Unbiased orthogonal designs

0.21 MB·

by  Hadi Kharaghani

Unbiased orthogonal designs
Hadi Kharaghani∗ Sho Suda†
January 19, 2016
6
1
0
2
Abstract

n
a Thenotion of unbiased orthogonal designs is introduced as a general-
J ization among unbiased Hadamard matrices, unbiased weighing matrices

8 andquasi-unbiasedweighingmatrices. Weprovideupperboundsandsev-
1 eralconstructionsformutuallyunbiasedorthogonaldesigns. Asanappli-
cation,mutuallyquasi-unbiasedweighingmatricesforvariousparameters

] are obtained.
O
C
1 Introduction
.
h
t
a A Hadamard matrix of order n is ann n (1, 1)-matrixH such that HH⊤ =
m nI ,whereH⊤ denotesthetransposeof×H and−I denotestheidentitymatrixof
n n
[ ordern. Aweighing matrix of order n and weight k isann n(0,1, 1)-matrix
W suchthatWW⊤ =kI . RecentlyunbiasedHadamardm×atricesan−dunbiased
1 n
v weighingmatriceshavebeenstudied[3,10,13]. TwoHadamardmatricesH and
6 K ofordern are calledunbiased ifHK⊤ =√nL for some Hadamardmatrix L.
1 Two weighing matrices H and K of order n and weight k are called unbiased if
4 HK⊤ =√kLforsomeweighingmatrixLofweightk[3,10]. Mutuallyunbiased
4
weighing matrices of weight 4 naturally arise in the minimum vectors of root
0
. lattices admitting a decomposition of disjoint orthogonal bases [14, Theorem
1
3.5]. In the paper [3], Best, Kharaghani and Ramp posed the question for a
0
6 constructionof22t+1HadamardmatricesH1,...,H22t+1 oforder22t+1suchthat
1 the entries of HiHj⊤ are 0,±2t+1 for any distinct i,j. In order to answer their
: question and consider more general situations, the concept of quasi-unbiased
v
i weighing matrices was given in [14], see Section 2 for the definition. An answer
X
wasobtainedbyconsideringtheBCHcodeswithcosetsbythefirstorderReed-
r Muller code[14,Theorem4.4]. Bothobjectsarerelatedtoa spreadinapartial
a
geometry[4]orinastronglyregulargraph[9]andyieldasymmetricassociation
scheme [18].
A generalized concept for a Hadamard matrix and a weighing matrix is an
orthogonal design, see Section 2 for the definition. In this paper, as a unify-
ingwaytostudy unbiasedHadamardmatrices,unbiasedweighingmatricesand
quasi-unbiased weighing matrices, the concept of unbiased orthogonal designs
∗DepartmentofMathematicsandComputerScience,UniversityofLethbridge,Lethbridge,
Alberta,T1K3M4,Canada. [email protected]
†DepartmentofMathematicsEducation,AichiUniversityofEducation,1Hirosawa,Igaya-
cho,Kariya,Aichi448-8542, Japan. [email protected]
1
is introduced. Connecting unbiased orthogonal designs with unbiased weigh-
ing matrices, we obtain the upper bound for the number of mutually unbiased
orthogonal designs. We provide various constructions of unbiased orthogonal
designs to use direct sum and tensor product for matrices, mutually suitable
Latin squares.
The main result of the paper is Theorem 4.8, which constructs mutually
unbiased orthogonaldesigns from a weighing matrix and an orthogonaldesign.
The significance of the construction provided here is that any weighing matrix
and any orthogonal design of the same order can be used to construct unbi-
ased orthogonaldesigns. Furthermore we demonstrate how the plug-in method
provides mutually quasi-unbiased weighing matrices from unbiased orthogonal
designs with Goethals-Seidel matrices and Williamson type matrices.
The organization of the paper is as follows. In Section 2 we prepare some
notations ans results needed later. In Section 3 we introduce the concept of
unbiased orthogonal designs, and extend constructions for unbiased Hadamard
matricesandrelatedtopicstounbiasedorthogonaldesigns. InSection4wepro-
vide a new construction of quasi-unbiased weighing matrices from a finite ring
with unity. By use of this construction, we obtain mutually unbiased orthog-
onal designs. Applications are also provided. In Section 5, we investigate the
propertiesforsomemutuallyquasi-unbiasedweighingmatricesconstructedfrom
Theorem4.8, andfinally we discuss unbiasedness for unit orthogonaldesigns in
Section 6.
2 Preliminaries
In this section, we present notations and results to be used throughout the
paper.
Definition 2.1. An orthogonal design of order n and type (s ,...,s ) in vari-
1 u
ables x ,...,x is a (0, x ,..., x )-matrix D, where x ,...,x are distinct
1 u 1 u 1 u
commutingindeterminat±es,sucht±hatDD⊤ =(s x2+ +s x2)I . Wedenote
1 1 ··· u u n
it by OD(n;s ,...,s ).
1 u
Letting (0,1, 1)-matrices W ,...,W be such as D = u x W , it holds
− 1 u i=1 i i
that Wi is a weighing matrix of order n and weight si for aPny i.
Werecalltheexistenceoforthogonaldesignsoforder2t,tapositiveinteger.
ThereexistorthogonaldesignsDoforder2,4,8andtype(1,1),(1,1,1,1),(1,1,1,1,
1,1,1,1) respectively as follows:
x x x x
1 2 3 4
D = x1 x2 ,D =−x2 x1 x4 −x3,
(cid:18) x2 x1(cid:19) x3 x4 x1 x2
− − − 
 x x x x 
4 3 2 1
− − 
2
x x x x x x x x
1 2 3 4 5 6 7 8
 x2 x1 x4 x3 x6 x5 x8 x7
− − − −
x x x x x x x x
3 4 1 2 7 8 5 6
− − − − 
D =−x4 x3 −x2 x1 x8 x7 −x6 −x5.
 x x x x x x x x 
− 5 − 6 7 − 8 1 2 − 3 4 
 x6 x5 x8 x7 x2 x1 x4 x3 
− − − − 
 x x x x x x x x 
− 7 − 8 − 5 6 3 − 4 1 2 
 x8 x7 x6 x5 x4 x3 x2 x1 
− − − − 
For t > 3, there exists an orthogonal design D of order 2t and type (s )2t =
i i=1
(1,1,1,1,2,2,4,4,...,2t−2,2t−2) [15]. Thus we have:
Lemma 2.2. (1) For any positive integer t, there exists an orthogonal design
of order 2t and type (s ,...,s ) where
1 u
u=2,(s )2 =(1,1) if t=1,
i i=1
u=4,(s )4 =(1,1,1,1) if t=2,
i i=1
u=8,(s )8 =(1,1,1,1,1,1,1,1) if t=3,
i i=1
u=2t,(s )2t =(1,1,1,1,2,2,4,4,...,2t−2,2t−2) if t>3.
i i=1
(2) For any positive integers t,k such that k 2t, there exists a weighing
≤
matrix of order 2t and weight k.
Proof. (1) is already seen. It holds
s =S 1,...,u = k Z 1 k 2t , (2.1)
i∈S i |∅6 ⊂{ } { ∈ | ≤ ≤ }
(cid:8)P (cid:9)
from which we obtain (2) by substituting 1,0 into suitable variables.
The concept of quasi-unbiased weighing matrices was introduced in [14] as
a generalization of unbiased weighing matrices [3, 10].
Definition 2.3. Two weighing matrices W ,W of order n and weight k are
1 2
saidtobequasi-unbiasedforparameters(n,k,l,a)if(1/√a)W W⊤isaweighing
1 2
matrix of order n and weight l. Weighing matrices W ,...,W of order n and
1 f
weight k are mutually quasi-unbiased for parameters (n,k,l,a) if any distinct
two of them are quasi-unbiased for the parameters.
If there exist quasi-unbiased weighing matrices for parameters (n,k,l,a),
then it holds that l = k2/a. Using this equality, it is easily shown that for
weighing matrices W ,W of order n and weight k, W ,W are quasi-unbiased
1 2 1 2
for parameters (n,k,l,a) if and only if (1/√a)W W⊤ is a (0,1, 1)-matrix.
1 2 −
The case for the parameters (n,n,l,a) was studied in [2] from the viewpoint of
coding theory.
A(symmetric) association scheme of class d withvertex setX ofsize n is a
set of non-zero (0,1)-matrices A ,...,A , which are called adjacency matrices,
0 d
with rows and columns indexed by X, such that:
(1) A =I .
0 n
d
(2) A =J , J is the all-one matrix of order n.
i=0 i n n
P
(3) For any i 0,1,...,d , A⊤ =A .
∈{ } i i
3
(4) For any i,j 0,1,...,d , A A = d pkA for some pk’s.
∈{ } i j k=0 ij k ij
P
The vector space over R spanned by A ’s forms a commutative algebra,
i
denotedby andcalledadjacency algebra. Thereexists abasisof consisting
A A
ofprimitiveidempotents,sayE =(1/n)J ,E ,...,E . Since A ,A ,...,A
0 n 1 d 0 1 d
{ }
and E ,E ,...,E aretwobasesof ,thereexistthechange-of-basesmatrices
0 1 d
{ } A
P =(P )d , Q=(Q )d so that
ij i,j=0 ij i,j=0
d d
1
A = P E , E = Q A .
j ij i j ij i
n
Xi=0 Xi=0
ThematrixP (Qrespectively)issaidtobe thefirst (second respectively) eigen-
matrix.
3 Unbiased orthogonal designs
Definition3.1. LetD ,D beorthogonaldesignsofordernandtype(s ,...,s )
1 2 1 u
invariablesx ,...,x . TheorthogonaldesignsD ,D areunbiasedwithparam-
1 u 1 2
eter αif αis apositive realnumberandthereexists a(0,1, 1)-matrixW such
−
that
s x2+ +s x2
D D⊤ = 1 1 ··· u uW.
1 2 √α
Orthogonal designs D ,...,D are mutually unbiased with parameter α if any
1 f
distinct two of the orthogonaldesigns are unbiased with parameter α.
Remark 3.2. (1) Note that the (0,1, 1)-matrix W in Definition 3.1 must be
−
a weighing matrix of weight α, thus α must be a positive integer.
(2) If α = n, then the matrix W in Definition 3.1 is a Hadamard matrix of
order n.
Proposition 3.3. Suppose there exist unbiased orthogonal designs of order n
andtype(s ,...,s )withparameterα. Thenthereexistquasi-unbiasedweighing
1 u
matrices for the parameters (n, s ,α,( s )2/α)) for any nonempty
i∈S i i∈S i
subset S 1,...,u . P P
⊂{ }
Proof. Let D ,D be unbiasedorthogonaldesigns withthe desiredparameters.
1 2
Substituting1ifj Sand0otherwiseintox inD andD yieldquasi-unbiased
j 1 2
∈
weighing matrices for the desired parameters.
Remark 3.4. Proposition 3.3 shows that we have:
u
unbiased Hadamard matrices if α=n and s =n,
• i=1 i
P
unbiasedweighingmatricesifα<nandS 1,...,u suchthat s =
• ⊆{ } i∈S i
α, P
u
quasi-unbiased Hadamard matrices if α<n and s =n,
• i=1 i
P
quasi-unbiasedweighingmatricesifS 1,...,u suchthat s =α.
• ⊆{ } i∈S i 6
P
4
Thus, unbiased orthogonal designs is a unified concept for various unbiased
matrices.
Assume that D ,...,D are mutually unbiased orthogonal designs of order
1 f
n and type (s ,...,s ) with parameter α. By Proposition 3.3 with S = 1 ,
1 u
{ }
we obtain quasi-unbiased weighing matrices W ,...,W for the parameters
1 f
(n,s ,α,s2/α). Then (√α/s )W W⊤,...,(√α/s )W W⊤ are f 1 mutually
1 1 1 1 2 1 1 f −
unbiased weighing matrices of weight α. Applying [3, Corollary 9] to these, we
obtain the following upper bound.
Proposition 3.5. Let D ,...,D be mutually unbiased orthogonal designs of
1 f
order n and type (s ,...,s ) with parameter α. Then the following holds.
1 u
(1) f (n−1)(n+2) +1.
≤ 2
(2) If 3α (n+2) 0, then f α(n−1) +1.
− ≥ ≤ 3α−(n+2)
Problem 3.6. Find examples of mutually unbiased orthogonaldesigns attain-
ing the upper bounds in Proposition 3.5, or improve the upper bounds.
IntherestofthissectionweshowhowconstructionsofunbiasedHadamard/weighing
matrices are extended to those of unbiased orthogonaldesigns.
The direct product of matrices is used to give unbiased orthogonal designs.
Proposition 3.7. If there exist unbiased orthogonal designs D ,...,D of or-
1 f
der n with parameter α and type (s ,...,s ) and unbiased orthogonal designs
1 u
D′,...,D′ of order m and type (s ,...,s ) with parameter α, then D
D1′,...,Df D′ are unbiased ortho1gonal duesigns of order n + m and t1yp⊕e
1 f ⊕ f
(s ,...,s ) with parameter α.
1 u
Proof. Straightforward.
Thetensorproductofunbiasedorthogonaldesignsandquasi-unbiasedweigh-
ing matrices give unbiased orthogonaldesigns.
Proposition3.8. SupposethatthereexistunbiasedorthogonaldesignsD ,...,D
1 f
of order n and type (s ,...,s ) with parameter α and quasi-unbiased weighing
1 u
matrices W ,...,W for the parameters (m,k,l,a).
1 f
(1) D W ,...,D W are unbiased orthogonal designs of order nm and
1 1 f 1
⊗ ⊗
type (ks ,...,ks ) with parameter α.
1 u
(2) D W ,...,D W are unbiased orthogonal designs of order nm and
1 1 1 f
⊗ ⊗
type (ks ,...,ks ) with parameter lα.
1 u
Proof. Straightforward.
For an orthogonal design of order n and type (s ,...,s ) and a weighing
1 u
matrix of order n and weight k, we obtain n matrices as an extension of a part
of a result in [11].
Lemma 3.9. Let D be an orthogonal design D of order n and type (s ,...,s )
1 u
in variables x ,...,x with i-th column d , and W a weighing matrix of order
1 u i
n and weight k with i-th column w . Define C = w d⊤,W = w w⊤ for i
i i i i i i i ∈
1,...,n . Then the following hold.
{ }
5
(1) C C⊤ =O , 1 i=j n, where O is the zero matrix.
i j n ≤ 6 ≤ n
(2) C C⊤ =(s x2+ +s x2)W , 1 i n.
i i 1 1 ··· u u i ≤ ≤
n
(3) W =kI .
i=1 i n
P
The following lemma will be used in Proposition 3.11, which is an exten-
sion of a construction of Bush-type Hadamard matrices [11, Corollary 5], see
Section 5 for the definition of Bush-type Hadamard matrices.
Lemma 3.10. Let D,W,C be the same as Lemma 3.9, and L = (l(i,j))n
i i,j=1
a Latin square of order n. Then D˜ = (C )n is an orthogonal design of
l(i,j) i,j=1
order n2 and type (ks ,...,ks ).
1 u
Proof. The (i,j)-block of D˜D˜⊤ is
n
⊤
C C . (3.1)
l(i,m) l(j,m)
mX=1
When i = j, (3.1) is equal to k(s x2 + +s x2)I by Lemma 3.9 (2), (3).
When i = j, (3.1) is equal to O b1y1Lem·m··a 3.9u(u1).nThus D˜ is an orthogonal
n
6
design of order n2 and type (ks ,...,ks ).
1 u
NextweuseLatinsquares. TwoLatinsquaresL andL ofsizenonsymbol
1 2
set 1,2,...,n arecalledsuitableifeverysuperimpositionofeachrowofL on
1
{ }
each row of L results in only one element of the form (a,a). Latin squares in
2
whicheverydistinctpairofLatinsquaresissuitablearecalledmutually suitable
Latin squares. Note that the existence of f mutually suitable Latin squares is
equivalenttothe existence ofm mutually orthogonalLatinsquares[10, Lemma
9]. The following is an extension of [10, Theorem 13].
Proposition 3.11. If there exist an orthogonal design D of order n and type
(s ,...,s ), a weighing matrix W of order n and weight k and f mutually
1 u
suitable Latin squares L ,...,L of order n, then there exist f +1 unbiased
1 f
orthogonal designs of order n2 and type (ks ,...,ks ) with parameter α=1.
1 u
Proof. Let m ,m be distinct elements in 1,...,f . Let l(i,j),l′(i,j) denote
1 2
the (i,j)-entry of Lm1,Lm2 respectively. S{et D˜m1 =} (Cl(i,j)),D˜m2 = (Cl′(i,j)),
where C is defined in Lemma 3.9. By Lemma 3.10, each D˜ is an orthogonal
i i
design of order n2 and type (ks ,...,ks ).
1 u
First we claim D˜ ,D˜ are unbiased with parameter α=1. We calculate
m1 m2
the (i,j)-block of D˜ D˜⊤ as follows.
m1 m2
n
the (i,j)-block of D˜ D˜⊤ = C C⊤ . (3.2)
m1 m2 l(i,m) l′(j,m)
mX=1
There uniquely exists k 1,...,n such that l(i,k) = l′(j,k) = a, say, and
l(i,m)=l′(j,m) for any∈m{=k since}L ,L are suitable. Then (3.2) is
6 6 m1 m2
C C⊤ =C C⊤ =(s x2+ +s x2)W .
l(i,k) l′(j,k) a a 1 1 ··· u u a
Since W is a (0,1, 1)-matrix, D˜ ,D˜ are unbiased with parameter α=1.
a − m1 m2
6
Next we show that one more orthogonal design is added as follows. Define
a (0, x ,..., x )-matrix D′ to be (w d⊤)n . Then D′ is an orthogonal
± 1 ± u j i i,j=1
design. Indeed,
n
the (i,j)-block of D′D′⊤ = w d⊤d w⊤ =δ k(s x2+ +s x2)I .
m i j m ij 1 1 ··· u u n
mX=1
Next we show that D′ and D˜ are unbiased for any m 1,...,f . Letting
m
l′′(i,j) denote the (i,j)-entry of a Latin square L , ther∈e u{niquely}exists k
m
1,...,n such that l′′(j,k)=i, where . Then ∈
{ }
n
the (i,j)-block of D′D˜m⊤ = wmd⊤i dl′′(j,m)wl⊤′′(j,m)
mX=1
=(s x2+ +s x2)w w⊤.
1 1 ··· u u k i
Since w w⊤ is a (0,1, 1)-matrix, D′ and D˜ are unbiased with parameter
k i − m
α=1.
It is known that if there exist Hadamard matrices of order 4m,4n, then
there exists a Hadamard matrix of order 8mn [1]. This construction was used
to construct quasi-unbiased Hadamard matrices in [2]. We use this idea to
orthogonaldesigns in order to obtain unbiased orthogonal designs.
Proposition3.12. Ifthereexistanorthogonaldesignorder4moftype(s ,...,s )
1 u
and quasi-unbiased Hadamard matrices for parameters (4n,4n,l,a), then there
exist unbiased orthogonal designs of order 8mn and type (2ns ,...,2ns ) with
1 u
parameter α=16n2/a.
Proof. Let D be an orthogonal designs of order 4m and type (s ,...,s ) in
1 u
variables x ,...,x and H ,H be quasi-unbiased Hadamard matrices for the
1 u 1 2
parameters (4n,4n,l,a).
Let H (i,j = 1,2) be 4n 2n matrices and D (i = 1,2) be 2m 4m
i,j i
× ×
matrices such that
D
H = H H , D = 1 .
i i,1 i,2 (cid:18)D2(cid:19)
(cid:0) (cid:1)
We define D˜ (i=1,2) as
i
1 1
D˜ = (H +H ) D + (H H ) D .
i i,1 i,2 1 i,1 i,2 2
2 ⊗ 2 − ⊗
Then it is directly shown that D˜ (i = 1,2) are unbiased orthogonal designs of
i
order 8nm and type (2ns ,...,2ns ) with parameter α=16n2/a.
1 u
4 A construction and some applications using
the plug-in method
In this section, first we provide a construction of quasi-unbiased weighing ma-
trices, and then it will be used to construct unbiased orthogonaldesigns.
The following lemma is a construction of (0,1)-matrices from a finite ring
with unity, which satisfy Lemma 4.6. This lemma with Lemma 4.6 constructs
mutually quasi-unbiased weighing matrices satisfying Proposition 4.7.
7
Lemma 4.1. Let R be a finite ring with unity and n elements. If there exist
elements x ,...,x R such that x x is a unit in R for any distinct i,j,
1 m i j
∈ −
then there exist n n monomial (0,1)-matrices K , i,j 1,...,m such that
i,j
m K K⊤ is a×(0,1)-matrix for any distinct i,j. ∈{ }
l=1 i,l j,l
P
Proof. Assume that the additive group of R is isomorphic to Z Z .
n1 ×···× ns
Letr beanh hcirculantmatrixwiththefirstrow(0,1,0,...,0). Weidentify
h
elements in R×with elements in Z Z . Define a grouphomomorphism
φ:R→GLn(R) as φ((xi)si=1)=n⊗1ti×=1·r·nx·ii×. ns
Let α ,...,α be any distinct elements in R. Set K = φ(x α ) for i,j
1 m i,j i j
∈
1,...,m . Then eachK is clearly ann n monomial(0,1)-matrix. For any
i,j
{distincti,}j, m K K⊤ = m φ((x ×x )α )isa(0,1)-matrixsincex x
l=1 i,l j,l l=1 i− j l i− j
is a unit forPany distinct i,j.P
Example 4.2. Let m,p be positive integers such that p is the least prime
number dividing m. Then0,1,...,p 1satisfy the propertythatthe difference
of any distinct two elements is a unit−in Z , and p is the maximum number of
m
such elements by the pigeonhole principle.
Example 4.3. Let q be a prime power and F the finite field with q elements.
q
Then any distinct m elements in F satisfy that the difference of any distinct
q
two is a unit in F .
q
Example 4.4. Let p,s,m be positive integers such that p is prime, let h(x) be
a basic irreducible polynomial of degree m over Zps. The ring Zps[x]/(h(x)) is
called a Galois ring, denoted by GR(ps,psm). Write ξ = x+(h(x)). Then the
order of ξ is pm 1, and ξi ξj is a unit for any distinct i,j 0,...,pm 2
− − ∈{ − }
[19, Theorem 14.8].
We pose a problem in order to construct (0,1)-matrices in Lemma 4.1.
Problem 4.5. For a given finite ring R with unity, determine the largestposi-
tive integerm suchthat thereexistelements x ,...,x R insuchawaythat
1 m
∈
x x is a unit in R for any distinct i,j.
i j
−
Foranm mmatrixW =(w )m andn nmatricesK ,...,K ,denote
× ij i,j=1 × 1 m
by W (K ,...,K )
1 m
⊗
w K w K w K
11 1 12 2 1m m
···
w21K1 w22K2 w2mKm
 ... ... ·.·.·. ... .
 
wm1K1 wm2K2 wmmKm
 ··· 
The following lemma provides mutually quasi-unbiased weighing matrices
from (0,1)-matrices satisfying the assumptions of Lemma 4.1, and will be used
to construct unbiased orthogonaldesigns in Proposition 4.7.
Lemma 4.6. Let n,m,k be positive integers such that m n. Let W =
≤
(w )m beaweighing matrixof order mandweight k,K (i,j 1,...,m )
ij i,j=1 i,j ∈{ }
n nmonomial(0,1)-matricessuchthat m K K⊤ isa(0,1)-matrixfor any
× l=1 i,l j,l
distinct i,j 1,...,m ,andset Wi =WP (Ki,1,...,Ki,m)for i 1,...,m .
∈{ } ⊗ ∈{ }
Then the following hold.
8
(1) W is a weighing matrix of order nm and weight k.
i
(2) W ,...,W aremutuallyquasi-unbiasedweighingmatricesfor(nm,k,k2,1).
1 m
Proof. (1): Since the (a,b)-th block of W W⊤ is
i i
m m
⊤
(w K )(w K ) = w w I =δ kI ,
al i,l bl i,l al bl n ab n
Xl=1 Xl=1
W is a weighing matrix of the desired order and weight, where δ denotes the
i ab
Kronecker delta.
(2): It is enough to show that W W⊤ is a (0,1, 1)-matrix for any distinct
i j −
i,j. Letting i,j be distinct elements in 1,...,m ,the (a,b)-thblockofW W⊤
{ } i j
is
m m
⊤ ⊤
(w K )(w K ) = w w K K ,
al i,l bl j,l al bl i,l j,l
Xl=1 Xl=1
which is a (0,1, 1)-matrix,since m K K⊤ is a (0,1)-matrix and w w is
− l=1 i,l j,l al bl
0, 1. P
±
Finallyweprovideaconstructionforunbiasedorthogonaldesignsfromsome
quasi-unbiased weighing matrices and an orthogonal design.
Proposition 4.7. Let W ,...,W be mutually quasi-unbiased weighing ma-
1 f
trices for parameters (nm,k,k2,1). Assume that W (I J ) is a (0,1, 1)-
i n m
⊗ −
matrix for any i 1,...,f . Let K be an orthogonal design of order m and
∈ { }
type (s ,...,s ) in variables x ,...,x . Then there exist f mutually unbiased
1 u 1 u
orthogonal designs of order nm and type (ks ,...,ks ) with parameter α=k2.
1 u
Proof. Let D =W (I K) for any i 1,...,f .
i i n
⊗ ∈{ }
Each matrix D is clearly a (0, x ,..., x )-matrix. For i,j 1,...,f ,
i 1 u
± ± ∈{ }
u
D D⊤ =W W⊤(I KK⊤)=( s x2)W W⊤.
i j i j n⊗ k k i j
Xk=1
SinceW W⊤ =kI foranyi,D isanorthogonaldesignofordernmandtype
i i nm i
(ks ,...,ks ). Since W W⊤ is a (0,1, 1)-matrix for any distinct i,j, D ,D
1 u i j − i j
are unbiased with parameter α=k2.
We areready forthe mainresult. By Lemmas 4.1, 4.6, Propositions3.3, 4.7
and Example 4.3, we obtain the following result.
Theorem 4.8. Let q,m,k,s ,...,s be positive integers such that q is a prime
1 u
power and m q. Assume that there exist a weighing matrix of order m and
≤
weight k and an orthogonal design of order m and type (s ,...,s ). Then the
1 u
following hold.
(1) There exist m mutually unbiased orthogonal designs of order mq and type
(ks ,...,ks ) with parameter α=k2.
1 u
(2) There exist m mutually quasi-unbiased weighing matrices for parameters
(mq,k s ,k2,( s )2) for any nonempty subset S 1,...,u .
i∈S i i∈S i ⊂{ }
P P
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In particular, by taking q a power of 2 in Theorem 4.8 and by Lemma 2.2,
we obtain the following.
Corollary 4.9. Let t,k,u,s ,...,s ,mbe positive integers such that k,m 2t,
1 u
≤
u is 2t if t=1,2,3 and 2t if t>3 and
(s )2 =(1,1) if t=1,
i i=1
(s )4 =(1,1,1,1) if t=2,
i i=1
(s )8 =(1,1,1,1,1,1,1,1) if t=3,
i i=1
(s )2t =(1,1,1,1,2,2,4,4,...,2t−2,2t−2) if t>3.
i i=1
Then the following hold.
(1) There exist 2t mutually unbiased orthogonal designs of order 22t and type
(ks )2t with parameter α=k2.
i i=1
(2) There exist 2t mutually quasi-unbiased weighing matrices for the parame-
ters (22t,mk,k2,m2).
Inthe restofthis sectionwe usethe plug-in method inTheorem4.8inorder
to show some of the many applications of the construction there. In order to
use the plug-in method in Theorem 4.8 the variables should be replaced with
amicable matrices and in order to preserve the orthogonality of the designs,
the matrices should satisfy the sum property. For example, matrices A and B
replaces variables a and b, if A and B are amicable, i.e. ABt = BAt. The
sum property refers to the property that matrices A replacing variables a ,
i i
i = 1,2, ,k, should satisfy i=kA At = ℓI for some positive integer ℓ. We
··· i=1 i i
refer reader to [16] for the termPinologies not defined here. Our first application
relates to part (1) in Corollary 4.9, but we need to recall a result of Goethals
andSeidel[8]. There they showedthe existence oftwocirculantandsymmetric
(1, 1)-matrices I +R and S of order q = 1(p+1), p 1 (mod 4) a prime
pow−er such that RqR⊤+SS⊤ =pI . Note tha2t the existen≡ce of Goethals-Seidel
q
matrices imply the existence of Williamson matrices, see [8].
Proposition 4.10. There are two quasi-unbiased weighing matrices for the
parameters (4q,4q 2,4,(2q 1)2) for every q = 1(p+1), p 1 (mod 4) a
− − 2 ≡
prime power.
Proof. Replace the variables in part (1) of Corollary 4.9 by Goethals-Seidel
matrices of order q = 1(p+1), p 1 (mod 4) a prime power.
2 ≡
Our second application relates to part (2) in Corollary 4.9, where there are
four independent variables. Here we replace the variables by four Williamson
type matrices.
Proposition 4.11. There are four mutually quasi-unbiased Hadamard matri-
ces for the parameters (16n,16n,16,16n2) for every n which is the order of
Williamson type matrices.
Proof. Replacethevariablesinpart(2)ofCorollary4.9bytheWilliamsontype
matrices of order n.
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