Cryptography Reference
In-Depth Information
a given coset, we call this vector a
coset leader
. The coset leader need not be
unique, in which case we select one at random and call it the coset leader.
It is possible to form an arrayof vectors from
q
arranged as the cosets of
F
C
in the following fashion.
Slepian Standard Array
The following is an algorithm for setting up a
q
n
−
k
q
k
matrix for a linear
×
[
n,k
]-code, called a
Slepian standard array
. (See [265].)
1. List the codewords of
C
beginning with the zero vector
c
1
=
0 , as the first
row:
c
1
,
c
2
,...,
c
k
.
2. Choose anyone of the remaining
q
n
q
k
vectors of minimum weight,
a
1
as
the first element of the second row, and let
−
a
j
=
a
1
+
c
j
for
j
=2
,
3
,...,q
k
be the remaining elements of the second row.
3. Select an element
b
1
of minimum weight from the remaining
q
n
2
q
k
vectors
−
as the first element of the third row, and let
b
j
=
b
1
+
c
j
for
j
=2
,
3
,...,q
k
be the remaining elements of the third row.
4. Continue in the above fashion until all
q
n
−
k
rows are filled and everyvector
of
F
q
appears exactlyonce in one of those rows.
Example 11.10
Let
C
be the binary
[4
,
2]
-code with generating matrix,
G
=
1010
0111
.
Then
n
=4
,
k
=2
,
q
k
=2
2
,
q
n
=2
4
,
C
=
{
c
1
,
c
2
,
c
3
,
c
4
}
=
{
(0
,
0)
G,
(0
,
1)
G,
(1
,
0)
G,
(1
,
1)
G
}
=
{
0000
,
0111
,
1010
,
1101
}
.
This forms the first row of the Slepian array. The balance are given in the
q
n
−
k
×
q
k
=4
×
4
matrix,
0000 0111 1010 1101
0100 0011 1110 1001
0010 0101 1000 1111
0001 0110 1011 1100
,
wherein the first column consists of the coset leaders, and the rows represent the
elements in the cosets of
C
.
(
Notice that in the third row,
1000
could also have
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