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Fig. 2.7
Product
Construction for
CA
(
13
,
2
8
00000000
11111111
10011001
10101010
01010101
11001100
00110011
01100110
00001111
01111000
10110100
11010010
11100001
,
3
)
0000
1111
1001
1010
0101
1100
0011
0110
A
=CA
(
8
,
2
4
0000
0111
1011
1101
1110
B
=CA
(
5
,
2
4
,
2
)
,
3
)
2
4
Example 2.4
Suppose we are given two covering arrays,
A
is a CA
(
8
,
,
3
)
and
B
is
2
4
1
, which is
1
aCA
(
5
,
,
2
)
. Since
v
=
2, we have only one permutation
π
π
(
0
)
=
1
1
1
0. So
B
π
and
π
(
1
)
=
is actually obtained by permuting symbol '0' and '1' in
B
.
1
2
8
with
A
,
B
and
B
π
We can form a CA
(
13
,
,
3
)
, as shown in Fig.
2.7
.
2.2.2.3 Covering Arrays of Arbitrary Strength
For covering arrays of arbitrary strength, we have the following theorem [
1
]:
v
k
k
Theorem 2.8
If a
CA
(
N
,
,
t
)
and a
CA
(
M
,w
,
t
)
both exist, then a
CA
(
NM
,
k
(
v
w)
,
t
)
also exists.
v
k
k
(
,
,
)
(
,w
,
)
Suppose
A
is a CA
N
t
, and
B
is a CA
M
t
. We construct a series of
×
k
matrices
C
l
with entries
C
l
(
,
)
=
<
(
,
),
(
,
)>
≤
≤
N
i
j
A
i
j
B
l
j
, where 1
i
N
,
1
≤
j
≤
k
, and 1
≤
l
≤
M
. Using these matrices, we form an
NM
×
k
matrix
T
. Obviously
C
is a CA
k
C
=[
C
1
,...,
C
M
]
(
NM
,(
v
w)
,
t
)
.
2.2.3 Construction Based on Difference Covering Arrays
A difference covering array, or a DCA
with entries
from an Abelian group
G
of order
v
, such that for any two distinct columns
l
and
h
,
the difference list
(
k
,
n
;
v
)
is an
n
×
k
array
(
d
ij
)
δ
l
,
h
={
d
1
l
−
d
1
h
,
d
2
l
−
d
2
h
,...,
d
nl
−
d
nh
}
contains every element of
G
at least once.
Example 2.5
ThearrayinFig.
2.8
is a DCA
(
4
,
3
;
2
)
over
Z
2
[
10
].
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