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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

[email protected]

May 26, 2011


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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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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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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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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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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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The Basic Question

The basic question concerning OAs is: Does an OAexist for given parameters ?


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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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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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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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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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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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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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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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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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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.



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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.



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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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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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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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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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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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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.



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The induced sumu

G G of u copies of G iswritten as Gu.



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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.



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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).



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Any k (c/v) subarray of D is called its a fan, if itforms a DM

(t 1, k , v).



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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.



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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;



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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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Proposition 2.1 If a DM(t,k,v) exists, then so does anOA(t,k,v).



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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.



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Proposition 2.2 If a completely divisible DM(t,k,v)exists, then we have an OA(t, k + 1, v).



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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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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

.



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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

,



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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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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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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.



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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.



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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



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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

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



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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).



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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).



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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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Open Problem 1

Construct a DM(2, 4, 3p) with an adder for any primep 11.



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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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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.



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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.



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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).



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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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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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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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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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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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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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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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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



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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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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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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




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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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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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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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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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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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.

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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.

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Existence of

OA(3, 5, 4n+2)s

Nested OAs

An OA(2, 3, (v, w)) exists iff v 2w Evans (1960).

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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).

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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).

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yConstruc-tions and

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A Brief



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).

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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:



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Construc-tions and

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Existence of

OA(3, 5, 4n+2)s

Nested OAs


Maurin (1985) gave the numerically necessary condition

for the existence of an OA/IOA(t,k, (v, w)):v (k t + 1)w.

OrthogonalArrays:



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Existence of

OA(3, 5, 4n+2)s

Nested OAs


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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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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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.

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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.

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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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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:

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Existence of

OA(3, 5, 4n+2)s

Nested OAs


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.

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Existence of

OA(3, 5, 4n+2)s

Nested OAs


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:

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.

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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A BriefIntroduction

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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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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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A BriefIntroduction

toOrthogonalArrays


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.

OrthogonalArrays:

Construc-tions and


To my knowledge, the question of existence of anOA( )(t k (v w)) with < does not seem to have been


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Related

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contents

A BriefIntroduction

toOrthogonalArrays


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


Aloke obtained some series of nested orthogonal arrays


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A BriefIntroduction

toOrthogonalArrays


Existence of

OA(3, 5, 4n+2)s

Nested OAs

Aloke obtained some series of nested orthogonal arrays.

OrthogonalArrays:

Construc-tions and


Aloke obtained some series of nested orthogonal arrays


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A BriefIntroduction

toOrthogonalArrays


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:

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A BriefIntroduction

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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



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A BriefIntroduction

toOrthogonalArrays


Existence of

OA(3,

5,

4n

+2)s

Nested OAs

We advanced the existence by proving the followingresults.


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



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A BriefIntroduction

toOrthogonalArrays


Existence of

OA(3,

5,

4n

+2)s

Nested OAs


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

toOrthogonalArrays


Existence of

OA(3,

5,

4n

+2)s

Nested OAs

Thanks !

7shcc yin jianxing

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