A Short Course at Tamkang University Taipei, Taiwan, R.O.C. March 7-9 2006

1
An Application of Coding Theory into Experimental
Design

• A Short Course at Tamkang University Taipei,
  Taiwan, R.O.C. March 7-9 2006

Shigeichi Hirasawa
Department of Industrial and Management Systems
Engineering, School of Science and Engineering
, Waseda University
2
1.Introduction
??
3
1.1 Abstract
????
????
Experimental Design
Coding Theory
????
Error-Correcting Codes (ECCs)
Orthogonal Arrays (OAs)
close relation
Hamming codes, BCH codes RS codes etc.
??? L8
table of OA L8 etc.
ｷ relations between OAs and ECCs ｷ the table of
OAs and Hamming codes ｷ the table of OAs
allocation
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1.2 Outline
1.Introduction 2.Preliminary 3.Relation between
ECCs and OAs 4.Conclusion
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??
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2.Preliminary
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Experimental Design
?????
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2.1 Experimental Design (?????)
2.1.1 Experimental Design
Ex.)
??A
ｷ Factor A (materials)
A0(A company),A1(B company)
a Ratio of Defective Products
??B
ｷFactor B (machines)
B0(new),B1(old)
??C
ｷFactor C (temperatures)
C0(100?),C1(200?)
ｷHow the level of factors affects a ration of
defective products ? ｷWhich is the best
combination of levels ?
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????
Complete Array
experiments with all combination of levels
A B C
experiment with A0,B0,C0
??
0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1
1 0 1 1 1
Experiment ?
?
?
?
?
?
?
?
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????
Orthogonal Array (OA) OA(M, n, s,t) (s2)
????
subset (subspace) of complete array
A B C
011
001
0 0 0 0 1 1 1 0 1 1 1 0
Experiment ?
101
111
?
?
000
010
?
100
110
??
strength t2
every 2 columns contains each 2-tuple exactly
same times as row
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????
Orthogonal Array (OA) OA(M, n, s,t) (s2)
????
subset (subspace) of complete array
A B C
011
001
0 0 0 0 1 1 1 0 1 1 1 0
Experiment ?
101
111
?
?
000
010
?
100
110
??
strength t2
every 2 columns contains each 2-tuple exactly
same times as row
11
????
Orthogonal Array (OA) OA(M, n, s,t) (s2)
????
subset (subspace) of complete array
A B C
011
001
0 0 0 0 1 1 1 0 1 1 1 0
Experiment ?
101
111
?
?
000
010
?
100
110
??
strength t2
every 2 columns contains each 2-tuple exactly
same times as row
12
????
Orthogonal Array (OA) OA(M, n, s,t) (s2)
????
subset (subspace) of complete array
A B C
011
001
0 0 0 0 1 1 1 0 1 1 1 0
Experiment ?
101
111
?
?
000
010
?
100
110
??
strength t2
every 2 columns contains each 2-tuple exactly
same times as row
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2.1.2 Construction Problem of OAs
???
the number of factors n3
Parameters of OAs
A B C
???
ｷthe number of factors n
?
0 0 0
????
?
0 1 1
ｷthe number of runs M
????
the number of runs M4
?
1 0 1
??
ｷstrength t2t
trade off
?
1 1 0
this can treat t-th order interaction effect
??
strength t2
Construction problem of OAs is to construct OAs
with as few as possible number of runs, given the
number of factors and strength (n,t ? min M)
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2.1.3 Generator Matrix (????)
Generator Matrix of an OA G
Ex.)
orthogonal array 000 , 011 , 101 , 110
A
B
C
A
B
C
0 1 1
(?,?,?) (?,?)
1 0 1
OA
each k-tuple (k2) based on0,12 2kM
generator matrix G
To construct OAs is to construct generator matrix
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2.1.3 Generator Matrix (????)
Generator Matrix of an OA G
Ex.)
orthogonal array 000 , 011 , 101 , 110
A
B
C
A
B
C
0 1 1
( 0, 0, 0 ) ( 0,0 )
1 0 1
OA
each k-tuple (k2) based on0,12 2kM
generator matrix G
To construct OAs is to construct generator matrix
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2.1.3 Generator Matrix (????)
Generator Matrix of an OA G
Ex.)
orthogonal array 000 , 011 , 101 , 110
A
B
C
A
B
C
0 1 1
( 0, 1, 1 ) ( 1,0 )
1 0 1
OA
each k-tuple (k2) based on0,12 2kM
generator matrix G
To construct OAs is to construct generator matrix
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2.1.3 Generator Matrix (????)
Generator Matrix of an OA G
Ex.)
orthogonal array 000 , 011 , 101 , 110
A
B
C
A
B
C
0 1 1
( 1, 0, 1 ) ( 0,1 )
1 0 1
OA
each k-tuple (k2) based on0,12 2kM
generator matrix G
To construct OAs is to construct generator matrix
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2.1.3 Generator Matrix (????)
Generator Matrix of an OA G
Ex.)
orthogonal array 000 , 011 , 101 , 110
A
B
C
A
B
C
0 1 1
( 1, 1, 0 ) ( 1,1 )
1 0 1
OA
each k-tuple (k2) based on0,12 2kM
generator matrix G
To construct OAs is to construct generator matrix
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Parameters of OAs and Generator Matrix G
Ex.)
orthogonal arrays 000 , 011 , 101 , 110
the number of factors n3
3
ｷthe number of factors n3
0 1 1
G
ｷthe number of runs M4
2
1 0 1
the number of runs M22
ｷstrength t2
any 2 columns are linearly independent
strength t2
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Parameters of OAs and Generator Matrix
Ex.)
orthogonal arrays 000 , 011 , 101 , 110
the number of factors n3
3
ｷthe number of factors n3
0 1 1
G
ｷthe number of runs M4
2
1 0 1
the number of runs M22
ｷstrength t2
any 2 columns are linearly independent
0
1
0
strength t2

?
1
0
0
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Parameters of OAs and Generator Matrix
Ex.)
orthogonal arrays 000 , 011 , 101 , 110
the number of factors n3
3
ｷthe number of factors n3
0 1 1
G
ｷthe number of runs M4
2
1 0 1
the number of runs M22
ｷstrength t2
any 2 columns are linearly independent
0
1
0
strength t2

?
1
1
0
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Parameters of OAs and Generator Matrix
Ex.)
orthogonal arrays 000 , 011 , 101 , 110
the number of factors n3
3
ｷthe number of factors n3
0 1 1
G
ｷthe number of runs M4
2
1 0 1
the number of runs M22
ｷstrength t2
any 2 columns are linearly independent
1
1
0
strength t2

?
0
1
0
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OAs and ECCs HSS