Journal ArticleDOI

# Parameter Inequalities for Orthogonal Arrays with Mixed Levels

01 Nov 2004-Designs, Codes and Cryptography (Kluwer Academic Publishers)-Vol. 33, Iss: 3, pp 187-197
TL;DR: This paper will provide a generalization of an inequality developed by Bierbrauer for symmetric orthogonal arrays and utilize his algebraic approach to provide an analogous inequality for orthogsonal arrays having mixed levels and show that the bound obtained in this fashion is often sharper than Rao’s bounds.
Abstract: An important question in the construction of orthogonal arrays is what the minimal size of an array is when all other parameters are fixed. In this paper, we will provide a generalization of an inequality developed by Bierbrauer for symmetric orthogonal arrays. We will utilize his algebraic approach to provide an analogous inequality for orthogonal arrays having mixed levels and show that the bound obtained in this fashion is often sharper than Rao’s bounds. We will also provide a new proof of Rao’s inequalities for arbitrary orthogonal arrays with mixed levels based on the same method.

### 1 Introduction

• The authors will use |A| to denote the cardinality of a multiset A, taking into account how often each of its element occurs.
• (Such arrays are also often referred to as asymmetric orthogonal arrays.).

### 2 A bound for mixed-level arrays

• (1) Note that this is the dual of the Plotkin bound in the sense that upper bounds on the minimum distance of codes yields lower bounds on the size of orthogonal arrays.
• In addition to being much easier to compute than Rao’s bound, the inequality (1) leads to a larger minimum value for N than Rao’s bound for a number of parameter combinations.
• The authors will adapt his method to provide the following generalization of Theorem 2.1 for orthogonal arrays with mixed levels.
• The authors have not been able to determine the exact conditions under which the bound given in (2) is superior to Rao’s bounds.
• As is the case for symmetric orthogonal arrays, it is possible that even though the bound in (2) is greater than Rao’s bound, both still lead to the same minimum value of N .

### If t is odd, then

• (20) The proof of Theorem 3.1 indicated in Rao’s paper is valid only for simple orthogonal arrays (i.e. arrays that have no repeated columns) and is carried out in Beder [1].
• Bierbrauer [2] provided a proof for Rao’s inequalities in the symmetric case that is based on his technique for developing the lower bound in Theorem 2.1.
• The authors will extend Bierbrauer’s method to prove Rao’s inequalities for arrays with mixed levels.
• This proof also does not require simplicity.
• Thus for z ∈ Z0, the functions fz are pairwise orthogonal in L2(P ) and therefore linearly independent.

### 4 Concluding remarks

• The lower bound in (2) appears to be sharper than Rao’s bounds for a significant number of parameter combinations.
• Whether a mixed-level orthogonal array for a given parameter combination and having the minimum size prescribed by the bound actually exists is a question that has yet to be answered.
• Both Rao’s bounds and the linear programming bound for mixed-level orthogonal arrays developed in Sloane and Stufken [7] are much more difficult to compute than the one developed in this paper.
• A general comparison of the different bounds awaits further investigation.

### Acknowledgments

• I would like to thank the referees for a number of suggestions that greatly improved this paper.

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Parameter inequalities for orthogonal arrays with mixed levels
Wiebke S. Diestelkamp
Department of Mathematics
University of Dayton
Dayton, OH 45469-2316
wiebke@udayton.edu
Designs, Co des and Cryptography, Vol. 33 (2004), 187-197
Abstract
An important question in the construction of orthogonal arrays is what the minimal size of an
array is when all other parameters are ﬁxed. In this paper, we will provide a generalization
of an inequality developed by Bierbrauer for symmetric orthogonal arrays. We will utilize his
algebraic approach to provide an analogous inequality for orthogonal arrays having mixed levels
and show that the bound obtained in this fashion is often sharper than Rao’s bounds. We will
also provide a new pro of of Rao’s inequalities for arbitrary orthogonal arrays with mixed levels
based on the same method.
Key words. Mixed-level orthogonal array, group character, adjacency op erator.
AMS Subject Classiﬁcation: Primary 05B15, Secondary 62K15
1 Introduction
Among the ﬁrst lower bounds on the size of an orthogonal array were those developed by Rao
[5] for symmetric orthogonal arrays, and in Rao [6] for orthogonal arrays with mixed levels.
Various other bounds have been developed since then; see Hedayat et al. [3] for a survey of the
diﬀerent inequalities that are known thus far. Bierbrauer [2] introduced a lower bound on the
size of a symmetric orthogonal array which he found to be better than Rao’s for some values
of t (the strength of the array), although he provides no speciﬁcs about the kind of arrays for
which this would be the case. We will provide some examples of parameter combinations for
symmetric orthogonal arrays for which his bounds are better than Rao’s in Section 2.
We will then prove a generalization of Bierbrauer’s inequality for mixed-level orthogonal
arrays. The proof is based on Bierbrauer’s proof in the symmetric case. We will provide some
examples for parameter combinations for orthogonal arrays for which this inequality results in
better bounds on the size of the array than Rao’s inequalities do.
In the same paper, Bierbrauer also demonstrated how his method could be used to prove
Rao’s inequalities for symmetric orthogonal arrays. In Section 3, we provide an analogous
1

proof for Rao’s inequalities for mixed-level orthogonal arrays. Rao [6] suggests a method of
proof in which, as pointed out in Beder [1], only applies to simple arrays (i.e., arrays with no
repeated elements). Our pro of does not depend on the simplicity of the array and thus holds
for arbitrary orthogonal arrays with mixed levels.
We will use |A| to denote the cardinality of a multiset A, taking into account how often
each of its element occurs. For example, if A = {a, a, b, c}, then |A| = 4. Since any set is also a
multiset, we will use the same notation for the cardinality of a set. We will denote the multiset
consisting of n copies of A by nA. That is, for A as above, 2A = {a, a, a, a, b, b, c, c}. Finally,
Z
s
will denote the additive cyclic group of integers modulo s.
Deﬁnition 1.1 Let A
1
, . . . , A
k
be sets with s
1
, . . . , s
k
elements, respectively. An orthogo-
nal array of size N , k constraints (or factors), s
1
, s
2
, . . . , s
k
levels and strength t, denoted
OA(N, s
1
, . . . , s
k
, t), is a multiset O on A
1
× · · · × A
k
of cardinality N such that the following
holds: For any set I = {i
1
, . . . , i
t
} {1, . . . , k} there exists a number λ
I
such that the projection
of O onto A
i
1
× · · · × A
i
t
is the multiset λ
I
(A
i
1
× · · · × A
i
t
).
When the s
i
are not all equal, an orthogonal array of the form OA(N, s
1
, . . . , s
k
, t) is said to
have mixed levels. (Such arrays are also often referred to as asymmetric orthogonal arrays.) If
s
1
= . . . = s
k
= s, we will denote the array by OA(N, k, s, t) and call it a symmetric orthogonal
array.
In the proofs provided in this paper, we will make use of the fact that we can view an
orthogonal array O of the form OA(N, s
1
, . . . , s
k
, t) as a function on G = A
1
× · · · × A
k
.
Namely, for x G, O(x) equals the number of times x occurs in O.
Throughout the paper, we will assume that the sets A
i
are additive groups and utilize the
following deﬁnition:
Deﬁnition 1.2 Let x, y A
1
× · · · × A
k
. The Hamming weight of x, denoted wt(x), is the
number of nonzero components of x. We say that x and y are neighbors, denoted x _ y, if
wt(x y) = 1.
2 A bound for mixed-level arrays
Bierbrauer [2] provided the following bound on the size N of a symmetric orthogonal array:
Theorem 2.1 Assume that an orthogonal array of the form OA(N, k, s, t) exists. Then
N s
k
µ
1
(s 1) k
s (t + 1)
. (1)
Note that this is the dual of the Plotkin bound in the sense that upper bounds on the
minimum distance of codes yields lower bounds on the size of orthogonal arrays.
In addition to being much easier to compute than Rao’s bound, the inequality (1) leads
to a larger minimum value for N than Rao’s bound for a number of parameter combinations.
(Rao’s b ounds are stated in Section 3, Theorem 3.1.) A few examples are given in Table 1.
2

s = 3: t = 3, k = 5 s = 4: t = 4, k = 6
t = 4, k = 6, 7 t = 5, k = 7
t = 5, k = 7, 8 t = 6, k = 8 , 9
t = 6, k = 9, 10 t = 7, k = 9 , 10
t = 7, k = 9, 10, 11 t = 8, k = 10, 11
t = 8, k = 10, 11, 12 t = 9, k = 11, 12, 13
t = 9, k = 11, 12, 13, 14 t = 10, k = 12, 13, 14
t = 10, k = 12, 13, 14, 15, 16
Table 1: Examples of parameter combinations for OA(N, k, s, t) for
which the bound in (2) is sharp er than Rao’s.
Note that it is possible that Rao’s bounds and the bound derived from (1) take diﬀerent
values, but still lead to the same minimum value for N , since N also has to be a multiple of
s
t
. All examples listed in Table 1 are instances where the inequality in (1) leads to a greater
minimum value for N than Rao’s bounds. Early computational results suggest that (1) yields
a minimum value for N that is at least as good as Rao’s whenever s > 2 and t < k <
t(s+1)
s1
.
Bierbrauer [2] outlined an algebraic proof in which he regards the orthogonal array as a mul-
tiset on the group G = (Z
s
)
k
. We will adapt his method to provide the following generalization
of Theorem 2.1 for orthogonal arrays with mixed levels.
Theorem 2.2 Assume that an orthogonal array of the form OA(N, s
1
, . . . , s
k
, t) exists. Let
s
T
= s
1
+ . . . + s
k
, s
M
= max
j
(s
j
) and s
m
= min
j
(s
j
). Then
N (s
m
)
k
µ
1
s
T
k
s
T
+ (t k + 1)s
M
. (2)
Note that letting s = s
1
= · · · = s
k
yields the inequality in (1).
We have not been able to determine the exact conditions under which the bound given in
(2) is superior to Rao’s bounds. As is the case for symmetric orthogonal arrays, it is possible
that even though the bound in (2) is greater than Rao’s bound, both still lead to the same
minimum value of N. This is due to the fact that the size of the array must be a multiple of
s
i
1
s
i
2
· · · s
i
t
whenever 0 < i
1
< . . . < i
t
k. This immediately leads to the condition
N LCM{s
i
1
s
i
2
· · · s
i
t
: 0 < i
1
< · · · < i
t
k}. (3)
(In fact, N is a multiple of the LCM.)
In Table 2, we list just a few examples of orthogonal arrays for which the bound in (2)
yields a larger minimum value for N than both Rao’s bounds and (3).
Proof of Theorem 2.2. Note that if
t + 1
(s
M
1) k
s
M
,
then
s
T
k
s
T
+ (t k + 1)s
M
1,
3

s
1
= 5, s
2
= . . . = s
k
= 4: s
1
= 4, s
2
= . . . = s
k
= 3: s
1
= 3, s
2
= . . . = s
k
= 2:
k = 15, t = 12 k = 15, t = 11, 12 k = 19, t = 14
k = 16, t = 12, 13 k = 16, t = 12, 13 k = 20, t = 14, 15, 16
k = 17, t = 13, 14 k = 17, t = 12, 13, 14 k = 21, t = 14, 15, 16
k = 18, t = 14, 15 k = 18, t = 13, 14, 15 k = 22, t = 15, 16, 17, 18
k = 19, t = 15, 16 k = 19, t = 14, 15, 16 k = 23, t = 16, 17, 18
k = 20, t = 16, 17 k = 20, t = 15, 16, 17 k = 24, t = 16, 17, 18, 19, 20
k = 21, t = 16, 17, 18 k = 21, t = 15, 16, 17, 18 k = 25, t = 16, 17, 18, 19, 20
k = 22, t = 17, 18, 19 k = 22, t = 16, 17, 18, 19
k = 23, t = 18, 19, 20 k = 23, t = 17, 18, 19, 20
k = 24, t = 19, 20, 21 k = 24, t = 18, 19, 20, 21
k = 25, t = 20, 21, 22 k = 25, t = 18, 19, 20, 21, 22
Table 2: Examples of parameter combinations for OA(N, k, s
1
, . . . , s
k
, t) for which the
bound in (2) is sharper than b oth Rao’s bound and the condition in (3).
and inequality (2) is trivially true. Thus let us assume that
t + 1 >
(s
M
1) k
s
M
. (4)
Now, let O be an orthogonal array of the form OA(N, s
1
, . . . , s
k
, t), and let G = Z
s
1
× · · · ×
Z
s
k
. Then O can be viewed as a multiset on the additive group G. Let ς
i
be a primitive s
i
-th
root of unity (note that ς
i
is a complex number). For each z
i
Z
s
i
, the function φ
z
i
(g) = ς
gz
i
i
deﬁnes a character on the additive group Z
s
i
. The characters of G are then the functions
φ
z
(x) =
k
Y
i=1
φ
z
i
(x
i
) =
k
Y
i=1
ς
z
i
x
i
i
,
where z = (z
1
, . . . , z
k
) G and x = (x
1
, . . . , x
k
) G (cf. Ledermann [4]).
Consider the space L
2
(G) of complex-valued functions on G with scalar product h , i given
by
hf, gi =
1
|G|
X
xG
f(x)g ( x).
It is a well-known fact that the characters of G form an orthonormal basis of L
2
(G) (see [4]).
We will view O as a function on G as described in Section 1. Then we have
O (x) =
X
zG
µ
z
φ
z
(x) ,
where µ
z
C, and
µ
z
= hO, φ
z
i =
1
|G|
X
xG
O(x)
k
Y
i=1
ς
x
i
z
i
i
. (5)
Here the µ
z
are the Fourier coeﬃcients of O with respect to the orthonormal family {φ
z
: z G}.
4

Now ﬁx z G with wt(z) = t, and let I = {i
1
, . . . , i
t
} {1, . . . , k} be the index set
corresponding to the nonzero components of z. Let G
0
= Z
s
i
1
× · · · × Z
s
i
t
, and let π be the
projection of G onto G
0
. Denote by O
0
the orthogonal array derived from O by π as follows:
for all y G
0
,
O
0
(y) =
X
π(x)=y
O(x),
where the sum is taken over all x O for which π(x) = y. Then
µ
z
=
1
|G|
X
yG
0
X
π(x)=y
O(x)
Y
I
ς
y

z


=
1
|G|
X
yG
0
O
0
(y)
Y
I
ς
y

z


.
Since O has strength t, O
0
consists of all t-tuples (x
i
1
, . . . , x
i
t
) with i
j
I j, each occurring
λ
I
times. Thus we have O
0
(y) = λ
I
for all y G
0
, and so
µ
z
=
λ
I
|G|
X
yG
0
Y
I
ς
y

z


.
The function
Y
I
ς
y

z


of y = (y
i
1
, . . . , y
i
t
)
0
is a nontrivial character of G
0
, and thus
X
yG
0
Y
I
ς
y

z


= 0,
using the fact that
X
xH
χ(x) = 0 for all nontrivial characters χ of a ﬁnite abelian group H. (6)
Thus µ
z
= 0 when wt(z) = t.
Since O has strength d for every d {1, . . . , t}, we must have
µ
z
= 0 whenever 1 wt(z) t. (7)
Deﬁne the adjacency operator A : L
2
(G) L
2
(G) by
Af(x) =
X
y_x
f(y)
for f L
2
(G), x G. Then for any z G,
z
(x) =
X
y_x
φ
z
(y) =
k
X
j=1
X
yB
j,x
φ
z
(y), (8)
where B
j,x
= {y G : x and y diﬀer in only the j-th component} .
5

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##### References
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Book
22 Jun 1999
TL;DR: The Rao Inequalities for Mixed Orthogonal Arrays., 9.2 The Rao InEqualities for mixed Orthogonic Arrays.- 9.4 Construction X4.- 10.1 Constructions Inspired by Coding Theory.
Abstract: 1 Introduction.- 1.1 Problems.- 2 Rao's Inequalities and Improvements.- 2.1 Introduction.- 2.2 Rao's Inequalities.- 2.3 Improvements on Rao's Bounds for Strength 2 and 3.- 2.4 Improvements on Rao's Bounds for Arrays of Index Unity.- 2.5 Orthogonal Arrays with Two Levels.- 2.6 Concluding Remarks.- 2.7 Notes on Chapter 2.- 2.8 Problems.- 3 Orthogonal Arrays and Galois Fields.- 3.1 Introduction.- 3.2 Bush's Construction.- 3.3 Addelman and Kempthorne's Construction.- 3.4 The Rao-Hamming Construction.- 3.5 Conditions for a Matrix.- 3.6 Concluding Remarks.- 3.7 Problems.- 4 Orthogonal Arrays and Error-Correcting Codes.- 4.1 An Introduction to Error-Correcting Codes.- 4.2 Linear Codes.- 4.3 Linear Codes and Linear Orthogonal Arrays.- 4.4 Weight Enumerators and Delsarte's Theorem.- 4.5 The Linear Programming Bound.- 4.6 Concluding Remarks.- 4.7 Notes on Chapter 4.- 4.8 Problems.- 5 Construction of Orthogonal Arrays from Codes.- 5.1 Extending a Code by Adding More Coordinates.- 5.2 Cyclic Codes.- 5.3 The Rao-Hamming Construction Revisited.- 5.4 BCH Codes.- 5.5 Reed-Solomon Codes.- 5.6 MDS Codes and Orthogonal Arrays of Index Unity.- 5.7 Quadratic Residue and Golay Codes.- 5.8 Reed-Muller Codes.- 5.9 Codes from Finite Geometries.- 5.10 Nordstrom-Robinson and Related Codes.- 5.11 Examples of Binary Codes and Orthogonal Arrays.- 5.12 Examples of Ternary Codes and Orthogonal Arrays.- 5.13 Examples of Quaternary Codes and Orthogonal Arrays.- 5.14 Notes on Chapter 5.- 5.15 Problems.- 6 Orthogonal Arrays and Difference Schemes.- 6.1 Difference Schemes.- 6.2 Orthogonal Arrays Via Difference Schemes.- 6.3 Bose and Bush's Recursive Construction.- 6.4 Difference Schemes of Index 2.- 6.5 Generalizations and Variations.- 6.6 Concluding Remarks.- 6.7 Notes on Chapter 6.- 6.8 Problems.- 7 Orthogonal Arrays and Hadamard Matrices.- 7.1 Introduction.- 7.2 Basic Properties of Hadamard Matrices.- 7.3 The Connection Between Hadamard Matrices and Orthogonal Arrays.- 7.4 Constructions for Hadamard Matrices.- 7.5 Hadamard Matrices of Orders up to 200.- 7.6 Notes on Chapter 7.- 7.7 Problems.- 8 Orthogonal Arrays and Latin Squares.- 8.1 Latin Squares and Orthogonal Latin Squares.- 8.2 Frequency Squares and Orthogonal Frequency Squares.- 8.3 Orthogonal Arrays from Pairwise Orthogonal Latin Squares.- 8.4 Concluding Remarks.- 8.5 Problems.- 9 Mixed Orthogonal Arrays.- 9.1 Introduction.- 9.2 The Rao Inequalities for Mixed Orthogonal Arrays.- 9.3 Constructing Mixed Orthogonal Arrays.- 9.4 Further Constructions.- 9.5 Notes on Chapter 9.- 9.6 Problems.- 10 Further Constructions and Related Structures.- 10.1 Constructions Inspired by Coding Theory.- 10.2 The Juxtaposition Construction.- 10.3 The (u, u + ?) Construction.- 10.4 Construction X4.- 10.5 Orthogonal Arrays from Union of Translates of a Linear Code.- 10.6 Bounds on Large Orthogonal Arrays.- 10.7 Compound Orthogonal Arrays.- 10.8 Orthogonal Multi-Arrays.- 10.9 Transversal Designs, Resilient Functions and Nets.- 10.10 Schematic Orthogonal Arrays.- 10.11 Problems.- 11 Statistical Application of Orthogonal Arrays.- 11.1 Factorial Experiments.- 11.2 Notation and Terminology.- 11.3 Factorial Effects.- 11.4 Analysis of Experiments Based on Orthogonal Arrays.- 11.5 Two-Level Fractional Factorials with a Defining Relation.- 11.6 Blocking for a 2k-n Fractional Factorial.- 11.7 Orthogonal Main-Effects Plans and Orthogonal Arrays.- 11.8 Robust Design.- 11.9 Other Types of Designs.- 11.10 Notes on Chapter 11.- 11.11 Problems.- 12 Tables of Orthogonal Arrays.- 12.1 Tables of Orthogonal Arrays of Minimal Index.- 12.2 Description of Tables 12.1?12.3.- 12.3 Index Tables.- 12.4 If No Suitable Orthogonal Array Is Available.- 12.5 Connections with Other Structures.- 12.6 Other Tables.- Appendix A: Galois Fields.- A.1 Definition of a Field.- A.2 The Construction of Galois Fields.- A.3 The Existence of Galois Fields.- A.4 Quadratic Residues in Galois Fields.- A.5 Problems.- Author Index.

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### "Parameter Inequalities for Orthogon..." refers background in this paper

• ...An alternate proof that does not require the array to be simple is suggested in Hedayat et al. [ 3 ]....

[...]

• ...Various other bounds have been developed since then; see Hedayat et al. [ 3 ] for a survey of the dierent inequalities that are known thus far....

[...]

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441 citations

### "Parameter Inequalities for Orthogon..." refers background in this paper

• ...1 Introduction Among the first lower bounds on the size of an orthogonal array were those developed by Rao [5] for symmetric orthogonal arrays, and in Rao [6] for orthogonal arrays with mixed levels....

[...]

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• ...The proof is based on Bierbrauer’s proof in the symmetric case....

[...]

• ...Bierbrauer [2] outlined an algebraic proof in which he regards the orthogonal array as a multiset on the group G = (Zs)....

[...]

• ...We will extend Bierbrauer’s method to prove Rao’s inequalities for arrays with mixed levels....

[...]

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