,c
k
,c
k
+1
,...,c
n
)
,
where
k
c
k
+
i
=
m
j,i
c
j
for 1
≤
i
≤
n
−
k
j
=1
are the
check digits
, and the original
c
i
for 1
k
are the
message digits
.In
the example of the [8
,
5
,
1] binarylinear code given above,
≤
i
≤
c
=
cG
=(
c
1
,c
2
,c
3
,c
4
,c
5
,c
6
,c
7
,c
8
)=(1
,
1
,
1
,
1
,
1
,
1
,
1
,
1)
,
with
n
=8,
k
=5,
m
1
,
1
m
1
,
2
m
1
,
3
m
2
,
1
m
2
,
2
m
2
,
3
m
3
,
1
m
3
,
2
m
3
,
3
m
4
,
1
m
4
,
2
m
4
,
3
m
5
,
1
m
5
,
2
m
5
,
3
101
001
000
001
010
M
k,n
−
k
=
M
5
,
3
=
=
,
c
6
=
j
=1
m
j,
1
c
j
c
7
=
j
=1
m
j,
2
c
j
c
8
=
j
=1
m
j,
3
c
j
≡
≡
≡
1(mod 2).
Now we add an illustration for linear codes (see diagram 11.2) that supple-
ments Diagram 11.1.
Linear Code Encoding
Diagram 11.2
M
E
S
S
A
G
E
N
O
I
S
Y
message
−−−−−−−−−−−−→
c
=(
c
1
,...,c
k
)
codeword
−−−−−−−−−−−−→
c
=(
c
1
,...,c
n
)
Encoder
→
c
c
=
cG
C
H
A
N
N
E
L
S
O
U
R
C
E
Search WWH ::
Custom Search