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TRANSCRIPT
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OrthogonalArrays:
Construc-
tions andRelated
Structures
JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Orthogonal Arrays: Constructions andRelated Structures
Jianxing YIN
Department of Mathematics,Soochow University
May 26, 2011
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OrthogonalArrays:
Construc-
tions andRelated
Structures
JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Contents
1 A Brief Introduction to Orthogonal Arrays
2 OAs and Difference Matrices
3 Existence of OA(3, 5, 4n + 2)s
4 Nested OAs
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OrthogonalArrays:
Construc-
tions andRelated
Structures
JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
What Is an Orthogonal Array ?
An orthogonal array (OA) of index , strength t,degree k and order v, denoted by OA(t,k,v) (or
OA(t,k,v) if = 1), is a k vt array with entriesfrom a set V of v symbols such that in any t rows everyt 1 column vectors over V appears exactly times.
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OrthogonalArrays:
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tions andRelated
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JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
What Is an Orthogonal Array ?
An orthogonal array (OA) of index , strength t,degree k and order v, denoted by OA(t,k,v) (or
OA(t,k,v) if = 1), is a k vt array with entriesfrom a set V of v symbols such that in any t rows everyt 1 column vectors over V appears exactly times.
Here, we employ the definition from design theory. In
literature, statisticians prefer to represent an OA in thetransposed form, namely, a vt k array.
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OrthogonalArrays:
Construc-
tions andRelated
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JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Example 1.1
Take t = 3, k = 4, v = 2 and = 1. Then the followingbinary array:
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 10 1 1 0 1 0 0 1
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OrthogonalArrays:
Construc-
tions andRelated
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JianxingYIN
contents
A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Example 1.1
Take t = 3, k = 4, v = 2 and = 1. Then the followingbinary array:
0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 10 1 1 0 1 0 0 1
forms an OA(3, 4, 2) over Z2. It is easy to check that everybinary triple, e.g. (0, 1, 1)T, occurs in any 3 rows as acolumn exactly once.
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OrthogonalArrays:
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
The Basic Question
The basic question concerning OAs is: Does an OAexist for given parameters ?
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OrthogonalArrays:
Construc-
tions andRelated
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JianxingYIN
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
The Basic Question
The basic question concerning OAs is: Does an OAexist for given parameters ?
The concept of an OA is simple enough, yet thesolution to the question have involved innovativecombinatorial techniques as well as ingeniousapplications of methods from other area of
mathematics. The larger the strength t or the degree k,the more limited is our ability to find OAs.
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction
The concept of an OA was originated from a series ofseminal papers by C. R. Rao in the 1940s and termedby Bush (1950).
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OrthogonalArrays:
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tions andRelated
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction
The concept of an OA was originated from a series ofseminal papers by C. R. Rao in the 1940s and termedby Bush (1950).
Since their introduction, OAs have played a prominentrole in the design of experiments. In experimentalsetups, the rows of an OA represent k factors affectingresponse, in which the entries within each row indicateslevels for that factor. The columns then represent teststo be run, in which a value for each factor is dictated.
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
The study of OAs of strength t = 2 may go back to1782 when Euler made the conjecture
No OA(2, 4, 4n + 2) can exist.
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
The study of OAs of strength t = 2 may go back to1782 when Euler made the conjecture
No OA(2, 4, 4n + 2) can exist.
In 1922, MacNeish made an extended conjecture
An OA(2, k , v) can exist only if k 1
the minimum prime-power factor of v.
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
Although it was not neglected by mathematicians of
the day, Eulers conjecture remained unresolved untilBose, Parker and Shrikhande showed it to be false forn 2 in 1959-1960. The New York Times of April 26,1959 showed these three men working together on theconstruction of an OA(2, 4, 10).
A f O A
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OrthogonalArrays:
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tions andRelated
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
Over the past decades, both statisticians andmathematicians made the significant contributions to
this field.
1 A B i f I d i O h l A
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
Over the past decades, both statisticians andmathematicians made the significant contributions to
this field.
However, a glance at bibliography shows that openproblems loom large. For example, the conjecture
An OA(2, 5, 10) does not exist
made by Parker in 1991 and remains open to this day!
1 A B i f I t d ti t O th l A
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OrthogonalArrays:
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
Historical Introduction (Contd.)
Further, on the basis of our limited present knowledge,
some people guess that an OA(2, n + 1, n), orequivalently a projective plane of order n, can existonly if n is a prime power. This problem has beenproposed [Mullen (1995)] as a candidate for thenext Fermats Last Theorem.
1 A B i f I t d ti t O th l A
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
The Purpose of this Talk
The goal of this talk is to present some recent progresson OAs obtained by our research group at Soochow
University.
1 A B i f I t d ti t O th l A
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
1. A Brief Introduction to Orthogonal Arrays
The Purpose of this Talk
The goal of this talk is to present some recent progresson OAs obtained by our research group at Soochow
University.
The detailed information about OAs, not touched uponhere, can be found in
Colbourn-Dinitz (2007);
Hedayat-Slone-Stufken (1999).
2 OA d Diff M t i
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2. OAs and Difference Matrices
2.1 A Brief Outline of the Idea
To facilitate the construction by using algebraic tool,one often makes an assumption that the OA to beconstructed admits an automorphism group G, acting
on the columns of the OA regularly. It follows that thesearch of an OA can be simplified to find orbitrepresentatives.
2 OAs and Difference Matrices
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2. OAs and Difference Matrices
2.1 A Brief Outline of the Idea
To facilitate the construction by using algebraic tool,one often makes an assumption that the OA to beconstructed admits an automorphism group G, acting
on the columns of the OA regularly. It follows that thesearch of an OA can be simplified to find orbitrepresentatives.
This technique was first used by Bose and Bush
(1952) in the case strength t = 2, where the initialmatrix is by now well known as a difference matrix(DM).
2 1 A Brief Outline of the Idea (Contd )
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.1 A Brief Outline of the Idea (Contd.)
A DM of strength 2 with parameters (v,k,) is a k vmatrix over an additive group of order v such that thevector difference between any two distinct rows of the
array contains every group-element exactly times.
2 1 A Brief Outline of the Idea (Contd )
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.1 A Brief Outline of the Idea (Contd.)
A DM of strength 2 with parameters (v,k,) is a k vmatrix over an additive group of order v such that thevector difference between any two distinct rows of the
array contains every group-element exactly times.
The generalization of a DM from strength 2 to highstrength was first exhibited in Seiden (1954), andfurther studied by a number of authors from statistics
(see, for example, Mukhopadhyay (1981) andHedayat et al. (1996)).
2 2 Definitions and Notations
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.2 Definitions and Notations
To describe the definition, we will use the symbol G todenote an commutative group of order v whoseoperation is written as addition, throughout whatfollows.
2 2 Definitions and Notations
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.2 Definitions and Notations
To describe the definition, we will use the symbol G todenote an commutative group of order v whoseoperation is written as addition, throughout whatfollows.
The induced sumu
G G of u copies of G iswritten as Gu.
2 2 Definitions and Notations
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.2 Definitions and Notations
To describe the definition, we will use the symbol G todenote an commutative group of order v whoseoperation is written as addition, throughout whatfollows.
The induced sumu
G G of u copies of G iswritten as Gu.
By Gu0 we mean the subgroup of Gu being isomorphic
to G, which consists of all elements of the form(g,g, , g). The cosets of this subgroup in Gu will bedenoted by Gui , 0 i v
u1 1.
2 2 Definitions and Notations
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.2 Definitions and Notations
Now write c = vt1 and consider a k c matrix Dover G. If in every t c submatrix of D each coset Gti
(0 i vt1 1) is represented by exactly columns,then D is called a difference matrix of strength tand index , denoted by DM(t,k,v) (or DM(t,k,v)if = 1).
2.2 Definitions and Notations
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Nested OAs
2.2 Definitions and Notations
Any k (c/v) subarray of D is called its a fan, if itforms a DM
(t 1, k , v).
2.2 Definitions and Notations
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
2.2 Definitions and Notations
Any k (c/v) subarray of D is called its a fan, if itforms a DM
(t 1, k , v).
A DM(t,k,v) is said to be completely divisible, if itcan be partitioned into v fans.
2.2 Definitions and Notations
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
By definition, a DM(t,k,v) D over G must satisfy thefollowing two properties:
adding a fix element of G to all entries in a row or a
column of D, or permuting rows or columns of D, theresult is again a DM(t,k,v) over G;
2.2 Definitions and Notations
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
By definition, a DM(t,k,v) D over G must satisfy thefollowing two properties:
adding a fix element of G to all entries in a row or a
column of D, or permuting rows or columns of D, theresult is again a DM(t,k,v) over G;
D is also a DM(s,k,v) over G with = vts for
2 s t.
2.3 The Standard DM-Construction of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Proposition 2.1 If a DM(t,k,v) exists, then so does anOA(t,k,v).
2.3 The Standard DM-Construction of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Proposition 2.1 If a DM(t,k,v) exists, then so does anOA(t,k,v).
Proof. Let D = (dij ) be a DM
(t,k,v) over G. Then, underthe action of G, the development of D
(D + g0 | D + g1 | D + gv1)is an OA(t,k,v). Here, G = {g0 = 0, g1, , gv1} andD + g is the matrix obtained from D by adding g G to
each entry of D.
2.3 The Standard DM-Construction of OAs
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Proposition 2.2 If a completely divisible DM(t,k,v)exists, then we have an OA(t, k + 1, v).
2.3 The Standard DM-Construction of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Proposition 2.2 If a completely divisible DM(t,k,v)exists, then we have an OA(t, k + 1, v).
Proof. Add one more row to the development of the DM inan appropriate way to end up an OA(t, k + 1, v).
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2.3 The Standard DM-Construction of OAs
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
The principal idea for constructing OAs from DMs issimple, but quite powerful.
Example 2.1 Start with the completely divisible
DM2(3, 4, 3) over Z3 given by
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 1 2 0 1 2 0 1 2 0 1 2 0 1 2 0 1 22 1 0 1 0 2 0 2 1 2 1 0 1 0 2 0 2 12 0 2 1 1 0 0 1 0 2 2 1 1 2 1 0 0 2
.
2.3 The Standard DM-Construction of OAs
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Nested OAs
Example 2.1 (Contd.)
By applying Proposition 2.2, we end up an OA2(3, 5, 3)over Z3. It is the juxtaposition of the following three arrayscorresponding 3 fans of the DM:
0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2
0 1 2 0 1 2 1 2 0 1 2 0 2 0 1 2 0 12 1 0 1 0 2 0 2 1 2 1 0 1 0 2 0 2 12 0 2 1 1 0 0 1 0 2 2 1 1 2 1 0 0 20 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
,
0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 20 1 2 0 1 2 1 2 0 1 2 0 2 0 1 2 0 10 2 1 2 1 0 1 0 2 0 2 1 2 1 0 1 0 20 1 0 2 2 1 1 2 1 0 0 2 2 0 2 1 1 01 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
,
2.3 The Standard DM-Construction of OAs
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Nested OAs
Example 2.1 (Contd.)
0 0 0 0 0 0 1 1 1 1 1 1 2 2 2 2 2 2
0 1 2 0 1 2 1 2 0 1 2 0 2 0 1 2 0 11 0 2 0 2 1 2 1 0 1 0 2 0 2 1 2 1 01 2 1 0 0 2 2 0 2 1 1 0 0 1 0 2 2 12 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
.
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2.3 The Standard DM-Construction of OAs
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
RemarksDifference matrices with strength t = 2 has received a
great deal of attention in the literature. In contrast, notmuch is known about DMs of strength 3. The followingtheorem is an alternative version of the construction
presented in Ji-Zhu (2003) for arbitrary indices. Itprovides a possible way to obtain a DM of high strengthfrom low strength.
Theorem 2.1If a DM(2, 4, v) exists, then there exists a completelydivisible DM(3, 4, v), and hence an OA(3, 5, v).
2.4 Extended DM-Constructions of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Preliminaries Recently, we found some methods of constructing OAs
of strength 3. These constructions are established byusing a DM(2, 4, v) with some more restrictions, whichcan be viewed as an extension of the standard
DM-Construction of OAs.
2.4 Extended DM-Constructions of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Preliminaries Recently, we found some methods of constructing OAs
of strength 3. These constructions are established byusing a DM(2, 4, v) with some more restrictions, whichcan be viewed as an extension of the standard
DM-Construction of OAs.
Let D = (dij ) be a DM(2, 4, v) over G. Suppose thata = (a1, a2, , av) is a -fold permutation of theelements of G. If the matrix Da = (dij ) is also a
DM(2, 4, v)-DM over G, then a is termed an adder ofD. Here, dij = dij for i {1, 2} and d
ij = dij + aj ,otherwise.
2.4 Extended DM-Constructions of OAs
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Example 2.2
A DM(2, 4, 15) over Z15 with an adder:
a 0 9 1 4 2 14 7 12 6 8 10 5 11 3 13R1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0R2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
R3 0 2 7 1 11 4 10 13 3 6 12 14 5 9 8R4 0 10 9 8 1 7 4 2 14 5 13 12 11 6 3
2.4 Extended DM-Constructions of OAs
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Existence ofOA(3, 5, 4n+2)s
Nested OAs
Example 2.2
A DM(2, 4, 15) over Z15 with an adder:
a 0 9 1 4 2 14 7 12 6 8 10 5 11 3 13R1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0R2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
R3 0 2 7 1 11 4 10 13 3 6 12 14 5 9 8R4 0 10 9 8 1 7 4 2 14 5 13 12 11 6 3
Adding the corresponding adder to its row 3 and row 4 weobtain another DM(2, 4, 15) over Z15:
R1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0R2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14(R3 + a) 0 11 8 5 13 3 2 10 9 14 7 4 1 12 6(R4 + a) 0 4 10 12 3 6 11 14 5 13 8 2 7 9 1
2.4 Extended DM-Constructions of OAs
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A BriefIntroductiontoOrthogonalArrays
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Existence ofOA(3, 5, 4n+2)s
Nested OAs
Preliminary (Contd.)
Suppose again that D = (dij ) is a DM(2, 4, v) over G.
Write R1, R2, R3 and R4 for the row vectors of D. Ifthe vector difference (R1 + R4) (R2 + R3) containsevery element of G exactly times, then we say that Dis a DM*(2, 4, v).
2.4 Extended DM-Constructions of OAs
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Existence ofOA(3, 5, 4n+2)s
Nested OAs
Theorem 2.2 [Ji-Yin, J Combin Theory-A 117 (2010)]
If a DM(2, 4, v) with an adder exists, then there exists anOA
(3, 6, v).
2.4 Extended DM-Constructions of OAs
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Theorem 2.2 [Ji-Yin, J Combin Theory-A 117 (2010)]
If a DM(2, 4, v) with an adder exists, then there exists anOA
(3, 6, v).
Theorem 2.3 [Li-Ji-Yin, Des Codes Crypt 50 (2009)]
If a DM*(2, 4, v) exists, then there exists an OA(4, 6, v).
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2.4 Extended DM-Constructions of OAs
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Open Problem 1
Construct a DM(2, 4, 3p) with an adder for any primep 11.
2.4 Extended DM-Constructions of OAs
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Open Problem 1
Construct a DM(2, 4, 3p) with an adder for any primep 11.
Open Problem 2
Establish some classes of DM(t,k,v) with t 3 andnon-prime power values of v.
3. Existence of OA(3, 5, 4n + 2)s
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Recent Progress The construction of OA(3, 5, 4n + 2)s seems very
challenging. No concrete example of an OA(3, 5, 4n + 2)was found for over 60 years since the introduction of
OAs.
3. Existence of OA(3, 5, 4n + 2)s
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Construc-tions and
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contents
A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Recent Progress The construction of OA(3, 5, 4n + 2)s seems very
challenging. No concrete example of an OA(3, 5, 4n + 2)was found for over 60 years since the introduction of
OAs. As already noted, Euler conjecture remains open for
over a century. This shows the difficulty of theproblem, since an OA(3, 5, 4n + 2) implies the existenceof an OA(2, 4, 4n + 2) and much more. One reason forthe difficulty is that when using the known powerfulrecursions, one lacks of the small OAs to start with.
3. Existence of OA(3, 5, 4n + 2)s
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A Brief
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Recent Progress (Contd.)
Let me now turn to our recent paper:Yin-Wang-Ji-Li, J. Combin. Theory-A 118(2011),in which infinite many OA(3, 5, 4n + 2)s areconstructed. Among them, the smallest one is anOA(3, 5, 250).
3. Existence of OA(3, 5, 4n + 2)s
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Design-theoretic Background
Suppose that A is an OA(t,k,v) over symbol set Vand B an OA(t,k,w) over symbol set W, where W isa subset of V and B is a subarray of A
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Design-theoretic Background
Suppose that A is an OA(t,k,v) over symbol set Vand B an OA(t,k,w) over symbol set W, where W isa subset of V and B is a subarray of A
We say that the array obtained by removing B from Ais an IOA(t,k, (v, w)) (here the prefix I stands forincomplete). We also call W a hole of the IOA. Infact, the missing subarray need not exists. Clearly, the
number of columns of the IOA is (vt wt).
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A Brief
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Example 3.1An IOA(2,3, (4,2)) over Z4:
3 1 0 0 1 1 2 1 3 2 3 33 1 1 3 2 3 3 0 0 1 1 2
2 2 3 1 1 0 3 3 1 1 0 3 ,
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Example 3.1An IOA(2,3, (4,2)) over Z4:
3 1 0 0 1 1 2 1 3 2 3 33 1 1 3 2 3 3 0 0 1 1 2
2 2 3 1 1 0 3 3 1 1 0 3 ,
where the missing subarray is based on {0, 2} and is asfollows: 2 0 0 22 0 2 0
0 0 2 2
.
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Design-theoretic Background (Contd.)
Given that an OA(t,k,v) A over symbol set V, anyk vt1 subarray of A is referred to as a fan, if it
forms an OA(t 1, k , v).
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contents
A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Design-theoretic Background (Contd.)
Given that an OA(t,k,v) A over symbol set V, anyk vt1 subarray of A is referred to as a fan, if it
forms an OA(t 1, k , v).
Any k v subarray of A is termed a parallel class, ifeach row of the subarray forms a permutation ofsymbols of V.
O th l
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Construction Approach
Theorem 3.1 Suppose that A is an s-fan OA(3, k , g) wherethe s fans share a parallel class P in common and thecolumns not in P are pairwise distinct. Let non-negativeintegers m and mi (1 i s) be given. Write
w = si=1 mi. Suppose that further there exist(1) an OA(3, k , m);
(2) an IOA*(3, k, (m + mi, mi)) for 1 i s;
(3) an IOA(3, k, (m + w, w));
(4) an OA(3, k , m + w).Then there exists an OA(3,k,mg + w) that contains anOA(3, k , m + w) as a subarray.
O th l
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A Brief
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Remark to the Existence Applying Theorem 3.1 with an exhausted computer
search for ingredients, we obtain the first bulk ofconcrete examples of OA(3, 5, 4n + 2)s over 60 yearsmentioned earlier. The results provide infinitely manycounter-examples of Eulers and MacNeishs conjecturesin a stronger version.
Orthogonal
3. Existence of OA(3, 5, 4n + 2)s
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contents
A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Remark to the Existence Applying Theorem 3.1 with an exhausted computer
search for ingredients, we obtain the first bulk ofconcrete examples of OA(3, 5, 4n + 2)s over 60 yearsmentioned earlier. The results provide infinitely manycounter-examples of Eulers and MacNeishs conjecturesin a stronger version.
Remark that Blanchard (1995) and Wilson (2009)established an asymptotic existence of an OA(t,k,nqd)
with q a prime power, where q and d are required to besufficiently large (not specified).
Orthogonal
4. Nested OAs
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A Brief
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OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
4.1 Definition and Research Motivation
Currently, multiple experiments with different levels ofaccuracy and varying computational times, called nestedspace-filling designs, have received attention of statisticians.
See, for example,
Aloke, Discrete Math. 310 (2010);
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Orthogonal
4. Nested OAs
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OrthogonalArrays:
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contents
A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
4.1 Definition and Research Motivation
Currently, multiple experiments with different levels ofaccuracy and varying computational times, called nestedspace-filling designs, have received attention of statisticians.
See, for example,
Aloke, Discrete Math. 310 (2010);
Mukerjee et al., Discrete Math. 308 (2008);
Qian et al., The Annals of Statistics 37 ( 2009).
Orthogonal
4.1 Research Motivation
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
Consider an experimental setup that consists of twoexperiments, the expensive one of higher accuracy beingnested in a larger and relatively less expensive one of loweraccuracy. For example, the higher and lower accuracy
experiments can correspond to a physical versus a computerexperiment, or a detailed versus an approximate computerexperiment. The primary combinatorial object usedto generate aforementioned experiments is nested
OAs, that is, an OA containing a special subarray.
Orthogonal
4.2 Terminology
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
A nested OA, denoted by OA(, )(t,k, (v, w)), is anOA(t,k,v) having an OA(t,k,v) as a subarray.
Orthogonal
4.2 Terminology
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence ofOA(3, 5, 4n+2)s
Nested OAs
A nested OA, denoted by OA(, )(t,k, (v, w)), is anOA(t,k,v) having an OA(t,k,v) as a subarray.
It is obvious that the index of the subarray in anested OA cannot exceed the index of the largerarray.
Orthogonal
4.2 Terminology
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Whenever = , we simply write OA(t,k, (v, w)), foran OA(, )(t,k, (v, w)). If = 1, then the notationOA(t,k, (v, w)) is employed.
OrthogonalA
4.2 Terminology
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Whenever = , we simply write OA(t,k, (v, w)), foran OA(, )(t,k, (v, w)). If = 1, then the notationOA(t,k, (v, w)) is employed.
Remarkably, an OA(t,k, (v, w)) is closely related to anIOA(t,k, (v, w)) mentioned earlier. By simplyembedding an OA(t,k,w) (when it exists) into the holeof an IOA(t,k, (v, w)), one obtains an OA(t,k, (v, w))).
OrthogonalA
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
OA/IOA(t,k, (v, w))s are not only of practical use in
designing experiments, but also of significance in theircombinatorial interest. They are widely employed in theconstruction of designs. For instance, the well-known
Wilsons technique of constructing MOLS uses IOAs.
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
OA/IOA(t,k, (v, w))s are not only of practical use in
designing experiments, but also of significance in theircombinatorial interest. They are widely employed in theconstruction of designs. For instance, the well-known
Wilsons technique of constructing MOLS uses IOAs. The most important case = 1 for strength t = 2 has
been extensively studied in design theory, under thenames incomplete transversal designs and mutually
orthogonal Latin squares. We collect some knownexistence results below.
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
An OA(2, 3, (v, w)) exists iff v 2w Evans (1960).
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
An OA(2, 3, (v, w)) exists iff v 2w Evans (1960).
An IOA(2, 4, (v, w)) exists iff v 3w and (v, w) = (6, 1)= an OA(2, 4, (v, w)) if w {2, 6}Heinrich and Zhu (1985).
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
An IOA(2, 5, (v, w)) exists iff v 4w except when(v, w) = (6, 1) and possibly when (v, w) = (10, 1)= an OA(2, 5, (v, w)) if w {2, 3, 6, 10}
Abel, Colbourn, Yin and Zhang (1997).
OrthogonalArrays:
4.3 Some Progress to Strength 2
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yConstruc-tions and
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
An IOA(2, 5, (v, w)) exists iff v 4w except when(v, w) = (6, 1) and possibly when (v, w) = (10, 1)= an OA(2, 5, (v, w)) if w {2, 3, 6, 10}
Abel, Colbourn, Yin and Zhang (1997).
For k 6, the problem is far from completeColbourn-Dinitz (2007).
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Compared with strength 2, the existence ofOA(t,k, (v, w))s with t 3 is quite open.
OrthogonalArrays:
4.3 Some Progress to Strength 2
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Construc-tions and
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Compared with strength 2, the existence ofOA(t,k, (v, w))s with t 3 is quite open.
Maurin (1985) gave the numerically necessary condition
for the existence of an OA/IOA(t,k, (v, w)):v (k t + 1)w.
OrthogonalArrays:
4.3 Some Progress to Strength 2
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Compared with strength 2, the existence ofOA(t,k, (v, w))s with t 3 is quite open.
Maurin (1985) gave the numerically necessary condition
for the existence of an OA/IOA(t,k, (v, w)):v (k t + 1)w.
Quite recently, we found that the necessary condition isalso sufficient in the case k = t + 1 for any strength t.
OrthogonalArrays:
C
. e xistence pectrum oOA(t, t + 1, (v, w))s
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A Brief
IntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Theorem 4.1 [Wang-Yin (2011)]
Let t, w and v be positive integers. Then anOA(t, t + 1, (v, w)) exists if and only if v 2w.
OrthogonalArrays:
C t
4.5 Preliminary for the Proof of Theorem 4.1
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Let V be a set of v symbols. Let R(x1, x2, , xt) be at-ary operation defined on V which satisfies the uniquesolvability, that is, if values for any t variables are givenin the equation R(x1, x2, , xt) = xt+1, then the value
of the remaining variable is uniquely determined. Thepair (V, R) then is called a t-quasigroup of order v.
OrthogonalArrays:
Construc
4.5 Preliminary for the Proof of Theorem 4.1
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Let V be a set of v symbols. Let R(x1, x2, , xt) be at-ary operation defined on V which satisfies the uniquesolvability, that is, if values for any t variables are givenin the equation R(x1, x2, , xt) = xt+1, then the value
of the remaining variable is uniquely determined. Thepair (V, R) then is called a t-quasigroup of order v.
A t-quasigroup is called idempotent, if for any symbolx, R(x,x, , x) = x.
OrthogonalArrays:
Construc
4.5 Preliminary for the Proof of Theorem 4.1
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
When t = 2, it is just a quasigroup in the usual sense.When t = 1, R is just a bijection from V to V.Therefore, the image (y = R(x))xV is a permutationof symbols.
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OrthogonalArrays:
Construc-
4.5 Preliminary for the Proof of Theorem 4.1
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Constructions and
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A BriefIntroductiontoOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Teirlinck (1990) gave an concatenation construction ofquasigroups which we state in the following proposition.
OrthogonalArrays:
Construc-
4.5 Preliminary for the Proof of Theorem 4.1
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Existence of
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Nested OAs
Teirlinck (1990) gave an concatenation construction ofquasigroups which we state in the following proposition.
Proposition 4.1 Let V be an arbitrary set of v symbolsand W a fixed w-subset of V. Suppose that
(V, Ri) is a ti-quasigroup for i = 1, 2, , r and
t =
r
i=1ti
1.
OrthogonalArrays:
Construc-
4.5 Preliminary for the Proof of Theorem 4.1
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OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Teirlinck (1990) gave an concatenation construction ofquasigroups which we state in the following proposition.
Proposition 4.1 Let V be an arbitrary set of v symbolsand W a fixed w-subset of V. Suppose that
(V, Ri) is a ti-quasigroup for i = 1, 2, , r and
t =
r
i=1ti
1.
(V, R) is an (r 1)-quasigroup.
OrthogonalArrays:
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Proposition 4.1 (Contd.)
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Existence of
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Nested OAs
the notation [R1, , Rr; R] stands for the set of allcolumn vectors (x1, x2, , xt+1)
T Vt+1 such that(R1(x1, x2, , xt1 ), R2(xt1+1, xt1+2, , xt1+t2 ),
, Rr(x(
1ir1 ti)+1, , xt+1)) R.
OrthogonalArrays:
Construc-
Proposition 4.1 (Contd.)
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Existence of
OA(3, 5, 4n+2)s
Nested OAs
the notation [R1, , Rr; R] stands for the set of allcolumn vectors (x1, x2, , xt+1)
T Vt+1 such that(R1(x1, x2, , xt1 ), R2(xt1+1, xt1+2, , xt1+t2 ),
, Rr(x(
1ir1 ti)+1, , xt+1)) R.
Then (V, [R1, , Rr; R]) is a t-quasigroup. Further, if eachof the r + 1 quasigroups (V, Ri) (1 i r) and (V, R)contains a sub-quasigroup over the same subset W, then(V, [R
1, , R
r; R]) also contains a sub-quasigroup over W.
OrthogonalArrays:
Construc-i d
4.6 A Brief Outline of the Proof of Theorem 4.1
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Existence of
OA(3, 5, 4n+2)s
Nested OAs
From the previous discussion, we know that at-quasigroup (V, R) of order v and an OA(t, t + 1, v) are thetwo equivalent objects.
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OrthogonalArrays:
Construc-tions and
4.6 A Brief Outline of the Proof of Theorem 4.1
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OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Conversely, by definition, the property of an OA ispreserved if we change the order of its columns. Hence,we can regard an OA(t, t + 1, v) over V as a pair (V, R)where R is the set of vt column vectors of the array.The orthogonality of an OA guarantees that (V, R) is at-quasigroup of order v.
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OrthogonalArrays:
Construc-tions and
4.6 A Brief Outline of the Proof of Theorem 4.1
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Existence of
OA(3, 5, 4n+2)s
Nested OAs
Keeping the above equivalence in mind, we provedTheorem 4.1 by a careful application of the concatenation
construction given in Proposition. 4.1.
OrthogonalArrays:
Construc-tions and
4.7 Concluding Remarks
To my knowledge, the question of existence of anOA( )(t k (v w)) with < does not seem to have been
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Existence of
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Nested OAs
OA(, )(t,k, (v, w)) with < does not seem to have been
studied systematically. Motivated by designing nestedexperiments, Mukerjee et al. made an investigation in depthinto necessary conditions for the existence of nested OAs.
OrthogonalArrays:
Construc-tions and
4.7 Concluding Remarks
To my knowledge, the question of existence of anOA( )(t k (v w)) with < does not seem to have been
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A BriefIntroduction
toOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
OA(, )(t,k, (v, w)) with < does not seem to have been
studied systematically. Motivated by designing nestedexperiments, Mukerjee et al. made an investigation in depthinto necessary conditions for the existence of nested OAs.
Theorem 4.2 [Mukerjee et al. (2008)]
An OA(, )(t,k, (v, w)) can exists only if
vt
wtu
j=0
kj
(w1v 1)j , if t = 2u 2;
wt uj=0
kj (w1v 1)j + k1u (w1v 1)u+1 ,
if t = 2u + 1 3.
OrthogonalArrays:
Construc-tions and
4.7 Concluding Remarks
Aloke obtained some series of nested orthogonal arrays
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A BriefIntroduction
toOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Aloke obtained some series of nested orthogonal arrays.
OrthogonalArrays:
Construc-tions and
4.7 Concluding Remarks
Aloke obtained some series of nested orthogonal arrays
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A BriefIntroduction
toOrthogonalArrays
OAs andDifferenceMatrices
Existence of
OA(3, 5, 4n+2)s
Nested OAs
Aloke obtained some series of nested orthogonal arrays.
Theorem 4.3 [Aloke (2010)]
Let v > 2 be a power of 2. Then there exist
1 an OA(t, t + 1, (v, 2)) with t 2;
2 an OA(2, 3, (v, 4)) when v > 4;3 an OA(vu3, 2u3)(3, 2u, (v, 2)), where u 4 is an
integer;
4 an OA(v, 2)(4, 6, (v, 2));
5 an OA(2, 1)(2, v, (v, v 1)) provided that v > 3 so thatv 1 and v + 1 are both prime powers.
OrthogonalArrays:
Construc-tions and
4.7 Concluding Remarks
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Existence of
OA(3, 5, 4n+2)s
Nested OAs
We advanced the existence by proving the followingresults.
OrthogonalArrays:
Construc-tions and
R l d
4.7 Concluding Remarks
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A BriefIntroduction
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OAs andDifferenceMatrices
Existence of
OA(3,
5,
4n
+2)s
Nested OAs
We advanced the existence by proving the followingresults.
Theorem 4.4 [Wang-Yin (2011)]
If a (v + 1, k , )-DM containing columns of zeros and anOA(2, k + 1, v 1) both exist, then there exists anOA(2, )(2, k + 1, (v, v 1)).
OrthogonalArrays:
Construc-tions and
R l t d
4.7 Concluding Remarks
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OAs andDifferenceMatrices
Existence of
OA(3,
5,
4n
+2)s
Nested OAs
Theorem 4.5 [Wang-Yin (2011)]
Let v be an positive integer. Then
1 there exists an OA(2, 1)(2, 5, (v, v 1)), if v 5,v + 1 2 (mod 4) and v 1 {6, 10};
2 there exists an OA(2, 1)(2, 6, (v, v 1)); if v + 1 7 isodd, gcd(v + 1, 27) = 9 and v 1 {6, 10, 14, 18, 22};
3 there exist an OA(4, 1)(3, 6, (3q, q)) and anOA(2, 1)(3, 6, (6q, q)) for any prime power q 4.
OrthogonalArrays:
Construc-tions and
R l t d
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A BriefIntroduction
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OAs andDifferenceMatrices
Existence of
OA(3,
5,
4n
+2)s
Nested OAs
Thanks !