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Fig. 2.2
Construct an
⎛
⎝
⎞
⎠
1111
11
1111
1100
1010
1001
0000
0011
0101
0110
OA
(
8
,
2
4
,
3
)
from an
Hadamard matrix of order 4
−
−
1
1
−
−
1
11
1
1
−
1
−
11
−
1
−
1
−
1
−
1
−
1
−
111
−
11
−
11
−
111
−
1
2.1.4 Zero-Sum Construction
Theorem 2.4
Suppose Z
s
is the additive group of integers modulo s. For each of the
s
t
1
by adjoining in the last column
the negative of the sum of the elements in the first t columns. Use these vectors to
form an s
t
t -tuples over Z
s
, form a row vector of length t
+
s
t
s
t
+
1
×
(
t
+
1
)
array, and the array is an OA
(
,
,
t
)
.
2
5
Example 2.2
Assume that we would like to construct an OA
(
16
,
,
4
)
. Firstly, we
enumerate all tuples of length 4 over the set
{
0
,
1
}
, i.e., the 16 vectors ranging from
. Secondly, for each vector we add an extra element in the
end, so that the sum of all elements is divisible by 2. For instance,
0
,
0
,
0
,
0
to
1
,
1
,
1
,
1
0
,
0
,
0
,
0
is
extended to
. Finally, all
these vectors are used as row vectors to form the target array, as Fig.
2.3
illustrates.
0
,
0
,
0
,
0
,
0
, and
0
,
1
,
1
,
1
is extended to
0
,
1
,
1
,
1
,
1
2.1.5 Construction from Mutually Orthogonal Latin Squares
Theorem 2.5
There exist k mutually orthogonal Latin squares of order n (k-
MOLS(n)) if and only if there exists an OA(n
2
,n
k
+
2
,2).
2
5
Fig. 2.3
OA
(
16
,
,
4
)
00000
00011
00101
00110
01001
01010
01100
01111
10001
10010
10100
10111
11000
11011
11101
11110
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