AMICABLE MATRICES AND ORTHOGONAL DESIGNS
S M ERFANUL KABIR CHOWDHURY
Master of Science, University of Chittagong, 2010
A Thesis
Submitted to the School of Graduate Studies
of the University of Lethbridge
in Partial Fulfillment of the
Requirements for the Degree
MASTER OF SCIENCE
Department of Mathematics and Computer Science
University of Lethbridge
LETHBRIDGE, ALBERTA, CANADA
©c S M Erfanul Kabir Chowdhury, 2015
AMICABLE MATRICES AND ORTHOGONAL DESIGNS
S M ERFANUL KABIR CHOWDHURY
Date of Defense: June 24, 2015
Dr. Hadi Kharaghani
Supervisor Professor Ph.D.
Dr. Wolfgang Holzmann
Thesis Examination Committee Member Professor Emeritus Ph.D.
Dr. Mark Walton
Thesis Examination Committee Member Professor Ph.D.
Dr. Amir Akbary-Majdabadno
Chair, Thesis Examination Committee Professor Ph.D.
Dedication
To
my parents.
iii
Abstract
This thesis is mainly concerned with the orthogonal designs of Baumert-Hall array type,
OD(4n;n,n,n,n) where n = 2k, k is odd integer. For every odd prime power pr, we con-
struct an infinite class of amicable T-matrices of order n = pr + 1 in association with ne-
gacirculant weighing matrices W (n,n−1). In particular, for pr ≡ 1 (mod 4), we construct
amicable T-matrices of order n≡ 2 (mod 4) and application of these matrices allows us to
generate infinite class of orthogonal designs of type OD(4n;n,n,n,n) and OD(4n;2,2,2n−
2,2n−2) where n = 2k; k is odd integer. For a special class of T-matrices of order n where
each of Ti is a weighing matrix of weight wi;1 ≤ i ≤ 4 and Williamson-type matrices of
order m, we establish a theorem which produces four circulant matrices in terms of four
variables. These matrices are additive and can be used to generate a new class of orthogonal
design of type OD(4mn;w1s,w2s,w3s,w4s); where s = 4m. In addition to this, we present
some methods to find amicable matrices of odd order in terms of variables which have
an interesting application to generate some new orthogonal designs as well as generalized
orthogonal designs.
iv
Acknowledgments
I would start with a name who introduced me with Combinatorial Mathematics for the first
time. He is no one but professor Dr. Hadi Kharaghani. I do feel proud to pronounce his
generous support and goal oriented guideline all over the way that fix me to be a learner in
this field of study in upcoming days. My heartfelt gratitude to Hadi Kharaghani for letting
me the chance to work with him. Thanks-Hadi Kharaghani.
I am very thankful to Dr. Wolf Holzmann and Dr. Mark Walton for reading my thesis
carefully and providing me with valuable comments and suggestions. I also acknowledge
to Dr. Amir Akbary-Majdabadno (Chair, Thesis Examination Committee) for his kind
cooperation.
I would like to thank professor Dr. Soroosh Yazdani, who taught me two courses in this
couple of years.
I remember those people who helped me all over way inside the University. I should men-
tion two of them. One is Sara Sasani who was working with me and the other one is B.
Hodgson, administrative support. Thank you all for your kind co operation.
Finally, whose inspiration and support lead me to move forward is my family. I ac-
knowledge each and everybody of my family for their cordial support. I would like to
throw a big thank to all of them.
Thank you.
v
Contents
Contents vi
List of Tables viii
1 Introduction 1
2 Hadamard Matrices 3
2.1 Equivalence of Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Existence of Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 Hadamard Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Construction of Hadamard Matrices . . . . . . . . . . . . . . . . . . . . . 9
2.3.1 Kronecker Product Construction . . . . . . . . . . . . . . . . . . . 10
2.3.2 Paley construction . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Williamson Construction . . . . . . . . . . . . . . . . . . . . . . . 15
3 Sequences 18
3.1 T-Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Golay Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2.1 Symmetry Properties . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.2 Other Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Base Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Turyn Type Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4 Balanced Generalized Weighing Matrices 29
4.1 Difference Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Balanced Generalized Weighing Matrices . . . . . . . . . . . . . . . . . . 31
4.3 Negacirculant Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
5 Amicable T-Matrices 35
5.1 T-Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Amicable T-Matrices of Order n≡ 2 (mod 4) . . . . . . . . . . . . . . . . 39
5.2.1 Construction of Amicable T-Matrices of Order n = pr + 1; pr is
Odd Prime Power . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3 Multiplication Theorem for Amicable T-Matrices . . . . . . . . . . . . . . 42
6 New Orthogonal Designs 46
6.1 Orthogonal Designs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Orthogonal Designs From Amicable T-Matrices . . . . . . . . . . . . . . . 48
6.3 Orthogonal Designs From Special Class of T-Matrices . . . . . . . . . . . 56
vi
CONTENTS
6.4 Orthogonal Designs From Amicable Matrices With Composite Entries . . . 58
6.5 A New Method to Construct a Hadamard Matrix of Order 48 . . . . . . . . 63
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Bibliography 68
A 71
vii
List of Tables
2.1 No. of inequivalent Hadamard matrices . . . . . . . . . . . . . . . . . . . 6
3.1 Original Golay-sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 T-sequences (TS) from Golay-sequences (GS) . . . . . . . . . . . . . . . . 24
3.3 Base-sequences of odd lengths . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 T-sequences and Complementary sequences from Base sequences . . . . . 26
3.5 Base sequences (BS) from Turyn sequences (TrS) . . . . . . . . . . . . . . 27
6.1 Orthogonal Design of type OD(4n;n,n,n,n);n = 2k, k is odd . . . . . . . . 52
viii
Chapter 1
Introduction
While Hadamard conjecture has remained open for around a century, which methods would
be the good ones for constructing new Hadamard matrices? The answer must include or-
thogonal designs, a well known combinatorial object. A special category of orthogonal
designs is Baumert-Hall array which is an interesting area of study in the field of combina-
torial mathematics. In this thesis, our endeavor results in a new infinite class of Baumert-
Hall arrays, OD(4n;n,n,n,n) where n= 2k; k is odd integer and some Baumert-Hall-Welch
arrays with reduced parameter of order 4n;n is odd.
We cover necessary pre-requisite materials throughout the Chapters 2 to 4. In Chap-
ter 2 which laid the foundation of our research, we study some fundamental properties
of Hadamard matrices and some well known construction methods which include the
Kronecker product construction, Paley construction [36] and Williamson construction
[42]. Chapter 3 mainly focuses on some different (±1) sequences having aperiodic auto-
correlation function as well as periodic auto-correlation function zero. As T-sequences are
important tools to construct Hadamard matrices and orthogonal designs, we include some
basic rules and formulas [39] that generate this sequences from some other sequences
such as Golay sequences [18], Base sequences and Turyn type sequences [41]. Chapter
4 is mostly concerned with balanced generalized weighing matrices with classical param-
eters over cyclic groups. This class of matrices includes conference matrices, weighing
matrices, ω-circulant matrices and negacirculant matrices. In chapter 5, we mostly con-
centrate on amicable T-matrices of composite order. Using the idea of existence of ne-
gacirculant weighing matrices of order pr + 1 [15] for odd prime power pr, we construct
1
1. INTRODUCTION
an infinite class of amicable T-matrices of order n ≡ 2 (mod 4) where n = pr + 1. Also
we present a multiplication theorem for amicable T-matrices which enables us to gener-
ate composite order T-matrices of larger length. Chapter 6 includes the application of
amicable T-matrices which results in orthogonal design of type OD(4n;n,n,n,n). Using
Williamson-type matrices of order m and a special class of T-matrices of order n where
each of Ti is a weighing matrix of weight wi, a new multiplication theorem is established
which results in four circulant matrices A,B,C,D in terms of variables a,b,c,d satisfying
AAT +BBT +CCT +DDT = (w1sa2+w sb22 +w 23sc +w4sd2)Imn where s = 4m. This ma-
trices generates a new class of orthogonal design of type OD(4sn;w1m,w2m,w3m,w4m).
Additionally, sets of four circulant amicable matrices of odd order in terms of variables
are constructed and generalized orthogonal design has been discussed as consequences of
these matrices. Finally, we present a new method to construct a Hadamard matrix of order
48 found by standard constructions.
2
Chapter 2
Hadamard Matrices
Definition 2.1. A square matrix H of order n with the entries from the set {±1} satisfying
the property
HHT = nIn
is known as a Hadamard matrix. More explicitly, the inner product of any two distinct
rows or columns of this matrix is zero and all rows are linearly independent and all columns
are linearly independent. A few examples of Hadamard matrix are given below.
Example 2.2. H(2), H(4) and H(8) are the examples of Hadamard matrices of order 2, 4
and 8 respectively, where     1 − − − 1 1 − 1 − −
H(2  ) =  , H(4) =  ,
1 −  − − 1 − 
− − − 1
and
3
2.1. EQUIVALENCE OF HADAMARD MATRICES
 
 1 − − − 1 − − −

 − − − − − − 1 1
 − − 1 − − − 1 − 
H 8 

− − − 1 − − − 1 
( ) =

 .
1 − − − − 1 1 1 
  − 1 − − 1 − 1 1 

− − 1 − 1 1 − 1 
− − − 1 1 1 1 −
“−” stands for −1.
2.1 Equivalence of Hadamard Matrices
Though superficially Hadamard matrices seem to be a random arrangement of 1 and -1,
its configuration admits some basic matrix operations which yield new Hadamard matrices
of the same order. So from an existing Hadamard matrix one get many more of the same
kind.
Definition 2.3. Two Hadamard matrices H1 and H2 of the same order are called equiv-
alent if one can be obtained from the other by performing one or more of the following
operations.
(i) Permutation of the rows,
(ii) Permutation of columns,
(iii) Multiplying some row or column by −1.
Since every row or column in a Hadamard matrix H is orthogonal to all others, it fol-
lows the transpose of a Hadamard matrix is Hadamard matrix as we show now. The basic
4
2.1. EQUIVALENCE OF HADAMARD MATRICES
requirement, HHT = nIn leads us to write,
(HT )(HT )T = HT H
= (nH−1)(H)
= nIn.
This proves:
Proposition 2.4. If H is a Hadamard matrix then HT is also Hadamard matrix.
Among many of underlying features of Hadamard matrices, our interest is to present
one of these that restricts all of the entries in the first row and first column to be the same
kind.
Definition 2.5. If all of the entries in the first row and in the first column of a Hadamard
matrix H are +1, then this Hadamard matrix is called a normalized Hadamard matrix.
Example 2.6. For n = 4, the followingis a normalized H(4).
 1 1 1 1 1 1 − −
H(4) = 1 − 1 − 
 .
1 − − 1
In a normalized Hadamard matrix H of order 4n, we see that there are exactly equal
number of +1’s and −1’s in every row and every column except the first row and the first
column. Moreover, n number of −1’s in any row(column) overlap with n number of −1’s
in any other row(column) except the first row(column).
Multiplication of any row or column in a Hadamard matrix by −1 yields a new Hadamard
matrix. So one can easily transform an arbitrary Hadamard matrix into a normalized
Hadamard matrix by negating the rows and columns whose leading entry is −1. This
shows:
5
2.1. EQUIVALENCE OF HADAMARD MATRICES
Lemma 2.7. Every Hadamard matrix is equivalent to a normalized Hadamard matrix.
Example 2.8. Here H(4) is equivalent to H ′(4) which is normalized.  − 1 1 1  1 − − − 1 1 1 1 1 − 1 1  
 1 − 1 1   1 1 − − 
H(4) = ⇒
 1 1 − 1  
 ⇒ = H
′(4).
 1 1 − 1   1 − 1 − 
1 1 1 − 1 1 1 − 1 − − 1
As equivalence operations on a Hadamard matrix generate a class of new Hadamard
matrices, the number of inequivalent Hadamard matrices of a particular order is a good
topic of study. Searching those is very time consuming and more computational work.
H. Kharaghani and Tayfeh-Rezaie [30] were able to classify the highest order Hadamard
matrices of order 32. The following list presents the number of inequivalent Hadamard
matrices corresponding to all orders found up to the present.
Table 2.1: No. of inequivalent Hadamard matrices
Order (n) [Ref.] No. of matrices
1 1
2 1
4 1
8 1
12 1
16 [20] 5
20 [21] 3
24 [31] 60
28 [32] 487
32 [30] 13710027
6
2.2. EXISTENCE OF HADAMARD MATRICES
2.2 Existence of Hadamard Matrices
In 1867 [40], J. J. Sylvester was the first person to study (±1) matrices which would
later be known as Hadamard matrices. After around 25 years, J. Hadamard was trying
to find the maximal determinant of a matrix with entries from the complex unit disk and
eventually found two (±1) square matrices of orders 12 and 20 as a solution of the problem.
Though these solution matrices did not fit with matrices found by J. J. Sylvester, the internal
structure of both matrices were same. Later in 1893, J. Hadamard pointed out a hypothesis
about the existence of these solution matrices and this hypothesis became known as the
Hadamard conjecture. Now we recall a theorem from Sylvester’s paper, published in 1867
[40] which allows the recursive use of a existing Hadamard matrix of order n to construct
a new Hadamard matrix of order 2n. An immediate corollary of this theorem generates a
class of Hadamard matrice of order 2t ; t ≥ 0.
Theorem 2.9. [40] If H is a Hadamard matrix then  H H 
H −H
is also a Hadamard matrix.
( )
Starting with H = 1 and iterating this theorem we obtain
Corollary 2.10. There exist a Hadamard matrix of order 2t ; t ≥ 0.
2.2.1 Hadamard Conjecture
Conjecture 2.11. There exist a Hadamard matrix of order 4n for all n ∈ N.
This conjecture remains open since the last century. Solutions of some different open
problems in combinatorial design theory depend on a positive reply to this conjecture.
7
2.3. CONSTRUCTION OF HADAMARD MATRICES
Considering this conjecture as a sufficient condition for the existence of Hadamard matrices
we prove here a necessary condition for the existence of a Hadamard matrix.
Proposition 2.12. If there exist a Hadamard matrix of order n, then n = 1,2 or 4k;k ∈ N.
Proof. Suppose H is Hadamard matrix of order n and H ′ is a normalized Hadamard matrix
corresponding to H. Decompose H ′ into four blocks p,q,r,s and permute the columns of
H ′ in such a way that for the first three rows, each column in block p is (1,1,1), in block q
is (1,1,−), in block r is (1,−,1) and in block s is (1,−,−). So H ′ has form:
  1 1 · · · 1 1 1 1 · · · 1 1 1 1 · · · 1 1 1 1 · · · 1 1 

 1 1 · · · 1 1 1 1 · · · 1 1 − − ·· · − − − − ·· · − − 


 1 1 · · · 1 1 − − ·· · − − 1 1 · · · 1 1 − − ·· · − − 
 · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · 
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Since H ′ is Hadamard matrix, taking the orthogonality property of distinct rows into ac-
count we have
p+q+ r+ s = n
p+q− r− s = 0
p−q+ r− s = 0
p−q− r+ s = 0
A simple calculation yields the unique solution of this system of equations i.e, p = q = r =
s = n/4. Here n must be multiple of 4 because p,q,r,s and n are all integral.
8
2.3. CONSTRUCTION OF HADAMARD MATRICES
2.3 Construction of Hadamard Matrices
There are many existence results of Hadamard matrices which include different sub-
classes. In this section, we present some well known construction methods and start here
with some necessary definitions.
Definition 2.13. For a prime number p, a non-zero element in a Galois field, a ∈ Fp =
{1,2, · · · , p−1} is said to be quadratic residue if
x2 ≡ a (mod p)
for any x ∈ Fp. Otherwise it is called non-quadratic residue.
Definition 2.14. If a is any quadratic residue (mod p), then the Legendre symbol χ [36]
is defined by
0, if a ≡ 0 (mod p) ;
χ(a) =1, if a is quadratic ;−1, otherwise.
Definition 2.15. The Kronecker product of a m× n matrix A = [ai j] and a m′× n′ matrix
B = [bi j] is a mm′×nn′ block matrix given by 
 a11B a12B · · · a1nB

 a21B a

22B · · · a2nB 
A⊗B =  . . .. .. . . . ... 
am1B am2B · · · amnB
Definition 2.16. The Hadamard product of a m×n matrix, A = [ai j] and a m′×n′ matrix,
B = [bi j] is m×n matrix, given by
A∗B = [ai jbi j]
Definition 2.17. For a given n-dimensional vector V = (v0,v1, . . . ,vn−1), a square matrix
A = [ai j] obtained by circulating this row vector, whose entries are given by
9
2.3. KRONECKER PRODUCT CONSTRUCTION
ai j = v j−i;( j− i) (mod n)
is called circulant matrix and is denoted by circ(v0,v1, . . . ,vn−1).
2.3.1 Kronecker Product Construction
The earliest method, allowing the recurrence use of existing Hadamard matrices to con-
struct new Hadamard matrices is the Kronecker product construction. In 1893, J. Hadamard
used this method to produce a new Hadamard matrix from old ones.
Theorem 2.18. If H1 and H2 are two Hadamard matrices of order m and n respectively,
then H1⊗H2 is a Hadamard matrix of order mn.
Proof. Using a fundamental property of Kronecker product, (H1⊗H2)T = HT1 ⊗HT2 , we
have
(H1⊗H T T T2)(H1⊗H2) = H1H1 ⊗H2H2
= mIm⊗nIn
= mnImn
which proves the theorem.
Though Hadamard’s direct Kronecker product construction method is very simple to
generate Hadamard matrices of many new orders, it skips many orders in between the
order of new and old Hadamard matrices. In a lecture note published in 1985, Agayan-
Sharukhanyan [1] introduced a different way to use Kronecker products which allows
us to retrieve a Hadamard matrix of order reduced by half of the order Hadamard matrix
constructed by direct Kronecker product method.
Theorem 2.19. [1] Let H1 and H2 be Hadamard matrices of order 4h and 4k respectively.
Then there is a Hadamard matrix of order 8hk.
Proof. Decompose H1 and H2 into two 2×2 arrays
10
2.3. KRONECKER PRODUCT CONSTRUCTION
H 
 
 P Q 
 
1 =  and H2 = K L  .
R S M N
Since both H1 and H2 are Hadamard matrices, we have
PPT +QQT = 4hI2h KKT +LLT = 4kI2k
RRT +SST = 4hI2h MMT +NNT = 4kI2k
PRT +QST = 0 KMT +LNT = 0
RPT +SQT = 0 MKT +NLT = 0
Let
α = 1 12(P+Q)⊗K + 2(P−Q)⊗M
β = 12(P+Q)⊗L+
1
2(P−Q)⊗N
γ = 12(R+S)⊗K +
1
2(R−S)⊗M
δ = 1(R+S)⊗L+ 1( ) 2 2(R−S)⊗N
Now we define H = α βγ δ . To prove H is a Hadamard matrix we compute the following
quantities.
ααT
1
+ββT = [((P+Q)⊗K +(P−Q)⊗M)((P+Q)T ⊗KT +(P−Q)T ⊗MT ))
4
+((P+Q)⊗L+(P−Q)⊗N)((P+Q)T ⊗LT +(P−Q)T ⊗NT )]
1
= [(4hI2h +PQT +QPT )⊗4kI2k +(4hI2h−PQT −QPT )⊗4kI2k]4
1
= [8hI2h⊗4kI4 2k
]
= 8hkI4hk.
and
1
γαT +δβT = [((R+S)⊗K +(R−S)⊗M)((P+Q)T ⊗KT +(P−Q)T ⊗MT )+
4
11
2.3. KRONECKER PRODUCT CONSTRUCTION
((R+S)⊗L+(R−S)⊗N)((P+Q)T ⊗LT +(P−Q)T ⊗NT ).
1
= [(RQT +SPT )⊗4kI +(−RQT2k −SPT )⊗4kI4 2k
]
= 0.
Similarly, γγT +δδT = 8hkI4hk and αγT +βδT = 0. Therefore, we have
8hkI 0
HHT  4hk 

= = 8hkI8hk
0 8hkI4hk
i.e, H is a Hadamard matrix of order 8hk.
R. Craigen, J. Seberry and X. M Zhang gave a theorem in 1992 [9] which might be
regarded as an extension of Theorem 2.19. This theorem allows us to compute a Hadamard
matrices whose order is one fourth of the order of Hadamard matrices computed by using
theorem 2.19. Now we present this theorem.
Theorem 2.20. [9] If there exist Hadamard matrices of orders 4m,4m,4p,4q, then there
exist a Hadamard matrices of order 16mnpq.
Proof. Let H,K,L,M be four Hadamard matrices of order 4m,4n,4p,4q respectively. De-
composing these matrices into four blocks   
we write   
 H 1  
K1   L1   M1 
H =
H2     K2  K  
   
 =   
L 
 L2 
=  
M = M2 
H3 K3 L3  M3 
H4 K4 L4 M4
where j×4 j is the size of the blocks Hi,Ki,Li,Mi for 1 ≤ i≤ 4 and j ∈ {m,n, p,q}. Here
we note that ∑4 T Ti=1 HiHi = HH and HiHTj = 0; i 6= j which is also similarly applicable for
K,L and M. We define
1 T ⊗ 1R = (H1 +H2) K T2 1
+ (H −H ) ⊗K
2 1 2 2
12
2.3. PALEY CONSTRUCTION
1
S = (H +H )T3 4 ⊗
1
K + (H T
2 3 2 3
−H4) ⊗K4
1 1
U = (L1 +L2)T ⊗M1 + (L −L )T1 2 ⊗M2 2 2
1 1
V = (L3 +L4)T ⊗M3 + (L3−L )T4 ⊗M2 2 4
Now compute
1
RRT +SST = (H HT T T T
2 1 1
+H2H2 +H3H3 +H4H4 )⊗4nIn
1
= (4mI4m⊗4nI2 n
)
= 8mnI4mn.
and a simple calculation yields SRT = RST =UV T =VUT = 0. Here R,S,U,V are all (±1)
matrices. Suppose X = 12(R+S) and Y =
1
2(R−S) and then we obtain,
XXT
1
= (RRT +SST +SRT +RST )
4
= 2mnI4mn
Similarly YY T = 2mnI4mn.
Clearly X ,Y are disjoint (0,±1) matrices. Now defining H = X⊗U +Y ⊗V we get
HHT = (X⊗U +Y ⊗V )(X⊗U +Y ⊗V )T
= (XXT ⊗UUT +YY T ⊗VV T )
= 2mnI ⊗ (UUT4mn +VV T )
= 16mnpqI16mnpq
Therefore, H is a Hadamard matrices of order 16mnpq.
13
2.3. PALEY CONSTRUCTION
2.3.2 Paley Construction
Aside from Hadamard and Sylvester work, Paley’s work is well known for finding
Hadamard matrices of order 4n where n is odd. In 1933 [36], Paley constructed an infinite
class of Hadamard matrices of order (p+ 1) and 2(p+ 1) according as p ≡ 3 (mod 4)
and p≡ 1 (mod 4). To present these theorems with immediate examples we start here by
giving the definition of Jacobsthal matrix.
Definition 2.21. Let p be an odd prime power and χ be the Legendre symbol [36] over
the Galois field GF(p) = {a1,a2, . . . ,ap−1. Then a p× p matrix, Q = [pi j] with the entries
given by
pi j = χ(i− j);(i− j) (mod p)
is known as Jacobsthal matrix.
Theorem 2.22. [36] Let p≡ 3 (mod 4) be an odd prime power. Then
H 

 1 j 

= 
jT Q− I
is a Hadamard matrix of order p+1, where j is the all ones vector of length p and I is the
identity matrix of order p.
Example 2.23. Over Z7; 1, 2, 4 are non-zero squares and 3, 5, 6 are non squares. Then we
have the following Hadamard matrix of order 8.
  1 1 1 1 1 1 1 1 1 − − − 1 − 1 1  1 1 − − − 1 − 1 

 1 1 1 − − − 1 − H(8) =  1 − 1 1 − − − 1  1 1 − 1 1 − − − 1 − 1 − 1 1 − − 
1 − − 1 − 1 1 −
These type of Hadamard matrices are known an Paley-type Hadamard matrices.
14
2.3. WILLIAMSON CONSTRUCTION
Theorem 2.24. [36] Let p≡ 1(mod 4) be an odd prime power. Then
 1 j −1 j  

jT Q+ I jT Q− I 
H =   − − 1 j 1 − j 
jT Q− I − jT −Q− I
is a Hadamard matrix of order 2(p+1) where j is the all ones vector of length p and I is
the identity matrix of order p.
Example 2.25. Over Z5, 1, 2 are squares and 3, 4 are non-squares. Then we have the fol-
lowing Hadamard matrix of order 12.
  1 1 1 1 1 1 − 1 1 1 1 1

1 1 1 − − 1 1 − 1 − − 1
 

 1 1 1 1 − − 1 1 − 1 − −   1 − 1 1 1 − 1 − 1 − 1 − 

 1 − − 1 1 1 1 − − 1 − 1 1 1 − − 1 1 1 1 − 1 1 − 
H(12) =  − 1 1 1 1 1 − − − − − −    1 − 1 − − 1 − − − 1 1 −  1 1 − 1 − − − − − − 1 1   1 − 1 − 1 − − 1 − − − 1  1 − − 1 − 1 − 1 1 − − − 
1 1 − 1 1 − − − 1 1 − −
2.3.3 Williamson Construction
In 1944, Williamson found a class of Hadamard matrices by plugging a set of typical
matrices into a pre-designed array, called Williamson array. This method is very simple to
construct Hadamard matrices while the challenge is to find those typical plugging matrices
of order t ∈ N. Just before turning to this construction, we recall the definition of these
plugging matrices.
15
2.3. WILLIAMSON CONSTRUCTION
Definition 2.26. A set of n×n matrices {A,B,C,D} satisfying the following conditions
(i) all are symmetric and circulant with entries from the set {±1},
(ii) AAT +BBT +CCT +DDT = 4nIn.
are called Williamson matrices. However, if {A,B,C,D} is pairwise amicable for A,B,C,D
and AAT +BBT +CCT +DDT = 4nIn then these are known as Williamson-type matrices.
Williamson matrices are known for all orders n ≤ 63 [7] (section V ) while for n ∈
{35,47,53,59} there are no Williamson matrices [24].
Theorem 2.27. [42] If there exist n×n Williamson matrices A,B,C and D then there exist
a Hadamard matrix of order 4ngiven by  A B C D





−B A D −C 
−C −D A B 
−D C −B A
Example 2.28. Picking Williamson matrices as A= circ(1,−,−,−,−), B= circ(1,−,−,−,
−), C = circ(1,1,−,−,1) and D = circ(1,−,1,1,−) we get a Hadamard matrix of order
20.
16
2.3. WILLIAMSON CONSTRUCTION
  1 − − − − − 1 1 1 1 − − 1 1 − − 1 − − 1  − 1 − − − 1 − 1 1 1 − − − 1 1 1 − 1 − −   − − 1 − − 1 1 − 1 1 1 − − − 1 −

1 − 1 −  − − − 1 − 1 1 1 − 1 1 1 − − − − − 1 − 1 

 − − − − 1 1 1 1 1 − − 1 1 − − 1 − − 1 − 

 1 − − − − 1 − − − − − 1 − − 1 1 1 − − 1 

− 1 − − − − 1 − − − 1 − 1 − − 1 1 1 − −   − − 1 − − − − 1 − − − 1 − 1 − − 1 1 1 − 

− − − 1 − − − − 1 − − − 1 − 1 − − 1 1 1
 

  − − − − 1 − − − −1 1 − − − − 1 − − 1 1    1 1 − − 1 1 − 1 1 − 1 − − − − − 1 1 1 1   1 1 1 − − − 1 − 1 1 − 1 − − − 1 − 1 1 1    − 1 1 1 − 1 − 1 − 1 − − 1 − − 1 1 − 1 1  − − 1 1 1 1 1 − 1 1 − − − 1 − 1 1 1 − 1  

  1 − − 1 1 1 1 1 − 1 − − − − 1 1 1 1 1 −  

 1 − 1 1 − − − 1 1 − 1 − − − − 1 − − − − 

 − 1 − 1 1 − − − 1 1 − 1 − − − − 1 − − −

  1 − 1 − 1 1 − − − 1 − − 1 − − − − 1 − −   1 1 − 1 − 1 1 − − − − − − 1 − − − − 1 − 
− 1 1 − 1 − 1 1 − − − − − − 1 − − − − 1
Hadamard matrices obtained in this way are known as Williamson-type Hadamard ma-
trices. Employing this idea Baumert et.al. showed the existence of Hadamard matrices of
order 92 [4] and order 116 [2] which were unknown in the list of Hadamard matrices of
order less than 200 made by Paley. In addition to these methods, the Baumert Hall array is
another powerful tool for constructing Hadamard matrices which we cover in Chapter 6.
17
Chapter 3
Sequences
As far as study of different combinatorial objects like orthogonal designs and Hadamard
matrices are concerned, a set of sequences with commuting variables are of great interest.
Due to different generic properties, these sequences have versatile applications in computa-
tional mathematics, infrared multislit spectrometry [17], optical time-domain reflectometry
or orthogonal frequency division multiplexing (OFDM) [37] etc. These sequences include
Golay sequences, Base sequences, Turyn sequences, T-sequences etc. In this chapter we
study T-sequences among others.
3.1 T-Sequences
We start this section with some trivial properties of sequences.
Definition 3.1. If X = {x11, . . . ,x1n} is a sequence in commuting variables and is of length
n, then its aperiodic auto-correlation function is defined by
N n− jX( j) = ∑i=1 (x1,ix1,i+ j); 1≤ j ≤ n−1.
For a set of sequences Y = {{x11, . . . ,x1n},{x21, . . . ,x2n}, . . . ,{xm1, . . . ,xmn}} of length n
we define,
N j ∑n− jY ( ) = i=1 (x1,ix1,i+ j + x2,ix2,i+ j + · · ·+ xm,ixm,i+ j);1≤ j ≤ n−1.
Similarly, the periodic auto-correlation function is defined by
PY ( j) = ∑ni=1(x1,ix1,i+ j + x2,ix2,i+ j + · · ·+ xm,ixm,i+ j);1≤ j ≤ n−1.
where the second subscript is from the set of residues (mod n).
18
3.2. GOLAY SEQUENCES
Definition 3.2. A set of sequences having aperiodic auto-correlation function zero is called
complementary sequences (CS).
Here we note that the number of non zero entries in a sequence is called the weight
of that sequence and if at most one sequence is non zero at any position among all these
sequences then these sequences are called disjoint sequences (DS).
Example 3.3. A and B are disjoint complementary sequences of length 8.
A = (11000000)
B = (001−11−1)
where “−” stands for −1.
Definition 3.4. Four disjoint sequences M = {T1,T2,T3,T4}with entries from the set {0,±1}
and of length n satisfying
NM( j) = 0;1≤ j ≤ n−1
are called T-sequences (TS).
Example 3.5. Here T1,T2,T3 and T4 are T-sequences of length 9.
T1 = 100000000, T2 = 011000000,
T3 = 0001−11−1, T4 = 000000000.
These T-sequences have a close relation with many other sequences having aperiodic
auto-correlation function zero. In the next few sections, we study some of these sequences.
3.2 Golay Sequences
Definition 3.6. Two (±1) sequences {X = {x11, · · · ,x1n},Y = {y11, · · · ,y1n}} of length n
satisfying,
19
3.2. GOLAY SEQUENCES
NX( j)+NY ( j) = 0 for 1≤ j ≤ n−1
are called Golay sequences. Sometimes X , Y are also referred to as a Golay pair.
Example 3.7. X = 111− and Y = 11−1 are the Golay sequences of length 4.
Golay originally found four pairs of sequences of lengths 2, 10 and 26 which include
one pair of length 2, two pairs of length 10 and one pair of length 26. Now we present these
four pairs of sequences.
Table 3.1: Original Golay-sequences
Order(n) Sequences
2 (11);(1−).
10 (−11−1−111−),(−111111−−1).
(1−1−1111−−),(1111−11−−1).
26 (1−11−−1−−−−1−1−−−−11−−−1−1),
(−1−−11−1111−−−−−−−11−−−1−1).
These sequences have many properties some of which we present here.
3.2.1 Symmetry Properties
The symmetry properties are one of the most important characteristics of Golay se-
quences. This properties permits one or more of the following operations which were
originally given by Golay [18].
(i) Reversing either or both sequences.
(ii) Negating either or both sequences.
(iii) Transforming by x → (−1)ii xi,yi→ (−1)iyi.
(iv) Interchanging the sequences in the pair.
20
3.2. GOLAY SEQUENCES
3.2.2 Other Properties
Due to versatile importance in many engineering applications, many people are inter-
ested in characterizing Golay sequences. Here we mention some characteristics of Golay
sequences found by different authors [15], [18], [41].
(i) ∑n− ji=1 (xixi+ j + yiyi+ j) = 0; for 1≤ j 6= 0≤ n−1.
(ii) They exist for all lengths n ∈ 2a10b26c;a,b,c ≥ 0 [41]. No Golay sequences are
known to exist for other than these lengths.
(iii) Length n is even and is the sum of two squares [18].
(iv) xn−i+1 = eixi⇔ yn−i+1 =−eiyi where ei =±1 [15] .
S. Eliahou, M. Kervaire and B. Saffari’s [12] outstanding work is very helpful to study the
non existence of Golay sequence. They showed that there are no Golay sequences of length
n having a prime factor p ≡ 3 (mod 4). This work also includes the works of M. Griffin
[19] and C. Koukouvinos et.al. [35] where they proved that Golay sequences do not exist
for the lengths 2 ·9t and 2 ·49t respectively where t is a positive integer. In order to prove
this theorem here, we recall first the idea of Hall polynomial [7] and then a lemma whose
proof is given in the same paper of S. Eliahou et.al. [12].
Definition 3.8. If A = (a0,a1, · · · ,an−1) is a sequence of length n, then its associated Hall
polynomial is defined as
n−1
A(x) = ∑ a xii .
i=0
For a given Hall polynomial A(x), we define A(x−1)=∑n−1 a x−ii=0 i and |A|2 =A(x)A(x−1).
Then we get
( )( )
n−1 n−1
|A|2 = ∑ a xii ∑ aix−i
i=0 i=0
21
3.2. GOLAY SEQUENCES
n−1 n−1 n−1− j
= ∑ a2i + ∑ ∑ akak+ j(x j + x− j)
i=0 j=1 k=0
n−1 n−1
= ∑ a2k + ∑ N(k)(xk + x−k)
k=0 k=0
where N(k) is the aperiodic auto-correlation function. So for a (±1) sequence A of length n
we have, |A|2 = n+∑n−1k=0 N(k)(x
k +x−k) and for a pair of (±1)-complementary sequences
A and B of length n, we get |A|2 + |B|2 = 2n.
Lemma 3.9. [12] Let p be a prime number satisfying p≡ 3 (mod 4). Let N be a positive
integer ζ be the primitive pN-th root of unity. Suppose α,β satisfy the condition
α ·α+β ·β ∈ p2Z[ζ]
where bar denotes the complex conjugation. Then α,β ∈ pZ[ζ].
Theorem 3.10. [12] There are no Golay sequences of length n, divisible by a prime factor
p of the form p = 4k+3.
Proof. Assume that P and Q are Golay sequences of length n. If P(x) and Q(x) are the
corresponding Hall polynomials then we must have
P(x)P(x−1)+Q(x)Q(x−1) = 2n
in the Laurent polynomial ring Z[x,x−1]. Putting x = 1, we obtain
P(1)2 +Q(1)2 = 2n
i.e 2n is the sum of two integral squares which implies that if n is divisible by a prime p≡ 3
(mod 4), then n must be divisible by p2. Indeed, P(1)2 +Q(1)2 = 0 (mod pZ) is possible
only when P(1)≡ Q(1)≡ 0 (mod pZ). Then p2 must divide P(1)2 +Q(1)2.
Now by establishing a contradiction with the assumption n≡ 0 (mod p2); p≡ 3 (mod 4),
we prove the theorem.
Suppose q = pN ;N is large and ζ be a primitive q’th root of unity. Then we obtain
22
3.2. GOLAY SEQUENCES
P(ζ)P(ζ)+Q(ζ)Q(ζ) = 2n ∈ p2Z[ζ]
where P(ζ−1) = P(ζ) = P(ζ); ζ is the complex conjugation of ζ.
Therefore, using the lemma above, we can write P(ζ),Q(ζ) ∈ pZ[ζ]. However,
P(ζ) = a0 +a1ζ+ · · ·+an−1ζn−1 where ai =±1; for 0≤ i≤ n−1.
N−1
Now, for prime power q = pN , let Φ X p−1q( ) = ∑ i(p )i=0 X is a cyclotomic polynomial over
Z[X ] and so Z[ζ] is a free Z-module whose rank is Φ = φ(pN) = pN−1(p− 1). If P(ζ) ∈
pZ[ζ] then there exist integers s0,s1, . . . ,sn−1 ∈ Z such that
n−1 φ(q)−1
P(ζ) = ∑ aiζi = ∑ ps jζ j (3.1)
i=0 j=0
Assume that N is big enough and satisfies the inequality n < φ = φ(pN) = pN−1(p− 1).
Since {1,ζ,ζ2, · · · ,ζφ−1} form a basis over Z, we deduce from Equation 3.1
ai = psi;0≤ i≤ n−1
which is impossible since ai =±1. Hence the theorem follows.
To construct new Golay sequences, one way is to construct them from already existing
Golay sequences. Before shifting into this construction we review some basic operations
on sequences.
For a pair of sequences X = (x1,x2,x3, ...,xm) and Y = (y1,y2,y3, ...,ym),
(i) Concatenation is defined as:
X |Y = (x1,x2, ...,xm,y1,y2, ...,ym).
(ii) Interleaving is defined as:
X ∼ Y = (x1,y1,x2,y2, ...,xm,ym).
23
3.3. BASE SEQUENCES
If X = (x1,x2,x3, ...,xm) and Y = (y1,y2,y3, ...,ym) are Golay sequences of length n then
(X |Y,X |(−Y )) and (X ∼ Y,X ∼ (−Y )) are Golay sequences of length 2n [27]. Golay
showed how to double the length of existing Golay sequences, which is known as Golay’s
direct construction method [18]. Using this method one can compute a Golay pair of length
2n from a existing pair of length n. These sequences are very helpful to construct a set of
four complementary sequences. Here we mention a lemma for such a construction.
Lemma 3.11. If X ;Y are Golay sequences of length n1 and U ;V are Golay sequences of
length n2, then A = (X |U), B = (X |−U), C = (Y |V ) and D = (Y |−V ) are four comple-
mentary sequences of length (n1 +n2).
Since our interest is the construction of T-sequences, we mention here a theorem that
generates T-sequences from Golay sequences.
Theorem 3.12. If X and Y are Golay sequences of lengths n, and if φn denotes the vector
of n zeroes then T = {{1,φn},{φ, 12(X +Y )},{φ,
1
2(X −Y )},{φn+1}} are T-sequences of
length n+1.
Proof. Obvious.
See table 3.2 for examples.
Table 3.2: T-sequences (TS) from Golay-sequences (GS)
GS of length (n) TS length (n+1)
11,1− 100,010,001,000.
111−,11−1 10000,01100,
0001−,00000.
111−11−1,111−−−1− 100000000,0111−0000,
0000011−1,000000000.
24
3.3. BASE SEQUENCES
3.3 Base Sequences
Definition 3.13. If {X1,X2,X3,X4} are all {±1} sequences of length (m+ p),(m+ p),m,m
respectively, where p is an odd and if the aperiodic auto-correlation function, is zero then
the sequences Xi’s; 1≤ i≤ 4; form Base sequences (BS).
Here we list some Base sequences of odd lengths.
Table 3.3: Base-sequences of odd lengths
Lengths Sequences
3,3,2,2 111,11−,1−,1−.
5,5,4,4 11−11,1111−,11−−,1−1− .
7,7,6,6 11−1−11,11−−−1−,−111111,−−11−1.
9,9,8,8 1111−11−1,−111−11−1,111−−−1−,111−−−1−.
One can regard Golay sequences as appropriate tools to construct Base sequences. We
see, if X ,Y are Golay sequences of length n, then the set {{1,X},{1,−X},{Y},{Y}} forms
Base sequences of length (n+1),(n+1),n and n respectively. Base sequences exist for all
lengths with m+ p;m ∈ {1, . . . ,30}∪2a10b26c and p = 1.
Base sequences are also effective tools to find four complementary sequences(CS) as well
as T-sequences. To illustrate this we state the following theorem from the monograph by J.
Seberry et.al. [39].
Theorem 3.14. If there are Base sequences of lengths m+1,m+1,m,m, there are
(i) 4 T-sequences of length (2m+1).
(ii) 4 complementary sequences of length (2m+1).
Proof. Suppose A = {X ,Y,Z,W} are Base sequences of lengths (m + 1),(m + 1),m,m.
Then the set of sequences defined as:
25
3.4. TURYN TYPE SEQUENCES
{{X,W},{X,-W},{Y,Z},{Y,-Z}}
are 4-complementary sequences of length (2m+1) and the set of sequences defined as:
{{12(X +Y ),φm},{
1
2(X−Y ),φm},{φ ,
1
m+1 2(W +Z)},{φm+1,
1
2(W −Z)}}
are T-sequences of length (2m+1).
Table 3.4 lists examples.
Table 3.4: T-sequences and Complementary sequences from Base sequences
BS of lengths TS of length CS of length
m+1,m+1,m,m (2m+1) (2m+1)
111,11−,1−,1−. 11000,00100,0001−,00000. 1111−,111−1,11−1−,11−−1.
11−1,11−−, 11−0000,0001000, 11−1111,11−1−−−,
111,1−1. 0000101,0000010. 11−−1−1,11−−−1−.
3.4 Turyn Type Sequences
Definition 3.15. A set {X ,Y,Z,W} of four {±1} sequences of lengths n,n,n and (n− 1)
respectively is said to be of Turyn type if the following condition is holds.
NX( j)+NY ( j)+2NZ( j)+2NW ( j) = 0; j ≥ 1
Turyn type sequences first appeared in a paper of R. J. Turyn [41]. In that paper, he gave
three examples of these sequences for lengths n = 4,6,8. Except these lengths all other
Turyn type sequences were found by computer search. Koukouvinos et.al. constructed
these sequences for lengths n = 10,12, . . . ,24 in [33] and the credit for the other lengths
n = 26,28, . . . ,34 goes to Kounias et.al. [34]. H. Kharaghani and B. Tayfeh-Rezaie found
Turyn type sequences of length n = 36 and verified the existence of a Hadamard matrix
of order 428 [29]. In the same paper [41], Turyn established a theorem to extract Base
sequences from existing Turyn type sequences. Now we present this well known theorem.
26
3.4. TURYN TYPE SEQUENCES
Theorem 3.16. [41] If X ,Y,Z,W are Turyn type sequences of lengths n,n,n,(n−1), then
the sequences A = (Z|W ),B = (Z|−W ),C = X ,D =Y are Base sequences of lengths (2n−
1),(2n−1),n,n.
See Table 3.5 for examples.
Table 3.5: Base sequences (BS) from Turyn sequences (TrS)
TrS of lengths BS of lengths
n,n,n,(n-1) (2n-1),(2n-1),n,n
1−1−,1−−−,1−−1,111. 1−−1111,1−−1−−−,1−1−,1−−−.
1−11−1,111−−1, 1−111−1−−−−,1−111−−1111,
1−111−,1−−−−. 1−11−1,111−−1.
1−−1−−−1,111−−−−1, 1111−−1−111−111,1111−−1−−−−1−−−,
1111−−1−,111−111. 1−−1−−−1,111−−−−1.
Yang’s [43, 44] work on Base sequences is an crucial step to find odd order T-sequences
of larger lengths. He showed if there is a number y ∈ {3,7,13, ...,2g + 1} where g ∈
2a10b26c;a,b,c ≥ 0, termed as Yang’s number and there are Base sequences of lengths
m+ p,m+ p,m,m then there are T-sequences of length y(2m+ p) which is of great interest
for the case when (2m+ p) is odd.
Theorem 3.17. (Yang) Let A,B,C,D be sequences of lengths (m+ 1),(m+ 1),m,m, re-
spectively and G = (gk) and F = ( fk) be Golay sequences of lengths s. Then the following
Q,R,S,T are 4-complementary sequences. Using X∗ to denote the reverse of X:
Q = (A fs,Cg1;0,0,A fs−1,Cg2;0,0; , ...,A f1,Cgs;0,0;−B∗,0);
R = (B fs,Dgs;0,0;B fs−1,Dgs−1,0,0, ...,B f1,Dg1;0,0;A∗,0);
S = (0,0;Ags,−CF1;0,0,Ags−1,−C f2; ...,0,0;Ag1,−C fs;0,−D∗);
T = (0,0;Bg1,−D f1;0,0;Bg2,−D f2; ...;0,0;Bgs,−D fs;0,C∗);
27
3.4. TURYN TYPE SEQUENCES
Furthermore, if we define sequences
X1 = (Q+R)/2, X2 = (Q−R)/2, X3 = (S+T )/2, X4 = (S−T )/2,
then the sequences become T-sequences of lengths t(2s+1), t = (2m+ p).
Proof. Let P(x),Q(x),R(x),S(x) be the associated Hall polynomials of the sequences X1,X2,
X3 and X4, having length (2s+1)(2m+ p), and weights w1,w2,w3,w3 respectively.
Defining |P|2 = P(x)P(x−1), we compute
|P|2 + |Q|2 + |R|2 + |S|2 = (w1 +w2 +w3 +w4)
where P(x−1) is the involution of P(x) and all these sequences are disjoint {0,±1}. So the
theorem follows.
Although T-sequences can be obtained from many infinite class of sequences, they do
not exist for all lengths. At the end of this chapter, we present a conjecture relating the
existence of T-sequences [7].
Conjecture 3.18. There are T-sequences of lengths n for each odd integer n≥ 1.
This conjecture has been verified for all odd integers n < 200 [7] (section V ) except for
n = 97,103,109,113,137,139,149,151,157,163,167,173,179,181,191,193,197,199.
28
Chapter 4
Balanced Generalized Weighing Matrices
In the field of combinatroics, much research has been done on the multiple application of
balanced generalized weighing matrices. This class of matrices includes Hadamard ma-
trices and conference matrices. In this chapter, we study balanced generalized weighing
matrices with classical parameter that includes ω-circulant as well as negacirculant weigh-
ing matrices.
4.1 Difference Sets
We begin here with the notion of set theory over a finite cyclic group G.
Definition 4.1. Let G be a multiplicative group of order m. Then a k-subset S(m,k,µ) of G
is said to be a difference set over G if every g ∈ G\{id} can be expressed as s −11s2 exactly
µ ways for s1,s2 ∈ S.
A simple computation shows that there are exactly (k2−k) pairs of elements in S which
produce non-identity elements and so the difference set satisfies the following equation
k2− k = (m−1)µ
This becomes more clear from the following examples.
Example 4.2. {0,1,3} is a (7,3,1) difference set in Z7.
Example 4.3. Over Galois field, GF(23) = {0,1,x,x2,x+1,x2+1,x2+x,x2+x+1}, α =
x+ 1 is a primitive element of GF(23)∗. Let S = {α,α2,α4} is a 3-subset of GF(23)∗.
Then we compute
{s s−1}= {α−1,α−3i j ,α,α
−2,α3,α2}
29
4.2. BALANCED GENERALIZED WEIGHING MATRICES
= {x,x2,x+1,x2 +1,x2 + x,x2 + x+1};(1≤ i 6= j ≤ 3)
where α−1 =α6. Hence, S is a (7,3,1) difference set over the multiplicative group GF(23)∗.
Definition 4.4. Let H be a normal subgroup of order m of a group G of order mn. Then a
k-subset D = {g1,g2, ...,gk} of distinct elements of G with parameters (m,n,k,λ), is said
to be a relative difference set (RDS) if the set
{gig−1j ;1≤ i 6= j ≤ k}
contains each element of the set G\H exactly λ times and no element of H exists within
this set.
Example 4.5. Let S = {1,α,α2,α4} be 4-subset of GF(23)∗. Then we compute
D = {g g−1i j }= {α
−1,α−2,α−4,α,α−1,α−3,α2,α,α−2,α4,α3,α2}
= {α,α,α2,α2,α3,α3,α4,α4,α5,α5,α6,α6}
= {x,x,x2,x2,x+1,x+1,x2 +1,x2 +1,
x2 + x,x2 + x,x2 + x+1,x2 + x+1};(1≤ i 6= j ≤ 3)
Here |GF(23)∗| = 7 and D contains each element of GF(23)∗ exactly 2 times except the
identity element. Thus S is a (7, 1, 4, 2) RDS with respect to the trivial subgroup.
However, in case of trivial subgroup H in G, an RDS in G with respect to H is a
difference set in G. Though there are several kinds of relative difference sets, we limit
ourselves in the study of relative difference set with classical parameters.
Definition 4.6. If q is a prime power and d is positive integer then a RDS with parameters
((qd+1−1)/(q−1);q−1;qd;qd−1)
in a cyclic group is said to be RDS with classical parameters.
30
4.2. BALANCED GENERALIZED WEIGHING MATRICES
4.2 Balanced Generalized Weighing Matrices
Definition 4.7. A square m×m matrix W = (wi j) over a multiplicative group G is called
Balanced Generalized Weighing Matrix BGW (m,k,µ) [25] if
(i) wi j ∈ {G∪0},
(ii) There are exactly k non zero elements in each row and each column in W ,
(iii) The multiset {w −1aiwbi ;1≤ i≤m,wai,wbi =6 0} contains precisely µ/|G| copies of each
element of G for a,b ∈ {1,2, ...,m}.
Example 4.8. A BGW(5, 4, 3) over a cyclic group G =< ω> of order 3 is as follows:
 0 1 1 1 1
 1 0 1 ω ω
2 
W = 1 1 0 ω2 ω 


 1 ω ω2 0 1 
1 ω2 ω 1 0
Definition 4.9. If α is a generator of the cyclic group G and W = (wi j) is an m×m square
matrix over the set G∪{0} then W is called ω- circulant if wi+1, j+1 = wi, j and wi+1,1 =
ωwi,m for 1≤ j ≤ m−1, where ω = α−1.
This class of matrices include BGW matrices. Jungnikel et.al. [26] gave a method for
constructing these class of matrices over Galois field(GF).
Example 4.10. Over GF(32)∗, R = {α4,α5,α7} is (4,2,3,1) relative difference set with
respect to N = GF(3)∗ where α = x+ 1. Then the corresponding ω-circulant BGW(4, 3,
1) is
  4 4 α α 0 α
4


1 α4 α4 
W = 0   0 1 α4 α4 
1 0 1 α4
31
4.3. NEGACIRCULANT MATRICES
where ω = α4 = 2.
4.3 Negacirculant Matrices
A special feature of ω-circulant matrices covers the notion of negacirculant matrices.
This class of matrices is very useful because of its converging nature to balanced gener-
alized weighing matrices, conference matrices and many others. In particular, the theory
behind this class of matrices opens the door to our research.
Definition 4.11. A weighing matrix C of order ν with off diagonal elements from the set
{±1} and diagonal elements are all zero, satisfying
CCT = (ν−1)Iν
is known as a conference matrix.
Example 4.12. A conference matrix of order 6 and weight 5, C =C(6,5) is
 0 − − − − −  

− 0 − 1 1 − 


 − − 0 − 1 1 

C =


 − 1 − 0 − 1  

 − 1 1 − 0 − 
− − 1 1 − 0
For i = 0,1, . . . ,ν−1, define a square matrix N of order ν with entries given by :
Ni,i+1 = 1,
Nν−1,0 =−1and
Ni, j = 0;otherwise.
N generates a matrix group of order 2ν where Nν =−Iν, NT =−Nν−1 and NNT = Iν.
32
4.3. NEGACIRCULANT MATRICES
Definition 4.13. Any square matrix A of order ν is said to be negacirculant [10] if AN =
NA. For i = 0,1, ...,ν−2, its entries ai, j;0≤ i, j ≤ ν−1 are of the form:
ai+1,0 =−ai,ν−1
ai+1, j = ai, j−1; j = 1,2, ...,ν−1.
Example 4.14.  
 0 1 1 − 1 − − − − 1

− 0 1 1 − 1 − − − − 
   1 − 0 1 1 − 1 − − − 

  1 1 − 0 1 1 − 1 − − 

1 1 1 − 0 1 1 − 1 − 
N = 


1 1 1 1 − 0 1 1 − 1 


 − 1 1 1 1 − 0 1 1 − 
 1 − 1 1 1 1 − 0 1 1   − 1 − 1 1 1 1 − 0 1 
− − 1 − 1 1 1 1 − 0
In [36], Paley constructed so called CK-conference matrices of order q+1 over GF(q)
for an odd prime power q = pk. If K is the set of cardinal number q+1 that includes pair-
wise independent vectors from the 2- dimensional vector space V (2,q) then the associated
Paley matrix is defined [10] as
CK = [χdet(xi,x j);xi,x j ∈ K] for i, j ∈ {0,1, ...,q}
where χ is the Legendre symbol over GF(q). Over the GF(5), CK is the Paley matrix
associated with the set K = {(0,1),(1,0) (2,1),(3,1),(3,2),(1,1)} such that C CTK K = 5I6.
33
4.3. NEGACIRCULANT MATRICES
  0 1 − − − 1  

1 0 1 1 − 1 
   − 1 0 1 1 1

CK =   − 1 1 0 − −  − − 1 − 0 1 
1 1 1 − 1 0
34
Chapter 5
Amicable T-Matrices
Earlier in this thesis, we saw that Williamson matrices of order n imply the existence of
Hadamard matrices of order 4n. Like Williamson matrices, there is another interesting
class of matrices called T-matrices whose size is directly related to the size of many combi-
natorial objects such as orthogonal designs and Hadamard matrices. J. Cooper and J. Wallis
were the first in giving the formal definition of T-matrices in 1972 [8]. In this chapter, we
form an infinite class of amicable T-matrices of order n = q+ 1, where q is odd prime
power and in particular, of order n≡ 2 (mod 4). Moreover, we introduce a multiplication
theorem for amicable T-matrices which is very useful to lengthen the size of T-matrices.
5.1 T-Matrices
We start here with few basic definitions.
Definition 5.1. Two {0,±1} matrices A and B of the same size are said to be disjoint if
there is at most one nonzero element in the same position of the both matrices for every
position. More explicitly, if there is a nonzero entry at the (i, j)-th position of A then the
entry at the same position in B must be zero, i.e, A∗B = 0.
Example 5.2. A and B are disjoint.    1 0 1 0  0 1 0 1 0 1 0 0  
 

 1 0 0 1
A =  B = 

 0 0 1 1   1 1 0 0 
1 0 1 0 0 1 0 1
35
5.1. T-MATRICES
Definition 5.3. A set of matrices {A1,A2, · · · ,A2n} is said to be amicable [28] if it satisfies
∑n T Ti=1(A2i−1A2i−A2iA2i−1) = 0
If this set consisting of only two matrices then these matrices are called amicable i.e, if
A1AT2 = A2A
T
1 then A1,A2 are amicable. We call a pair of matrices (A1,A2), for which
amicability happens in a set, is amicable in matching (A1,A2). In the definition above, for
n = 4, A1,A2,A3,A4 are amicable in matching (A1,A2) and (A3,A4)
Definition 5.4. A set of quadruples {T1,T2,T3,T4} of square matrices of order n with entries
from the set {0,±1} are called T-matrices if
(i) they are pairwise disjoint,
(ii) the commutative property holds for every pair of matrices in the set
K = {T1,T ,T ,T ,T T ,T T T2 3 4 1 2 ,T3 ,T T4 },
(iii) there exist a monomial matrix (matrix having a single nonzero entry in each row and
each column) R of order n such that AR is amicable to B for all A,B ∈ K,
(iv) T T T T T1T1 +T2T2 +T3T3 +T4T4 = tIn for some t.
Example 5.5. T1,T2,T3 and T4 are the T-matrices of order 5.
 1 1 0 0 0 

 0 0 1 0 0   

 0 1 1 0 0 0 0 0 1 0 T1 = 0 0 1 1 0  T2 = 0 0 0 0 1   0 0 0 1 1  

1 0 0 0 0 
1 0 0 0 1 0 1 0 0 0
36
5.1. T-MATRICES
    0 0 0 1 −   0 0 0 0 0  −
  
0 0 0 1
 
  0 0 0 0 0 
T3 =
  
1 − 0 0 0  T4 = 0 0 0 0 0  0 1 − 0 0  
  0 0 0 0 0 
0 0 1 − 0 0 0 0 0 0
A set of T-matrices satisfying the condition in Definition 5.3 with matching (T1,T2) and
(T3,T4) are called amicable T-matrices. In a study [46], G.X Zuo and M. Y. Xia used the
matching (T1,T T T4 ) and (T3,−T2 ) in the definition of amicability and called them suitable
T-matrices.
Definition 5.6. A set of T-matrices {T1,T2,T3,T4} satisfying the amicablility condition with
matching (T1,T3 +T4) and (T2,T3−T4) i.e,
T1(T3 +T4)T +T2(T T T T3−T4) = (T3 +T4)T1 +(T3−T4)T2
is called weak amicable T-matrices [23].
In a paper published in 2006 [23], Holzmann and Kharaghani found weak amicable
T-matrices for orders {3,5, . . . ,21}. We use these matrices to construct amicable matrices
in terms of variables in Chapter 6.
Example 5.7. T1 = circ(1,0,0),T2 = circ(0,0,0),T3 = circ(0,1,0) and T4 = circ(0,0,1)
are weak amicable T-matrices of order 3 in matching (T1,T3 +T4) and (T2,T3−T4) .
Definition 5.8. A set of T-matrices {T1,T2,T3,T4} satisfying
(T1 +T2)∗ (T3 +T )T4 = 0
is called strongly disjoint (SD) T-matrices [46] where ∗ is the Hadamard product of two
matrices.
Example 5.9.
For n = 5, the following {T1,T2,T3,T4} are SD T-matrices.
37
5.1. T-MATRICES
    1 0 0 0 1   0 1 0 0 0 
  
1 1 0 0 0   0 0 1 0 0 T1 = 0 1 1 0 0  
 
, T2 = 0 0 0 1 0  , 0 0 1 1 0   0 0 0 0 1 

 0 0 0 1 1  
1 0 0 0 0 
 0 0 1 − 0 

 −  
 0 0 0 0 0  0 0 0 1  

0 0 0 0 0 
T3 =   − 0 0 0 1 
, T4 =
 
0 0 0 0 0  .
 1 − 0 0 0  
 

0 0 0 0 0
0 1 − 0 0 0 0 0 0 0
Definition 5.10. A set of amicable T-matrices with matching (T1,T T4 ) and (T2,T
T
3 ) is called
a proper amicable T-matrices [16]. Equivalently the condition is
(T T1T4 +T2T3) = (T1T4 +T2T3)
Example 5.11. T1 = circ(1,0,0),T2 = circ(0,−,0),T3 = circ(0,0,1) and T4 = circ(0,0,0)
is proper amicable in matching (T1,T T4 ) and (T ,T
T
2 3 ).
This class of T-matrices is known for order 3,5,7,9,11,13,17,21 [16].
T-matrices satisfying disjointedness and proper amicability properties have a useful role
in the construction of four circulant amicable matrices in terms of variables. But such a
set of T-matrices of odd order does not exist. Now we present a theorem from Hamed
Gholamiangonabadi’s M.Sc thesis [16] showing that it is not possible to find odd order
T-matrices that have both of the conditions.
Theorem 5.12. There are no proper amicable T-matrices of odd order that satisfy the
disjointedness property.
38
5.2. AMICABLE T-MATRICES OF ORDER N ≡ 2 (MOD 4)
Proof. Consider a set of T-matrices {T1,T2,T3,T4} of odd order n which meets the follow-
ing conditions simultaneously.
(T1T4 +T2T3) = (T1T T4 +T2T3)
(T +T )∗ (T +T )T1 2 3 4 = 0
Let X = T1 +T T4 and Y = T2 +T
T
3 , then we compute
XXT +YY T = (T1 +T T4 )(T
T
1 +T4)+(T2 +T
T
3 )(T
T
2 +T3)
= (T T T T T T T T T1T4 +T2T3 +T1 T4 +T2 T3 )+(T1T1 +T2T2 +T3T3 +T4T4 )
≡ In (mod 2)
where we use the fact of amicability in matching (T T1,T4 ) and (T2,T
T
3 ) and matrix A≡−A,
(mod 2). Since they are strongly disjoint we have, X +Y = J(mod 2) and so, Y = X +
J(mod 2). Then we compute
XXT +YY T = XXT +XXT +XJT + JXT + JJT ≡ J (mod 2).
since for circulant matrices, AJT = JAT . Hence, XXT +YY T = In = J, a contradiction.
In the next section, we make progress towards one of our main research objective and
show that amicable T-matrices exist for order n≡ 2 (mod 4).
5.2 Amicable T-Matrices of Order n≡ 2 (mod 4)
In a study [10], P. Delsarte et.al. determined that the only negacirculant W (n,n−1) of
order n < 1000 have n = pr + 1 where pr is an odd prime power. An important theorem
by Delsarte et.al. [10] states that there exist an infinite class of negacirculant weighing
matrix of order n+ 1 where n is odd prime power. Proof of this theorem come directly
from the proofs of a theorem and a corollary which are given in the paper [10]. We give
this significant theorem with an immediate example.
39
5.2. AMICABLE T-MATRICES OF ORDER N ≡ 2 (MOD 4)
Theorem 5.13. [15] There is a negacirculant W (pr +1, pr) whenever pr is an odd prime
power.
40
5.2. AMICABLE T-MATRICES OF ORDER N ≡ 2 (MOD 4)
Example 5.14.  
 0 1 − −   1 0 1 −

W (4,3) =  1 1 0 1 
− 1 1 0
5.2.1 Construction of Amicable T-Matrices of Order n = pr + 1; pr is Odd Prime
Power
We know that the sum of periodic auto-correlation function of a identity matrix and a
negacirculant weighing matrix of same order is zero. Let
T1 = In, T2 =W (n,n−1), T3 = 0n, T4 = 0n;
where n = pr + 1, (pr is odd prime power) and 0n is zero matrix of order n. All of Ti’s;
1 ≤ i ≤ 4; are disjoint and cyclic, so form a commutative set with the all its transpose and
meet the requirement of amicability condition in pair. Moreover they satisfy
T T T1 1 +T2T
T
2 +T3T
T
3 +T4T
T
4 = nIn.
Consequently, T1,T2,T3 and T4 form a set of amicable T-matrices.
Example 5.15.
    1 0 0 0 0 0   0 1 1 1 − 1 
  
0 1 0 0 0 0   − 0 1 1 1 −   0 0 1 0 0 0

T1 =  T2 = 
 1 − 0 1 1 1 
 0 0 0 1 0 0   − 1 − 0 1 1 

0 0 0 0 1 0   − − 1 − 0 1 
0 0 0 0 0 1 − − − 1 − 0
41
5.3. MULTIPLICATION THEOREM FOR AMICABLE T-MATRICES
    0 0 0 0 0 0   0 0 0 0 0 0  0 0 0 0 0 0   0 0 0 0 0 0 

 0 0 0 0 0 0   0 0 0 0 0 0 

T3 = T4 =   0 0 0 0 0 0   0 0 0 0 0 0  0 0 0 0 0 0  

0 0 0 0 0 0 
0 0 0 0 0 0 0 0 0 0 0 0
By the above and Theorem 5.13, we have
Theorem 5.16. For every odd prime power pr, there exist amicable T-matrices of order
n = pr +1.
A particular case for pr ≡ 1 (mod 4) in the above theorem gives
Corollary 5.17. There exist amicable T-matrices of order n≡ 2 (mod 4).
5.3 Multiplication Theorem for Amicable T-Matrices
In this section, we introduce a multiplication theorem that leads us to generate infinite
class of composite order T-matrices using amicable T-matrices. Yang used two differ-
ent types of complementary sequences to obtain T-sequences (equivalently T-matrices) of
larger order. Later Zuo and Xia [45] derived T-matrices of order t(2m+ p) from Base
sequences (BS) of lengths m+ p,m+ p,m,m (p odd) and suitable T-matrices of order t. In
2012, the same authors proved a theorem to find composite order T-matrices using directly
strongly disjoint T-matrices and suitable T-matrices [46]. Now we present a theorem that
allows generating new T-matrices using only amicability condition.
Theorem 5.18. Suppose T1,T2,T3,T4 are arbitrary T-matrices of order t and A1,A2,A3,A4
are amicable T-matrices of order n with matching (A1,A3) and (A2,A4). Then the following
matrices
C1 = T1⊗A1−T2⊗A3−T3⊗AT2 −T ⊗AT4 4
42
5.3. MULTIPLICATION THEOREM FOR AMICABLE T-MATRICES
C2 = T1⊗A3 +T2⊗A1 +T3⊗AT T4 −T4⊗A2
C3 = T1⊗A2−T2⊗A4 +T ⊗AT3 1 +T T4⊗A3
C T T4 = T1⊗A4 +T2⊗A2−T3⊗A3 +T4⊗A1
are T-matrices of order tn.
Proof. Since Ti’s and Ai’s are all T- matrices, it clearly follows that
Ci ∗C j = 0; for 1≤ i 6= j ≤ 4.
A simple calculation shows that
4
∑C TiCi = (T1T T1 +T T T2T2 +T3T3 +T T T )⊗ (A AT T T4 4 1 1 +A2A2 +A3A3 +A4AT4 )
i=1
+(T1T T2 −T2T T1 +T T T3T4 −T4T3 )⊗ (A AT T T T3 1 −A1A3 +A4A2 −A2A4 )
= tnItn.
Therefore the theorem follows.
The multiplication theorem by Xia et.al. [46] is useful to find odd composite order
T-matrices using suitable T-matrices but this type of T-matrices are only known for orders
n ∈ {3,5, · · · ,21,25} [45, 46]. However, our approach has greater scope since we impose
only the condition of amicability on one set of T-matrices keeping the other free from any
condition and showed the existence of an infinite class of amicable T-matrices in Theorem
5.16.
Example 5.19. An obvious use of Theorem 5.18 with T-matrices T1 = circ(1,0,0),T2 =
circ(0,1,0), T3 = circ(0,0,1), T4 = circ(0,0,0) and amicable T-matrices as in Example
43
5.3. MULTIPLICATION THEOREM FOR AMICABLE T-MATRICES
5.15 generate T-matrices of order 18.
  1 0 0 0 0 0 0 0 0 0 0 0 0 1 − 1 1 1 0 1 0 0 0 0 0 0 0 0 0 0 − 0 1 − 1 1   0 0 1 0 0 0 0 0 0 0 0 0 − − 0 1 − 1   0 0 0 1 0 0 0 0 0 0 0 0 − − − 0 1 − 0 0 0 0 1 0 0 0 0 0 0 0 1 − − − 0 1   0 0 0 0 0 1 0 0 0 0 0 0 − 1 − − − 0  0 1 − 1 1 1 1 0 0 0 0 0 0 0 0 0 0 0 

− 0 1 − 1 1 0 1 0 0 0 0 0 0 0 0 0 0 
 − − 0 1 − 1 0 0 1 0 0 0 0 0 0 0 0 0 C1 = − − − 0 1 − 0 0 0 1 0 0 0 0 0 0 0 0  1 − − − 0 1 0 0 0 0 1 0 0 0 0 0 0 0 

− 1 − − − 0 0 0 0 0 0 1 0 0 0 0 0 0 


0 0 0 0 0 0 0 1 − 1 1 1 1 0 0 0 0 0 


0 0 0 0 0 0 − 0 1 − 1 1 0 1 0 0 0 0 
  0 0 0 0 0 0 − − 0 1 − 1 0 0 1 0 0 0  0 0 0 0 0 0 − − − 0 1 − 0 0 0 1 0 0 0 0 0 0 0 0 1 − − − 0 1 0 0 0 0 1 0 
0 0 0 0 0 0 − 1 − − − 0 0 0 0 0 0 1
  0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 
 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 

0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
 

 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 

 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 



0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 
C2 = 
 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 

  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1  1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 

0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0
44
5.3. MULTIPLICATION THEOREM FOR AMICABLE T-MATRICES
  0 1 1 1 − 1 0 0 0 0 0 0 1 0 0 0 0 0− 0 1 1 1 − 0 0 0 0 0 0 0 1 0 0 0 0  1 − 0 1 1 1 0 0 0 0 0 0 0 0 1 0 0 0 

− 1 − 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 
− − 1 − 0 1 0 0 0 0 0 0 0 0 0 0 1 0  − − − 1 − 0 0 0 0 0 0 0 0 0 0 0 0 1 

 1 0 0 0 0 0 0 1 1 1 − 1 0 0 0 0 0 0  0 1 0 0 0 0 − 0 1 1 1 − 0 0 0 0 0 0 0 0 1 0 0 0 1 − 0 1 1 1 0 0 0 0 0 0C3 = 

 0 0 0 1 0 0 − 1 − 0 1 1 0 0 0 0 0 0   0 0 0 0 1 0 − − 1 − 0 1 0 0 0 0 0 0   0 0 0 0 0 1 − − − 1 − 0 0 0 0 0 0 0   0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 1 − 1 

 0 0 0 0 0 0 0 1 0 0 0 0 − 0 1 1 1 −  0 0 0 0 0 0 0 0 1 0 0 0 1 − 0 1 1 1  0 0 0 0 0 0 0 0 0 1 0 0 − 1 − 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 − − 1 − 0 1 
0 0 0 0 0 0 0 0 0 0 0 1 − − − 1 − 0
 
 0 0 0 0 0 0 0 1 1 1 − 1 0 0 0 0 0 0 0 0 0 0 0 0 − 0 1 1 1 − 0 0 0 0 0 0

  0 0 0 0 0 0 1 − 0 1 1 1 0 0 0 0 0 0   0 0 0 0 0 0 − 1 − 0 1 1 0 0 0 0 0 0   0 0 0 0 0 0 − − 1 − 0 1 0 0 0 0 0 0 

  0 0 0 0 0 0 − − − 1 − 0 0 0 0 0 0 0   0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 − 1  0 0 0 0 0 0 0 0 0 0 0 0 − 0 1 1 1 −

  0 0 0 0 0 0 0 0 0 0 0 0 1 − 0 1 1 1 C4 =

0 0 0 0 0 0 0 0 0 0 0 0 − 1 − 0 1 1 
  0 0 0 0 0 0 0 0 0 0 0 0 − − 1 − 0 1 

 0 0 0 0 0 0 0 0 0 0 0 0 − − − 1 − 0   0 1 1 1 − 1 0 0 0 0 0 0 0 0 0 0 0 0  − 0 1 1 1 −

0 0 0 0 0 0 0 0 0 0 0 0 


1 − 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 


− 1 − 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 
− − 1 − 0 1 0 0 0 0 0 0 0 0 0 0 0 0 
− − − 1 − 0 0 0 0 0 0 0 0 0 0 0 0 0
45
Chapter 6
New Orthogonal Designs
In the literature that we reviewed in the previous chapters, our purpose was to find some
helpful tools like T-matrices, amicable matrices for constructing some well known arrays
called orthogonal designs. These arrays capture BH-arrays, BHW-arrays and essentially
result in Hadamard matrices. This chapter mainly includes the application of amicable T-
matrices as well as amicable matrices in terms of variables. We begin this chapter with the
definition of orthogonal design.
6.1 Orthogonal Designs
Definition 6.1. Any square matrix A of order n, with entries from a set of commuting
variables, X = {±x1,±x2, . . . ,±xk} satisfying the property
AAT = ∑k 2i=1(sixi )In
is called an orthogonal design of type (n;s1,s2, ...,sk) and is denoted by OD(n;s1,s2, ...,sk)
where we call {s1,s2, . . . ,sn} is the parameter set.
Example 6.2. A is a orthogonal design of type (4;1,1,1) where
 
 0 x1 x2 x3 
A =

−x1 0 −x

3 x2 
 −x2 x3 0 −x1 
−x3 −x2 x1 0
46
6.1. ORTHOGONAL DESIGNS
The Williamson array is a well known example of an orthogonal design of type
(4;1,1,1,1). Depending on the look of the parameter set, orthogonal design has different
name such as Baumert-Hall arrays, Baumert-Hall-Welch arrays, Plotkin arrays etc. As our
research revolves around the construction of Baumert-Hall arrays, we recall the definition
here.
Definition 6.3. An orthogonal design with four variables and of same parameter set is
called a Baumert Hall array or BH-array. More explicitly, an array A of size 4n;n ∈ N will
be called a Baumert Hall array if it satisfies
AAT = n(x2 2 21 + x2 + x3 + x
2
4)I4n.
and is denoted by OD(4n;n,n,n,n).
Definition 6.4. A special class of Baumert-Hall array, consisting of 16 circulant matrices
is known as Baumert-Hall-Welch array and is denoted by BHW-array.
In 1965, Baumert and Hall [3] gave the idea of such array by introducing an orthogonal
design of type (12;3,3,3,3) for the first time. The generalization of the idea of Baumert-
Hall arrays remained incomplete until the discovery of T-matrices, originally found by R.
J. Turyn. Now we present well known Goethals-Seidal array which has an extensive use in
generating orthogonal designs.
  A BR CR DR

−BR A −DT R CT R 
GS = 

−CR DT R A −BT R 
−DR −CT R BT R A
where R is the back diagonal identity matrix of order n i.e. R = [ri j] where ri j = 1 if
i+ j = n+1 and 0 otherwise. In addition, if A,B,C,D are four circulant amicable matrices
with matching (A,C) and (B,D) then the following array H [16] generates a BH-array,
consisting of 16 block circulant matrices.
47
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
 
 A C B D

  −C A −D B

H = −BT DT AT −CT 

−DT −BT CT AT
Now we present a theorem of Cooper et.al. [15] with a slight variation which has an
extensive use in constructing orthogonal designs, especially BHW-arrays in our approach.
Theorem 6.5. (Cooper-J. Wallis [15]) Let Ti’s; 1 ≤ i ≤ 4 be T-matrices of order n and
a,b,c and d be commuting variables. Then
A = aT1 +bT2 + cT3 +dT4
B =−bT1 +aT2 +dT3− cT4
C =−cT1−dT2 +aT3 +bT4
D =−dT1 + cT2−bT3 +aT4
can be used in Goethals-Seidal array to obtain a Baumert Hall array of order n.
In the next few sections, we pay attention to some new orthogonal designs such as BH-
arrays and BHW-arrays using amicable T-matrices formed in Chapter 5 and explore some
circulant amicable matrices in terms of variables that allow us to generate new orthogonal
designs.
6.2 Orthogonal Designs From Amicable T-Matrices
Using the amicable T-matrices of order n ≡ 2 (mod 4) in Theorem 6.5 we obtain cir-
culant matrices A,B,C,D in terms of variables a,b,c and d. Then plugging these matrices
into Goethals-Seidal array generates our desired orthogonal design of type (4n;n,n,n,n)
where n = 2k; k is odd.
48
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
Moreover, Seberry et.al. showed that BHW-array can be used to construct an infinite class
of orthogonal designs [38, 41].
Theorem 6.6. (Seberry-Yamada-Turyn [38, 41]) Suppose there are T-matrices of order
t and an orthogonal design OD(4s;u1, . . . ,un) constructed from 16 circulant s× s blocks
in variables x1, . . . ,xn. Then there is an OD(4st; tu1, . . . , tun). In particular, if there is
a BH-array, OD(4s;s,s,s,s) constructed from 16 circulant s× s blocks then there is an
OD(4st;st,st, st,st).
Proof. Let Ni j;1≤ i, j ≤ 4, be the 16 circulant blocks of the orthogonal design
OD(4s;u1, . . . ,un). Since this design is orthogonal then we have,∑4 u x2 k=1 k k
Is ; i = j
Ni1NTj1 +N
T T T
i2N j2 +Ni3N j3 +Ni4N j4 =0 ; i 6= j.
Suppose T1,T2,T3,T4 are the T-matrices of order t. Then we form the following matrices
P = T1⊗N11 +T2⊗N21 +T3⊗N31 +T4⊗N41,
Q = T1⊗N12 +T2⊗N22 +T3⊗N32 +T4⊗N42,
R = T1⊗N13 +T2⊗N23 +T3⊗N33 +T4⊗N43,
S = T1⊗N14 +T2⊗N24 +T3⊗N34 +T4⊗N44
which yields PPT +QQT +RRT + SST = t ∑4k=1 ukx2kIst . Plugging these matrices into the
Goethals-Seidal array produce OD(4st; tu1, · · · , tun).
However, amicability in T-matrices allows us to find circulant amicable matrices using
Theorem 6.5. These matrices have a major role in constructing BHW-arrays and BH-array
of order 4s and 4st.
Theorem 6.7. [16] Suppose T1,T2,T3,T4 are amicable T-matrices of order n. Then there
is a BHW-array of type OD(4n;n,n,n,n) consisting of 16 circulant matrices.
49
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
Proof. Assume T1,T2,T3,T4 are amicable T-matrices of order n with matching (T1,T3) and
(T2,T4). We have
A = aT1 +bT2 + cT3 +dT4,
B =−bT1 +aT2 +dT3− cT4,
C =−cT1−dT2 +aT3 +bT4,
D =−dT1 + cT2−bT3 +aT4
which are Seberry-Cooper matrices [15]. Now we compute
ACT −CAT +BDT −DBT = (a2 +b2 + c2 +d2)(T T T T T1T3 −T3T1 +T2T4 −T4T2 ) = 0
and
AAT +BBT +CCT +DDT = n(a2 +b2 + c2 +d2)In
which reveals that A,B,C,D are amicable with matching (A,C) and (B,D). Plugging these
matrices into the array H we obtain HHT = (AAT +BBT +CCT +DDT )I4n. Therefore, H
is a BHW-array of type OD(4n;n,n,n,n), consisting of 16 circulant matrices.
Now we have the following result
Theorem 6.8. For every odd prime power pr ≡ 1, there is a BHW-array of type
OD(4n;n,n,n,n) where n = pr + 1 and also a BH-array of type OD(4nt;nt,nt,nt,nt) for
any T-matrices of order t.
Proof. For every odd prime power pr ≡ 1, we established that there is a set of amicable
T-matrices of order n = pr + 1 (Corollary 5.17). Then the Theorem 6.7 gives a BHW-
array of type OD(4n;n,n,n,n). Suppose T1,T2,T3,T4 are any T-matrices of order t and
then the Theorem of Seberry-Yamada-Turyn (Theorem 6.6) gives four matrices P,Q,R,S
50
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
of order nt. Plugging these matrices into Goethals-Seidal array result in a BH-array of type
OD(4nt;nt,nt,nt,nt).
J. Bell and D. Z. Djokovic [6] found OD(4n;n,n,n,n) where n= 2k;k∈{1,3,5, · · · ,17}.
All these are a subclass of Baumert-Hall array, known as BHW-array. Now we tabulate
some new Baumert-Hall arrays corresponding to T-matrices of order n≡ 2 (mod 4)< 100,
which are different from those found by Bell and Djokovic [6].
51
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
Table 6.1: Orthogonal Design of type OD(4n;n,n,n,n);n = 2k, k is odd
Order n = Order of T-matrices OD(4n; n, n, n, n)
n pr +1 from Theorem 5.18 n=2k; k is odd
6 5+1 2*3 OD(24; 6, 6, 6, 6)
10 9+1 2*5 OD(40; 10, 10, 10, 10)
14 13+1 2*7 OD(56; 14, 14, 14, 14)
18 17+1 6* 3 OD(72; 18, 18, 18, 18)
22 2*11 OD(88; 22, 22, 22, 22)
26 25+1 2*13 OD(104; 26, 26, 26, 26)
30 29+1 6*5 OD(120; 30, 30, 30, 30)
34 2*17 OD(136: 34, 34, 34, 34)
38 37+1 2*19 OD(152; 38, 38, 38, 38)
. 42 41+1 6*7 OD(168; 42, 42, 42, 42)
46 2*23 OD(184; 46, 46, 46, 46 )
50 49+1 10*5 OD(200; 50, 50, 50, 50)
54 53+1 6*9 OD(216; 54, 54, 54, 54)
62 61+1 OD(248; 62, 62, 62, 62)
66 6*11 OD(264; 66, 66, 66, 66)
70 10*7 OD(280; 70, 70, 70, 70)
74 73+1 OD(296; 74, 74, 74, 74)
78 6*13 OD(312; 78, 78, 78, 78)
82 2*41 OD(328; 82, 82, 82, 82)
90 89+1 10*9 OD(360; 90, 90, 90, 90)
98 97+1 OD(392; 98, 98, 98, 98)
52
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
Moreover, this type of amicable T-matrices of order n ≡ 2 (mod 4) can be used to
generate orthogonal designs of type OD(4n;2,2,2n−2,2n−2).
Theorem 6.9. For every odd prime power pr, there is an orthogonal design of type
OD(4n;2,2,2n−2,2n−2) where n = pr +1.
Proof. For the amicable T-matrices constructed as in Section 5.3, set
A = aT1 +bT2,
B =−aT1 +bT2,
C = cT1 +dT2,
D =−cT1 +dT2
and plugging these circulant matrices into Goethals-Seidal array yields an orthogonal de-
sign of type OD(4n;2,2,2n−2,2n−2) where n = pr +1.
Since there are no odd order amicable T-matrices [5], finding odd order circulant am-
icable matrices in terms of variables is a challenging job. However, we employ a different
approach which help us to construct odd order amicable matrices in two variables and also
in four variables. Here we use the amicability of Williamson matrices to construct four
circulant amicable matrices in order to generate infinite class of orthogonal designs of BH-
array type.
Theorem 6.10. Suppose W1,W2,W3,W4 are Williamson-matrices of order n. Then there is
an orthogonal design of type OD(4n;2n,2n) and also an orthogonal design of type
OD(4nt;2nt ,2nt) for any T-matrices of order t.
Proof. Let W1,W2,W3,W4 be Williamson matrices of order n and a,b be two indeterminate.
Setting T1 = (W1 +W2)/2, T2 = (W1−W2)/2, T3 = (W3 +W4)/2, T4 = (W3−W4)/2 and
A = aT1 +bT2,
53
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
B = bT1−aT2,
C = aT3−bT4,
D = bT3 +aT4
we obtain circulant amicable matrices A,B,C,D with matching (A,C) and (B,D). Then
substitution of these matrices into the array H produces OD(4n;2n,2n) and plugging the
new matrices formed with A,B,C,D and T-matrices as in Theorem 6.6, into Goethals-Seidal
array leads an orthogonal design of type OD(4nt;2nt,2nt).
Example 6.11. Let W1 = circ(1,1,1), W2 = circ(−,1,1), W3 = circ(−,1 ,1), W4 = circ(−,
1,1) be Williamson matrices of order 3. Then T1 = circ(0,1,1), T2 = circ(1,0,0), T3 =
circ(−,1,1), T4 = circ(0,0,0) are four (0,±1) circulant amicable matrices. Thus we have b a a  −a b b   A = a b a   , B = b −a b  ,
a a b b b −a
   
−a a a   −b b b C  = a −a a  , D = b −b b  .
a a −a b b −b
A simple calculation shows ACT −CAT +BDT −DBT = 0 and AAT +BBT +CCT +DDT
= 6(a2 +b2)I3 and putting these matrices into the array H produces an OD(12;6,6).
In Theorem 6.5, we saw that amicability in A,B,C,D happens if T1,T2,T3,T4 are amica-
ble which is not possible for odd order T-matrices in general. However, we notice if T1 and
T4 are all zero matrices then the amicability with matching (T1,T3) and (T2,T4) is trivial.
But it is hard to find such a set of T-matrices while it is easy to find a set of T-matrices
where T1 is identity and T4 is all zero matrix. Since IIT = I, we substitute 1 by 0 in T1 = I
54
6.2. ORTHOGONAL DESIGNS FROM AMICABLE T-MATRICES
and so that the T-matrices T1 = In,T2,T3,T4 = 0 of order n allows us to construct four cir-
culant amicable matrices of order n. We know this class of T-matrices exist for order n+1
where n ∈ 2a10b26c;a,b,c≥ 0.
Theorem 6.12. There is an orthogonal design of type OD(4(n+1);n,n,n,n) consisting of
16 circulant matrices for every Golay length n. Moreover, for any T-matrices of order t
there exist an OD(4t(n+1);nt,nt,nt,nt).
Proof. Suppose P,Q be Golay sequences of length n. Then circulating the sequences
(1,0n), (0, P+Q), (0, P−Q2 2 ), (0n+1), we have T-matrices T1,T2,T3,T4 of order n+ 1 where
0n is the zero sequence of length n. Substituting 1 by 0 in T1 and then applying Theorem
6.5, we obtain four circulant amicable matrices of order n+ 1. Using these matrices into
the array H and in Theorem 6.6 suffices to the proof.
Example 6.13. Let P = (11);Q = (1−) be Golay sequences of length 2. Then we form
T1 = circ(0,0,0),T2 = circ(0,1,0),T3 = circ(0,0,1), T4 = circ(0,0,0) and obtain 0 b c   0 −d a A = c 0 b  , B = a 0 −d  ,
 b c 0  −d a 0  0 a d   0 −c bC =  d 0 a  , D = b 0 −c  .
a d 0 −c b 0
Since A,B,C,D yield ACT −CAT +BDT −DBT = 0 and AAT +BBT +CCT +DDT =
2(a2+b2+c2+d2)I3, putting these matrices into the array H produces an OD(12;2,2,2,2).
Repeat that (p,q,r,s) = −(a,b,c,d). In Appendix A, we list OD(72;18,18,18,18),
OD(88;22,22,22,22) and OD(104;26,26,26,26). These arise from the Theorem 6.8 in a
manner similar to Example 6.13.
55
6.3. ORTHOGONAL DESIGNS FROM SPECIAL CLASS OF T-MATRICES
Aside from Golay sequences of lengths n ∈ 2a10b26c;a,b,c≥ 0, Georgiou et.al. found
periodic Golay sequences of lengths n ∈ {34,50,58,74,82,122,136,202,226} [13] in
2014. In same year, Djokovic et.al. [11] published a paper giving a unique example
of periodic Golay sequences of length 72. At present these are the only periodic Golay
sequences having an order with factor congruent to 3 modulo 4. From these periodic Golay
sequences we can construct amicable T-matrices of size n as follows
T = circ(A+B1 2 ), T2 = circ(
A−B
2 ), T3 = 0n, T4 = 0n
where A, B are periodic Golay sequences of length n and T3,T4 are zero matrices of or-
der n. Using these matrices we have orthogonal designs of type OD(4n;n,n,n,n); n ∈
{34,50,58,74,82,122,136,202,226}
6.3 Orthogonal Designs From Special Class of T-Matrices
For any T-matrices of order n, Cooper-Seberry matrices [15] A,B,C,D satisfy the prop-
erty
AAT +BBT +CCT +DDT = q(a2 +b2 + c2 +d2)In (6.1)
where each of the variables appears same number of times. In this section, we introduce
a theorem which results in four circulant matrices in terms of four variables satisfying the
property 6.1 where each variable appears varied number of times.
Theorem 6.14. Let {T1,T2,T3,T4} be a set of T-matrices of order t where each Ti is a
weighing matrix of weight wi and A,B,C,D are Williamson-type matrices of order n. Then
the following matrices
C1 = aA⊗T1 +bB⊗T2 + cC⊗T3 +dD⊗T4,
C2 =−bA⊗T2 +aB⊗T1 +dC⊗T4− cD⊗T3,
C3 =−cA⊗T3−dB⊗T4 +aC⊗T1 +bD⊗T2,
C4 =−dA⊗T4 + cB⊗T3−bC⊗T2 +aD⊗T1
56
6.3. ORTHOGONAL DESIGNS FROM SPECIAL CLASS OF T-MATRICES
satisfy ∑4 C CTi=1 i i = 4n(w a21 +w2b2 +w3c2 +w 44d )Int .
Proof. Since A,B,C,D are pairwise amicable, we have
4
∑C CT = w a2(AAT +BBTi i 1 +CCT +DDT )⊗ It +
i=1
w b2(AAT2 +BBT +CCT +DDT )⊗ It +
w 2 T3c (AA +BBT +CCT +DDT )⊗ It +
w4d2(AAt +BBT +CCT +DDT )⊗ It .
= (w 2 2 2 41a +w2b +w3c +w4d )
(AAT +BBT +CCT +DDT )⊗ It
= 4n(w 2 21a +w2b +w c23 +w4d4)In⊗ It
= 4n(w1a2 +w2b2 +w 2 43c +w4d )Int .
This Theorem is very useful in generating a new and interesting class of orthogonal
designs in four variables where each variable appears varied number of times.
Corollary 6.15. Suppose m be the order of Williamson-type matrices and {T1,T2,T3,T4}
be a set of T-matrices of order n where each Ti is a weighing matrix of weight wi. Then
there exist an orthogonal design of type OD(4mn;w1s,w2s,w3s,w4s) where s = 4m.
Proof. Constructing Ci’s with T-matrices matrices of weight w1,w2,w3,w4 and plugging
these matrices into Goethals- Seidal array yields orthogonal design of type
OD(4n;w1s,w2s,w3s,w4s) where s = 4m.
Example 6.16. Let T1 = circ(1,0,0), T2 = circ(0,1,0), T3 = circ(0,0,1), T4 = circ(0,0,0)
and A,B,C,D be Williamson-type matrices of order 3. Then
∑4 Ti=1CiCi = 12(a2 +b2 + c2)
57
6.4. ORTHOGONAL DESIGNS FROM AMICABLE MATRICES WITH COMPOSITE
ENTRIES
and generate an OD(36;12,12,12).
Moreover for T1 = circ(0,0,−,0,−,−,1), T2 = circ(0,0,0,1,0,0,0),
T3 = circ(0,1,0,0,0,0,0), T4 = circ(−,0,0,0,0,0,0) and A,B,C,D are Williamson-type
matrices of order 3, we have
∑4i=1C CTi i = (48a2 +12b2 +12c2 +12d2)
and generate an OD(84;12,12,12,48). Also if A,B,C,D are Williamson-type matrices of
order 5 then we have
∑4 C CTi=1 i i = (80a2 +20b2 +20c2 +20d2)
and generate an OD(140;20,20,20,80). In Appendix A we include the design
OD(84;12,12,12,48).
Remark 6.17. Corollary 6.15 produces new classes of orthogonal designs, so it would be
an interesting topic for future research to classify T-matrices where each of Ti is a weighing
matrix of weight wi.
6.4 Orthogonal Designs From Amicable Matrices With Composite Entries
This section is quite different from the preceding one. Here we construct circulant
amicable matrices with composite entries, specially with linear combination of variables.
Application of these matrices produces generalized orthogonal designs. However, gener-
alized orthogonal design is another interesting topic for research and is beyond our present
research scope. To present the application of these amicable matrices, we gently start this
section with the formal definition of generalized orthogonal design.
Definition 6.18. Suppose {x1,x2, . . . ,xt} be a set of commuting variables and D be a m×n
matrix whose entries are of the form ±ki jxi for ki j ≥ 0, i = 1,2, . . . , t; j = 1,2, . . . ,ui (ui is
the number of times xi appears) and ∑ti=0 ui = n where u0 is the number of zeros in each row
or column. Set s = ∑uii j=1 k
2
i j. Then D is said to be generalized orthogonal design (GOD) if
58
6.4. ORTHOGONAL DESIGNS FROM AMICABLE MATRICES WITH COMPOSITE
ENTRIES
DDT = (∑t 2i=1 sixi )Im
and is denoted by
D = GOD(m;n;k1,1,k1,2, . . . ,k1,u1;k2,1,k2,2, . . . ,k2,u2; . . . ;kt,1,kt,2, . . . ,kt,ut ).
Also an alternative way to denote GOD is
D = GOD(m;n;< k1,1,a1,1 >,. . . ,< k1,u1,a1,u1 >; . . . ;< kt,1,at,1 >,. . . ,< kt,ut ,at,ut >
where ki j denotes the times of repetition of the variables in the coefficient ai j. If m = n then
the generalized orthogonal design is denoted by
GOD(n;k11,k12, . . . ,k1u1; . . . ;kt1,kt2, . . . ,ktut ).
Moreover, if m = n and ki j = 1 for all i = 1,2, . . . , t ; j = 1,2, . . . ,ui then generalized or-
thogonal design turns into orthogonal design OD(n;u1,u2, . . . ,ut). Thus orthogonal designs
are a special case of generalized orthogonal designs. Here we present here two different
examples of generalized orthogonal design.
Example 6.19. The following array D is a GOD(4;1,1;1;1) as well as OD(4;2,1,1) with
n = 4, u1 = 2, u2 = u3 = 1 and a11= a12 = a21 = a31 = 1.  b a b c

−a b c −b
D = 

 −b −c b a 
−c b −a b
Furthermore, 


3a −2b −c −2a −7b −c −3b −4a
 −

2b 3a 2a c −c 7b 4a −3b
 

D = c 2a 3a −2b −3b −4a 7b c  

 2a −c 2b 3a −4c 3b −c 7b 
7b c 3b 4a 3a −2b −c −2a
59
6.4. ORTHOGONAL DESIGNS FROM AMICABLE MATRICES WITH COMPOSITE
ENTRIES
is a GOD(5;8;2,3,4;2,3,7;1,1) with t = 3,u1 = u2 = 3,u3 = 2.
Although amicability in odd order T-matrices is not possible, Holzmann and Kharaghani
gave the definition of weak amicability in T-matrices and found ones for odd orders {3,5, . . .
,21} [23]. Use of weak amicable T-matrices in Theorem 6.5 with a substitution T3 =
T3+T4 and T4 = T3−T4 help us to compute circulant amicable matrices of odd order some
of whose entries are linear combination of variables. Application of these matrices gener-
ates an infinite class of generalized orthogonal designs.
Theorem 6.20. If T1,T2,T3,T4 are weak amicable T-matrices of order n then there is a
generalized orthogonal design of type GOD(4n;k11, . . .k1u1 ;k21, . . . ,k2u2). Furthermore,
for any T-matrices of order t there exist a generalized orthogonal design of type
GOD(4nt;k11, . . .k1u1;k21, . . . ,k2u2).
Proof. Suppose T1,T2,T3,T4 are weak amicable T-matrices of order n with matching
(T1,T3 +T4), (T2,T3−T4). Then the following matrices
A = aT1 +bT2 + c(T3 +T4)+d(T3−T4),
B =−bT1 +aT2−d(T3 +T4)+ c(T3−T4),
C =−cT1 +dT2 +a(T3 +T4)−b(T3−T4),
D =−dT1− cT2 +b(T3 +T4)+a(T3−T4)
are amicable in matching (A,C) and (B,D) with some composite entries. Plugging these
matrices into the array H yields an orthogonal array of order 4n. An appropriate substitution
in the variables leading each composite entry into either 0 or in the form ±ki jxi turns the
array H into generalized orthogonal design of type GOD(4n;k11, . . .k1u1 ;k21, . . . ,k2u2).
The second part is obvious in accordance the Theorem 6.8.
Example 6.21. We have T1 = circ(1,0,0),T2 = circ(0,0,0), T3 = circ(0,1,0),
T4 = circ(0,0,1) are weak amicable T-matrices with matching (T1,T3 +T4) and (T2,T3−
60
6.4. ORTHOGONAL DESIGNS FROM AMICABLE MATRICES WITH COMPOSITE
ENTRIES
T4). Then we compute    a c+d c−d   −b d− c d + cA   B  

= c−d a c+d , = d + c −b d− c  ,
c+d c−d a d− c d + c −b
    −c a+b a−b   −d a−b −b−aC = a−b −c a+b 

  , D = −b−a −d a−b  .
a+b a−b −c a−b b−a −d
Here A,B,C,D satisfy ACT −CAT +BDT −DBT = 0 and AAT + BBT +CCT +DDT =
5(a2 + b2 + c2 + d2)I3. Plugging these matrices into the array H produces an orthogonal
array of order 12.
Over and above, array H in the above Theorem can be used to generate an infinite class
of weighing matrices.
Corollary 6.22. If W1,W2,W3,W4 are Williamson matrices of order n and T1,T2,T3,T4 are
weak amicable T-matrices of order t then there exist a weighing matrix of weight 4nt−2n.
Proof. In the array constructed in the proof of Theorem 6.20, an appropriate substitution
of each entry by any of (W1 +W2)/2, (W1−W2)/2, (W3 +W4)/2 and (W3−W4)/2 such
that each of the composite entries must one of ±{W1,W2,W3,W4}, leads a weighing matrix
of weight 4nt−2n.
Amicability is a generic feature in Williamson-type matrices that allows us to generate
circulant amicable matrices where each entry is a linear combination of four variables.
Theorem 6.23. If W1,W2,W3,W4 are Williamson-type matrices order n then there is a gen-
eralized orthogonal design of type GOD(4n;k11, . . . ,k1u1). Further, for any T-matrices of
order t there exist generalized orthogonal design of type GOD(4nt;k11, . . . ,k1u1).
61
6.4. ORTHOGONAL DESIGNS FROM AMICABLE MATRICES WITH COMPOSITE
ENTRIES
Proof. Let W1,W2,W3,W4 be Williamson-type matrices order n and a,b,c,d be four vari-
ables. Then the following matrices
A = aW1 +bW2 + cW3 +dW4,
B =−bW1 +aW2−dW3 + cW4,
C =−cW1 +dW2 +aW3−bW4,
D =−dW1− cW2 +bW4 +aW4
are four circulant amicable matrices with matching (A,C) and (B,D) where each entry is a
linear combination of variables a,b,c,d. Plugging these matrices into the array H produce
an orthogonal array of order 4n and theorem 6.8 gives an orthogonal array of order 4nt.
Then an appropriate substitution in variables of these orthogonal arrays where each entry
turns into either 0 or in the form ±ki jxi, turns the array H into a generalized orthogonal
design of type GOD(4n;k11, . . . ,k1u1) and GOD(4nt;k11, . . . ,k1u1) respectively.
Example 6.24. Using the Williamson-type matrices W1 = circ(1,1,1), W2 = circ(−,1,1),
W3 = circ(−,1,1), W4= circ(−,1,1) of order 3 we get, 
 a−b− c−d a+b+ c+d a+b+ c+d A = a+b+ c+d a−b− c−d a+b+ c+d  ,
 a+b+ c+d a+b+ c+d a−b− c−d  −a−b+ c−d a−b− c+d a−b− c+d B =

a−b− c+d −a−b+ c−d a−b− c+d  ,
 a−b− c+d a−b− c+d −a−b+ c−d 
 −a−b− c+d a+b− c−d a+b− c−dC  

= a+b− c−d −a−b− c+d a+b− c−d  ,
a+b− c−d a+b− c−d −a−b− c+d
62
6.5. A NEW METHOD TO CONSTRUCT A HADAMARD MATRIX OF ORDER 48
  −a+b− c−d a−b+ c−d a−b+ c−d D  

= a−b+ c−d −a+b− c−d a−b+ c−d 
a−b+ c−d a−b+ c−d −a+b− c−d
that produce ACT −CAT +BDT −DBT = 0 and AAT +BBT +CCT +DDT =
12(a2 +b2 + c2 +d2)I3.
These types of generalized orthogonal designs have extensive use in statistics, specially
in construction of orthogonal latin hypercube designs (LHD) [14] and in coding theory
[22].
However each orthogonal array of order 4n, constructed in Theorem 6.23, can be split
into four coefficient matrix of order 4n with respect to variables. More explicitly, H =
aA + bB + cC + dD where A,B,C,D are pairwise anti-amicable. Use of a pair of anti-
amicable coefficient matrices in the following array generates a orthogonal design of order
8n in two variables.   aA bBK = 
bB aA
In addition to this, using these anti-amicable coefficient matrices in the following array we
obtain orthogonal design of order 16n in four variables consist of 16 blocks, where each
block is constructed from 16 symmetric circulant matrices.
 
 aA bB cC dD




bB aA dD cC 
M =  cC dD aA bB 
dD cC bB aA
6.5 A New Method to Construct a Hadamard Matrix of Order 48
A Hadamard matrix H can be constructed by H = A⊗ B where A and B are two
Hadamard matrices of order 12 and 4 respectively. Using negacirculant weighing ma-
63
6.5. A NEW METHOD TO CONSTRUCT A HADAMARD MATRIX OF ORDER 48
trix of order 4, equivalently amicable T-matrices of order 4 we construct an orthogonal
design of type OD(48;12,12,12,12). Substituting the variables of this design by 1 we get
a Hadamard matrix K of order 48. Using computer algebraic software MAGMA we de-
termined that the Hadamard matrix H(48) is inequivalent to K(48). Thus the Hadamard
matrix K(48), arising from OD(48;12,12,12,12) is inequivalent to the Hadamard matrices
constructed from Kronecker product method using Hadamard matrices of order 12 and 4
respectively.
Since we started the thesis with three examples of long known Hadamard matrices, it seems
appropriate to end the thesis with this example of a new construction for a Hadamard matrix
of order 48.
64
6.5. A NEW METHOD TO CONSTRUCT A HADAMARD MATRIX OF ORDER 48
 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 1 − − − − 1 1 1 − 1 1 1 1 − − − 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 1 1 − 1 − − 1 −  1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − − − − 1 1 1 1 − 1 1 1 − 1 1 1 − − − − 1 1 1 1 − 1 1 1 − − − − 1 1 − − − − 1 1 1 − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − 1 − − − − 1 1 1 1 − − − 1 − − −  − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − − 1 − − 1 − 1 1 − 1 − − − 1 − − − − 1 − 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − − − 1 − 1 1 − 1 − − 1 − − − 1 −  − − − 1 1 1 1 − 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 − − − 1 1 1 1 − − − − 1 − − − 1 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 1 − − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − − − 1 − − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − − − − 1 − − − 1 1 1 1 − 1 1 1 − 1 1 1 − − − − 1 − − − 1 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − − − − 1  1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − 1 − − − 1 − − − − 1 1 1 1 − − − 1 − − − 1 − − − 1 − − − − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 − − − 1 − −  − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − − − 1 − − − 1 − 1 1 − 1 − − 1 − − − 1 − − − 1 − − − 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 − − − 1  1 − − − 1 − − − 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 1 − − − 1 − − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − − 1 − − 1 − 1 1 − 1 − − − − 1 − − − 1 − − − 1 − − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − − − 1 − 1 1 − 1 − − 1 − − − − 1 − − − 1 − − − 1 − − − 1 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − − − − 1 − − − 1 1 1 1 − − − − 1 1 − − − 1 − − − 1 − − − − 1 1 1 1 − − − − 1 1 1 − 1 1 1 1 − − − − 1 1 1 − 1 1 1 1 − − − 1 − − − − 1 − − − 1 − − − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 1 − 1 1 − 1 − − − 1 − − − − 1 − − − 1 − − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 1 1 − 1 − − 1 − − − 1 −H  − − − 1 − − − 1 − − − 1 1 1 1 − − − − 1 1 1 1 − 1 1 1 − − − − 1 1 1 1 − 1 1 1 − − − − 1 − − − 1= − 1 1 1 1 − − − − 1 1 1 1 − − − 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 1 − − − 1 − − − 1 − − −  1 − 1 1 − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 − 1 − − − 1 − − − 1 − − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − − − 1 − − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − − − 1 − − − 1 − 1 1 1 − − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 1 1 1 − 1 1 1 − 1 1 1 − − − − 1 − − − 1 − − − 1  1 − − − 1 − − − − 1 1 1 1 − − − 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 − 1 − − − 1 − − 1 − 1 1 − 1 − − − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1  − − 1 − − − 1 − 1 1 − 1 − − 1 − − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 1 1 1 − − − − 1 − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 −  1 − − − − 1 1 1 1 − − − 1 − − − 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 − − 1 − 1 1 − 1 − − 1 − − − 1 − − − 1 − − − 1 − 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 − − − 1 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 − 1 1 1 −

− 1 1 1 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − − 1 1 1 − 1 1 1 1 − − −
 1 − 1 1 − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 1 − 1 − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − 1 1 − 1 1 1 − 1 − − 1 − 1 1 1 − − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 − − − 1 1 1 1 − 1 1 1 − − − − 1 1 − − − 1 − − − − 1 1 1 − 1 1 1 1 − − − 1 − − − − 1 1 1 1 − − − 1 − − − 1 − − − − 1 1 1 − 1 1 1 − 1 − − − 1 − − 1 − 1 1 1 − 1 1 − 1 − − − 1 − − 1 − 1 1 − 1 − − − 1 − − − 1 − − 1 − 1 1 1 − 1 1 − − 1 − − − 1 − 1 1 − 1 1 1 − 1 − − 1 − − − 1 − 1 1 − 1 − − 1 − − − 1 − − − 1 − 1 1 − 1 1 1 − 1  − − − 1 − − − 1 1 1 1 − 1 1 1 − − − − 1 − − − 1 1 1 1 − − − − 1 − − − 1 − − − 1 1 1 1 − 1 1 1 −− 1 1 1 1 − − − 1 − − − − 1 1 1 − 1 1 1 1 − − − − 1 1 1 − 1 1 1 1 − − − − 1 1 1 1 − − − − 1 1 11 − 1 1 − 1 − − − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 1 − 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1
1 1 − 1 − − 1 − − − 1 − 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 1 1 − 1 − − 1 − 1 1 − 1 − − 1 − 1 1 − 1
1 1 1 − − − − 1 − − − 1 1 1 1 − 1 1 1 − − − − 1 1 1 1 − 1 1 1 − − − − 1 1 1 1 − − − − 1 1 1 1 −
65
6.6. CONCLUSION
  1 1 − − 1 1 − − 1 − − 1 − 1 1 − − − 1 1 1 1 − − − 1 1 1 1 1 − 1 − − 1 − − 1 1 1 − − 1 − − − 1 − 1 1 1 − 1 1 1 − − 1 − − 1 1 − 1 − 1 1 1 1 − − − 1 1 1 1 1 − 1 − − 1 − 1 1 1 1 1 − 1 − 1 − 1 − 1 1 1 1 1 1 1 1 1 1 − 1 − 1 − 1 − 1 1 1 1 − − − − 1 1 1 − − 1 − − 1 − 1 1 1 1 1 − 1 − 1 1 1 − 1 1  − 1 1 1 − 1 1 1 1 1 − 1 − 1 − − 1 1 1 − − − − 1 1 1 − − 1 − − 1 − 1 1 − 1 1 − − − 1 1 − − 1 1 −

1 − − 1 1 1 − − 1 1 − − − − 1 1 1 1 − − − 1 1 − 1 1 − 1 − − 1 − − 1 1 1 − − 1 − − − 1 − − 1 1 1
 

 − 1 − − 1 1 1 − 1 1 1 − − 1 1 1 1 − − − 1 1 − 1 1 − 1 − − 1 − 1 1 1 1 1 − 1 − 1 − 1 − 1 1 1 1 1 1 − 1 − 1 1 1 1 1 1 1 1 1 1 1 1 − − − − 1 − 1 − − 1 − − 1 − 1 1 1 1 1 − 1 − 1 1 1 − 1 1 1 1 1 −  1 1 − 1 − 1 1 1 − 1 1 1 1 1 1 − − − − 1 − 1 − − 1 − − 1 − 1 1 − 1 1 − − − 1 1 − − 1 1 − 1 1 − − 1 1 − − 1 − − 1 1 1 − − 1 1 − − − 1 1 − − − 1 1 − − 1 − − 1 1 1 1 1 − 1 − − 1 − − 1 1 1 − − 1 −  1 1 1 − − 1 − − 1 1 1 − 1 − − − 1 1 − 1 − 1 1 1 − 1 − 1 1 1 1 1 1 − 1 − − 1 − 1 1 1 1 1 − 1 − 1

1 1 1 1 1 − 1 − 1 1 1 1 − − − − 1 − 1 − 1 1 1 1 1 − 1 1 1 1 1 − − 1 − − 1 − 1 1 1 1 1 − 1 − 1 1
 − 1 1 1 1 1 − 1 − 1 1 1 − − − 1 − 1 − − 1 1 1 − − 1 1 − 1 1 − − 1 − − 1 − 1 1 − 1 1 − − − 1 1 −  1 − − 1 1 1 − − − − 1 1 1 1 − − 1 1 − − 1 − − 1 1 − − 1 1 1 − − 1 − − 1 1 − − 1 − − 1 1 − 1 1 − − − 1 − 1 − − − − 1 1 1 1 1 1 − 1 1 1 − − 1 − − − − 1 − 1 − − − − − 1 − − − 1 − − 1 1 1 1 1 − 1 − 1 − 1 − − − − 1 1 1 1 1 1 1 1 1 1 1 1 1 − 1 − − 1 − 1 − − − − − 1 − 1 − 1 − 1 1 1 1 1 1 − 1 − 1 − 1 1 − − − 1 1 1 1 − − 1 1 1 − 1 1 1 1 1 − 1 1 − 1 1 − − − 1 1 − 1 1 1 − 1 1 1 1 1 − − 1 − − 1 1 − − − − 1 1 1 − − 1 1 − − 1 1 1 − − 1 1 − − 1 1 − − 1 − − 1 1 − − 1 − − 1 1 − 1 1 − 1 − − 1  1 − − − − 1 1 1 − − 1 − − 1 − − 1 1 1 − 1 1 1 − 1 − − − − − 1 − − − 1 − − 1 1 1 1 1 − 1 − − 1 − 

 − − − − 1 1 1 1 − 1 − 1 1 − 1 − 1 1 1 1 1 1 1 1 − − − − − 1 − 1 − 1 − 1 1 1 1 1 1 − 1 − − 1 − 1 − − − 1 1 1 1 − 1 − 1 1 1 1 − 1 − 1 1 1 − 1 1 1 − − − 1 1 − 1 1 1 − 1 1 1 1 1 − − 1 − − 1 − 1 1  − − 1 1 1 − − 1 1 1 − − 1 1 − − 1 − − 1 1 1 − − 1 − − 1 1 − − 1 1 1 − − − 1 1 − 1 − − 1 − − 1 1  − 1 1 1 − − 1 − 1 − − − 1 1 1 − − 1 − − 1 1 1 − − − 1 − − − 1 − 1 − − − 1 1 − 1 − − 1 − − 1 1 1  1 1 1 1 − 1 − 1 − − − − 1 1 1 1 1 − 1 − 1 1 1 1 − 1 − 1 − 1 − 1 − − − − 1 − 1 − − 1 − 1 1 1 1 1K  1 1 1 − 1 − 1 1 − − − 1 − 1 1 1 1 1 − 1 − 1 1 1 1 − 1 1 1 − 1 1 − − − 1 − 1 − − 1 − 1 1 1 1 1 −= 1 − − − − − 1 − 1 1 − 1 − 1 1 − − − 1 1 − 1 1 − 1 1 − − 1 1 − − 1 − − 1 1 − − − 1 1 − 1 − 1 1 1  − − − − − 1 − 1 1 − 1 − 1 1 − 1 − 1 1 1 1 1 − 1 1 1 1 − 1 1 1 − − 1 − − − − − − 1 − 1 − 1 1 1 1  − − − 1 1 − 1 1 − 1 − − 1 − 1 − 1 1 1 1 1 − 1 − 1 1 1 1 1 1 1 1 1 − 1 − − − − 1 − 1 − − 1 1 1 −  − − 1 1 − 1 1 − 1 − − 1 − 1 − − 1 1 1 − − 1 − − − 1 1 1 − 1 1 1 1 1 − 1 − − 1 1 1 − − 1 1 1 − − − − 1 − 1 1 − 1 1 − − − − − 1 1 − 1 1 − − 1 1 − 1 − − 1 1 1 − − 1 1 − − 1 1 − 1 − 1 1 1 1 − − −  − 1 − 1 1 − 1 − − − − − − 1 1 1 1 1 − 1 1 1 − 1 − 1 − − 1 1 1 − 1 1 1 − 1 − 1 − 1 1 1 1 − − − −  1 − 1 1 − 1 − − − − − 1 1 1 1 1 1 − 1 − 1 − 1 − 1 − 1 − 1 1 1 1 1 1 1 1 − 1 − − 1 1 1 − − − − 1 − 1 1 − 1 − − 1 − − 1 1 1 1 1 − − 1 − − − 1 − − 1 1 − 1 − 1 1 1 − 1 1 1 1 − − 1 1 1 − − − − 1 1

1 1 − 1 1 − − − − − 1 − − 1 1 − − 1 1 − − − 1 1 1 1 − − 1 − − 1 1 1 − − − 1 1 1 1 − − − 1 1 − 1
 1 − 1 − − − − − − 1 − 1 1 1 − 1 1 1 − 1 − 1 1 1 1 1 1 − − 1 − − 1 1 1 − 1 1 1 1 − − − − 1 − 1 −

− 1 − − − − − 1 1 − 1 1 1 − 1 − 1 − 1 − 1 1 1 1 1 1 1 1 1 − 1 − 1 1 1 1 1 1 1 − − − − 1 − 1 − −


1 − − 1 − − 1 1 − 1 1 − − 1 − − − 1 − − 1 1 1 − − 1 1 1 1 1 − 1 − 1 1 1 1 1 − − − − 1 1 1 − − 1
 1 − − − 1 1 − 1 1 1 − 1 − 1 1 − 1 1 − − 1 − − 1 − 1 1 1 − − 1 − 1 − − − 1 1 − − 1 1 − − 1 − − 1 

− − − − 1 − 1 − 1 − 1 − 1 1 − 1 1 − − − − − 1 − 1 1 1 1 − 1 − 1 − − − − 1 1 1 − 1 1 1 − − 1 − −
  − − − − − − − − − − − − − − − − − − − − − − − −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 − − 1 1 1 − − 1 1 − − 1 − 1 − − − − − 1 1 − 1 1 1 1 − − − 1 1 − − − 1 1 − 1 1 1 − 1 1 1 1 1 − 1 1 1 − 1 1 1 − 1 1 − − − 1 1 − − 1 − − 1 − 1 1 − − − 1 − 1 − − − − 1 1 1 1 − − 1 1 1 − − 1 1 − − 1 − 1 − 1 − 1 − − − − − 1 − − − − − 1 − 1 1 − 1 − 1 − 1 − − − − 1 1 1 1 − 1 − − 1 1 1 − 1 1 1 − 

− 1 − − − 1 − − − − − 1 − − − − − 1 − 1 1 − 1 − 1 − 1 1 − − − 1 1 1 1 − 1 − 1 − 1 1 1 1 1 1 1 1
  1 − − 1 1 − − 1 − − 1 1 − − − 1 1 − 1 1 − 1 − − − 1 1 − − − 1 1 1 1 − − 1 1 − 1 − 1 1 1 − 1 1 1 1 1 − 1 1 − − − 1 1 − 1 1 − − 1 − 1 1 − 1 1 − − 1 − − − − 1 1 1 − − 1 − 1 1 − − 1 − − 1 1 1 − −1 − 1 − − − − − 1 − 1 − − − 1 − 1 1 − 1 1 − − − − − − − 1 1 1 1 − 1 − 1 1 1 1 − − 1 − − 1 1 1 −
− 1 − − − − − 1 − 1 − − − 1 − 1 1 − 1 − − − − − − − − 1 1 1 1 − 1 − 1 1 1 1 1 1 1 − 1 − 1 1 1 1
1 − − 1 − − 1 1 1 − − 1 1 − 1 1 − 1 − − − − − 1 − − 1 1 1 1 − − − 1 1 − − 1 1 1 1 1 − 1 − 1 1 1
66
6.6. CONCLUSION
6.6 Conclusion
As we discussed in abstract, we established that of for every odd prime power, pr ≡
1 (mod 4) there exist a BHW-array of type OD(4n;n,n,n,n); n = pr + 1, which can be
used to generate BH-array of type OD(4nt;nt,nt,nt,nt) for any T-matrices of order t and
an orthogonal design of type OD(4n;2,2,2n− 2,2n− 2). A set of T-matrices of order n
where each Ti is a weighing matrix of weight wi (1 ≤ i ≤ 4), can be used to generate an
orthogonal design of type OD(4nm;w1s,w2s,w3s,w4s) where s = 4m;m is the order of any
Williamson-type matrices. Moreover, we constructed generalized orthogonal designs of
type GOD(4n;k11, . . .k1u1;k21, . . . ,k2u2) and GOD(4n;k11, . . .k1u1) from weak amicable T-
matrices of order n and Williamson-type matrices order n respectively where each GOD is
consisting of 16 circulant matrices.
67
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70
Appendix A
Here we include some orthogonal designs where (p,q,r,s) =−(a,b,c,d). a c c c r c b d d d s d c a p a a a q q q b q s c r c c c a s d s s s q r r r c r a q b q q q d a p a a a r s s s d s b a p a a a r b q b b b s r a c c c r s b d d d s p c a p a a q q b q s b r c c c a r d s s s q d r r c r a c b q q q d b p a a a r p s s d s b d p a a a r p q b b b s q c r a c c c d s b d d d p p c a p a q b q s b b c c c a r c s s s q d s r c r a c c q q q d b q a a a r p a s d s b d d a a a r p a b b b s q b r c r a c c s d s b d d p p p c a p b q s b b b c c a r c r s s q d s d c r a c c c q q d b q b a a r p a p d s b d d d a a r p a p b b s q b q r r c r a c s s d s b d a p p p c a q s b b b q c a r c r r s q d s d d r a c c c r q d b q b b a r p a p p s b d d d s a r p a p p b s q b q q r r r c r a s s s d s b p a p p p c s b b b q b a r c r r r q d s d d d a c c c r c d b q b b b r p a p p p b d d d s d r p a p p p s q b q q qc a p a a a a c c c r c b d d d s d c r c c c a s d s s s q q q q b q s q b q q q d a p a a a r r r r c r a a p a a a r b q b b b s s s s d s b p c a p a a r a c c c r s b d d d s r c c c a r d s s s q d q q b q s b b q q q d b p a a a r p r r c r a c p a a a r p q b b b s q s s d s b d p p c a p a c r a c c c d s b d d d c c c a r c s s s q d s q b q s b b q q q d b q a a a r p a r c r a c c a a a r p a b b b s q b s d s b d d p p p c a p r c r a c c s d s b d d c c a r c r s s q d s d b q s b b b q q d b q b a a r p a p c r a c c c a a r p a p b b s q b q d s b d d d a p p p c a r r c r a c s s d s b d c a r c r r s q d s d d q s b b b q q d b q b b a r p a p p r a c c c r a r p a p p b s q b q q s b d d d s p a p p p c r r r c r a s s s d s b a r c r r r q d s d d d s b b b q b d b q b b b r p a p p p a c c c r c r p a p p p s q b q q q b d d d s d b d d d s d c a p a a a a c c c r c s d s s s q q q q b q s c r c c c a a p a a a r r r r c r a q b q q q d b q b b b s s s s d s b a p a a a r s b d d d s p c a p a a r a c c c r d s s s q d q q b q s b r c c c a r p a a a r p r r c r a c b q q q d b q b b b s q s s d s b d p a a a r pd s b d d d p p c a p a c r a c c c s s s q d s q b q s b b c c c a r c a a a r p a r c r a c c q q q d b q b b b s q b s d s b d d a a a r p a s d s b d d p p p c a p r c r a c c s s q d s d b q s b b b c c a r c r a a r p a p c r a c c c q q d b q b b b s q b q d s b d d d a a r p a p s s d s b d a p p p c a r r c r a c s q d s d d q s b b b q c a r c r r a r p a p p r a c c c r q d b q b b b s q b q q s b d d d s a r p a p p s s s d s b p a p p p c r r r c r a q d s d d d s b b b q b a r c r r r r p a p p p a c c c r c d b q b b b s q b q q q b d d d s d r p a p p p b b b q b d r c r r r p d s d d d b a c c c r c b d d d s d c a p a a a a a a p a c s d s s s q b b b q b d b b b q b d c r c c c a p p p a p r b b q b d q c r r r p c s d d d b s r a c c c r s b d d d s p c a p a a a a p a c p d s s s q d b b q b d q b b q b d q r c c c a r p p a p r a b q b d q q r r r p c r d d d b s d c r a c c c d s b d d d p p c a p a a p a c p p s s s q d s b q b d q q b q b d q q c c c a r c p a p r a a q b d q q q r r p c r c d d b s d s r c r a c c s d s b d d p p p c a p p a c p p p s s q d s d q b d q q q q b d q q q c c a r c r a p r a a ab d q q q b r p c r c c d b s d s s r r c r a c s s d s b d a p p p c a a c p p p a s q d s d d b d q q q b b d q q q b c a r c r r p r a a a p d q q q b q p c r c c c b s d s s s r r r c r a s s s d s b p a p p p c c p p p a p q d s d d d d q q q b q d q q q b q a r c r r r r a a a p a r c r r r p d s d d d b b b b q b d c a p a a a a c c c r c b d d d s d s d s s s q b b b q b d a a a p a c c r c c c a p p p a p r b b b q b d c r r r p c s d d d b s b b q b d q p c a p a a r a c c c r s b d d d s d s s s q d b b q b d q a a p a c p r c c c a r p p a p r a b b q b d q r r r p c r d d d b s d b q b d q q p p c a p a c r a c c c d s b d d d s s s q d s b q b d q q a p a c p p c c c a r c p a p r a a b q b d q q r r p c r c d d b s d s q b d q q q p p p c a p r c r a c c s d s b d d s s q d s d q b d q q q p a c p p p c c a r c r a p r a a a q b d q q q r p c r c c d b s d s s b d q q q b a p p p c a r r c r a c s s d s b d s q d s d d b d q q q b a c p p p a c a r c r r p r a a a p b d q q q b p c r c c c b s d s s s d q q q b q p a p p p c r r r c r a s s s d s b q d s d d d d q q q b q c p p p a p a r c r r r r a a a p a d q q q b qd s d d d b b b b q b d r c r r r p b d d d s d c a p a a a a c c c r c b b b q b d a a a p a c s d s s s q p p p a p r b b b q b d c r c c c a s d d d b s b b q b d q c r r r p c s b d d d s p c a p a a r a c c c r b b q b d q a a p a c p d s s s q d p p a p r a b b q b d q r c c c a r d d d b s d b q b d q q r r r p c r d s b d d d p p c a p a c r a c c c b q b d q q a p a c p p s s s q d s p a p r a a b q b d q q c c c a r cd d b s d s q b d q q q r r p c r c s d s b d d p p p c a p r c r a c c q b d q q q p a c p p p s s q d s d a p r a a a q b d q q q c c a r c rd b s d s s b d q q q b r p c r c c s s d s b d a p p p c a r r c r a c b d q q q b a c p p p a s q d s d d p r a a a p b d q q q b c a r c r r b s d s s s d q q q b q p c r c c c s s s d s b p a p p p c r r r c r a d q q q b q c p p p a p q d s d d d r a a a p a d q q q b q a r c r r r c c c r c p b q b b b s p a p p p c p p p a p r d s d d d b q q q b q s a c c c r c b d d d s d c a p a a a c c c r c p q b q q q d s s s d s b c c r c p r q b b b s q a p p p c a p p a p r a s d d d b s q q b q s b r a c c c r s b d d d s p c a p a a c c r c p r b q q q d b s s d s b dc r c p r r b b b s q b p p p c a p p a p r a a d d d b s d q b q s b b c r a c c c d s b d d d p p c a p a c r c p r r q q q d b q s d s b d d r c p r r r b b s q b q p p c a p a a p r a a a d d b s d s b q s b b b r c r a c c s d s b d d p p p c a p r c p r r r q q d b q b d s b d d d c p r r r c b s q b q q p c a p a a p r a a a p d b s d s s q s b b b q r r c r a c s s d s b d a p p p c a c p r r r c q d b q b b s b d d d s p r r r c r s q b q q q c a p a a a r a a a p a b s d s s s s b b b q b r r r c r a s s s d s b p a p p p c p r r r c r d b q b b b b d d d s d b q b b b s p a p p p c c c c r c p d s d d d b q q q b q s p p p a p r c a p a a a a c c c r c b d d d s d q b q q q d s s s d s b c c c r c p q b b b s q a p p p c a c c r c p r s d d d b s q q b q s b p p a p r a p c a p a a r a c c c r s b d d d s b q q q d b s s d s b d c c r c p r b b b s q b p p p c a p c r c p r r d d d b s d q b q s b b p a p r a a p p c a p a c r a c c c d s b d d d q q q d b q s d s b d d c r c p r r b b s q b q p p c a p a r c p r r r d d b s d s b q s b b b a p r a a a p p p c a p r c r a c c s d s b d d q q d b q b d s b d d d r c p r r rb s q b q q p c a p a a c p r r r c d b s d s s q s b b b q p r a a a p a p p p c a r r c r a c s s d s b d q d b q b b s b d d d s c p r r r c s q b q q q c a p a a a p r r r c r b s d s s s s b b b q b r a a a p a p a p p p c r r r c r a s s s d s b d b q b b b b d d d s d p r r r c r p a p p p c c c c r c p b q b b b s q q q b q s p p p a p r d s d d d b b d d d s d c a p a a a a c c c r c s s s d s b c c c r c p q b q q q d a p p p c a c c r c p r q b b b s q q q b q s b p p a p r a s d d d b s s b d d d s p c a p a a r a c c c r s s d s b d c c r c p r b q q q d b p p p c a p c r c p r r b b b s q b q b q s b b p a p r a a d d d b s d d s b d d d p p c a p a c r a c c c s d s b d d c r c p r r q q q d b q p p c a p a r c p r r r b b s q b q b q s b b b a p r a a a d d b s d s s d s b d d p p p c a p r c r a c c d s b d d d r c p r r r q q d b q b p c a p a a c p r r r c b s q b q q q s b b b q p r a a a p d b s d s s s s d s b d a p p p c a r r c r a c s b d d d s c p r r r c q d b q b b c a p a a a p r r r c r s q b q q q s b b b q b r a a a p a b s d s s s s s s d s b p a p p p c r r r c r a b d d d s d p r r r c r d b q b b bd d d s d q p a p p p c q b q q q d q q q b q s r c r r r p a a a p a c r r r c r a b q b b b s d d d s d q a c c c r c b d d d s d c a p a a a d d s d q s a p p p c a b q q q d b q q b q s b c r r r p c a a p a c p r r c r a c q b b b s q d d s d q s r a c c c r s b d d d s p c a p a a d s d q s s p p p c a p q q q d b q q b q s b b r r r p c r a p a c p p r c r a c c b b b s q b d s d q s s c r a c c c d s b d d d p p c a p a s d q s s s p p c a p a q q d b q b b q s b b b r r p c r c p a c p p p c r a c c c b b s q b q s d q s s s r c r a c c s d s b d d p p p c a pd q s s s d p c a p a a q d b q b b q s b b b q r p c r c c a c p p p a r a c c c r b s q b q q d q s s s d r r c r a c s s d s b d a p p p c a q s s s d s c a p a a a d b q b b b s b b b q b p c r c c c c p p p a p a c c c r c s q b q q q q s s s d s r r r c r a s s s d s b p a p p p c p a p p p c q b q q q d d d d s d q r c r r r p a a a p a c q q q b q s b q b b b s d d d s d q r r r c r a c a p a a a a c c c r c b d d d s d a p p p c a b q q q d b d d s d q s c r r r p c a a p a c p q q b q s b q b b b s q d d s d q s r r c r a c p c a p a a r a c c c r s b d d d sp p p c a p q q q d b q d s d q s s r r r p c r a p a c p p q b q s b b b b b s q b d s d q s s r c r a c c p p c a p a c r a c c c d s b d d d p p c a p a q q d b q b s d q s s s r r p c r c p a c p p p b q s b b b b b s q b q s d q s s s c r a c c c p p p c a p r c r a c c s d s b d d p c a p a a q d b q b b d q s s s d r p c r c c a c p p p a q s b b b q b s q b q q d q s s s d r a c c c r a p p p c a r r c r a c s s d s b d c a p a a a d b q b b b q s s s d s p c r c c c c p p p a p s b b b q b s q b q q q q s s s d s a c c c r c p a p p p c r r r c r a s s s d s bq b q q q d d d d s d q p a p p p c a a a p a c q q q b q s r c r r r p d d d s d q r r r c r a b q b b b s b d d d s d c a p a a a a c c c r cb q q q d b d d s d q s a p p p c a a a p a c p q q b q s b c r r r p c d d s d q s r r c r a c q b b b s q s b d d d s p c a p a a r a c c c rq q q d b q d s d q s s p p p c a p a p a c p p q b q s b b r r r p c r d s d q s s r c r a c c b b b s q b d s b d d d p p c a p a c r a c c c
q q d b q b s d q s s s p p c a p a p a c p p p b q s b b b r r p c r c s d q s s s c r a c c c b b s q b q s d s b d d p p p c a p r c r a c c
q d b q b b d q s s s d p c a p a a a c p p p a q s b b b q r p c r c c d q s s s d r a c c c r b s q b q q s s d s b d a p p p c a r r c r a c
d b q b b b q s s s d s c a p a a a c p p p a p s b b b q b p c r c c c q s s s d s a c c c r c s q b q q q s s s d s b p a p p p c r r r c r a
OD(72; 18, 18, 18, 18)
71
A. APPENDIX A
 a a c p b b d a b q b b b d q p p r b p a p b q b a d b b p c a a a p a q c a a b s q q c r c d p c c s q d d d s d r q d d c a r r d s d r q d d c a r r r c r s a r r d b s sb a a c p b b d a b q p b b d q p p r b p a q b a d b b p c a a b p a q c a a b s q q a r c d p c c s q d d c s d r q d d c a r r d s d r q d d c a r r d c r s a r r d b s s rq b a a c p b b d a b a p b b d q p p r b p b a d b b p c a a b q a q c a a b s q q a p c d p c c s q d d c r d r q d d c a r r d s d r q d d c a r r d s r s a r r d b s s r cb q b a a c p b b d a p a p b b d q p p r b a d b b p c a a b q b q c a a b s q q a p a d p c c s q d d c r c r q d d c a r r d s d r q d d c a r r d s d s a r r d b s s r c ra b q b a a c p b b d b p a p b b d q p p r d b b p c a a b q b a c a a b s q q a p a q p c c s q d d c r c d q d d c a r r d s d r q d d c a r r d s d r a r r d b s s r c r sd a b q b a a c p b b r b p a p b b d q p p b b p c a a b q b a d a a b s q q a p a q c c c s q d d c r c d p d d c a r r d s d r q d d c a r r d s d r q r r d b s s r c r s ab d a b q b a a c p b p r b p a p b b d q p b p c a a b q b a d b a b s q q a p a q c a c s q d d c r c d p c d c a r r d s d r q d d c a r r d s d r q d r d b s s r c r s a r b b d a b q b a a c p p p r b p a p b b d q p c a a b q b a d b b b s q q a p a q c a a s q d d c r c d p c c c a r r d s d r q d d c a r r d s d r q d d d b s s r c r s a r r p b b d a b q b a a c q p p r b p a p b b d c a a b q b a d b b p s q q a p a q c a a b q d d c r c d p c c s a r r d s d r q d d c a r r d s d r q d d c b s s r c r s a r r d c p b b d a b q b a a d q p p r b p a p b b a a b q b a d b b p c q q a p a q c a a b s d d c r c d p c c s q r r d s d r q d d c a r r d s d r q d d c a s s r c r s a r r d ba c p b b d a b q b a b d q p p r b p a p b a b q b a d b b p c a q a p a q c a a b s q d c r c d p c c s q d r d s d r q d d c a r r d s d r q d d c a r s r c r s a r r d b sb b d q p p r b p a p a a c p b b d a b q b a p a q c a a b s q q b q b a d b b p c a a d s d r q d d c a r r c r c d p c c s q d d r c r s a r r d b s s d s d r q d d c a r r p b b d q p p r b p a b a a c p b b d a b q p a q c a a b s q q a q b a d b b p c a a b s d r q d d c a r r d r c d p c c s q d d c c r s a r r d b s s r s d r q d d c a r r da p b b d q p p r b p q b a a c p b b d a b a q c a a b s q q a p b a d b b p c a a b q d r q d d c a r r d s c d p c c s q d d c r r s a r r d b s s r c d r q d d c a r r d sp a p b b d q p p r b b q b a a c p b b d a q c a a b s q q a p a a d b b p c a a b q b r q d d c a r r d s d d p c c s q d d c r c s a r r d b s s r c r r q d d c a r r d s d b p a p b b d q p p r a b q b a a c p b b d c a a b s q q a p a q d b b p c a a b q b a q d d c a r r d s d r p c c s q d d c r c d a r r d b s s r c r s q d d c a r r d s d r r b p a p b b d q p p d a b q b a a c p b b a a b s q q a p a q c b b p c a a b q b a d d d c a r r d s d r q c c s q d d c r c d p r r d b s s r c r s a d d c a r r d s d r q p r b p a p b b d q p b d a b q b a a c p b a b s q q a p a q c a b p c a a b q b a d b d c a r r d s d r q d c s q d d c r c d p c r d b s s r c r s a r d c a r r d s d r q d p p r b p a p b b d q b b d a b q b a a c p b s q q a p a q c a a p c a a b q b a d b b c a r r d s d r q d d s q d d c r c d p c c d b s s r c r s a r r c a r r d s d r q d dq p p r b p a p b b d p b b d a b q b a a c s q q a p a q c a a b c a a b q b a d b b p a r r d s d r q d d c q d d c r c d p c c s b s s r c r s a r r d a r r d s d r q d d cd q p p r b p a p b b c p b b d a b q b a a q q a p a q c a a b s a a b q b a d b b p c r r d s d r q d d c a d d c r c d p c c s q s s r c r s a r r d b r r d s d r q d d c ab d q p p r b p a p b a c p b b d a b q b a q a p a q c a a b s q a b q b a d b b p c a r d s d r q d d c a r d c r c d p c c s q d s r c r s a r r d b s r d s d r q d d c a rq b q p s q q a r p p p a p b r p p q d b b a a c p b b d a b q b b b d q p p r b p a p c p r s s b c s 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c r c d d q s c c p c s d s c c p r s s b r d s d r r a c d d q d c r c d d q s c c p q d b b p a p b r p p a r p p q b q p s q q p p r b p a p b b d q b b d a b q b a a c p c r c d d q s c c p d s d s c c p r s s b c d s d r r a c d d q r c r c d d q s c c p dd b b p a p b r p p q r p p q b q p s q q a q p p r b p a p b b d p b b d a b q b a a c r c d d q s c c p d c d s c c p r s s b c s s d r r a c d d q r d r c d d q s c c p d cb b p a p b r p p q d p p q b q p s q q a r d q p p r b p a p b b c p b b d a b q b a a c d d q s c c p d c r s c c p r s s b c s d d r r a c d d q r d s c d d q s c c p d c rb p a p b r p p q d b p q b q p s q q a r p b d q p p r b p a p b a c p b b d a b q b a d d q s c c p d c r c c c p r s s b c s d s r r a c d d q r d s d d d q s c c p d c r c r c r s a r r d b s s s d s c b s s r p c c r a c d d q r d s d r s b d r r a s r c r s a a c p b b d a b q b b b d q p p r b p a p p r a q q s p q b q p b d q p p r b p a p b c r s a r r d b s s r d s c b s s r p c c s a c d d q r d s d r r b d r r a s r c r s s b a a c p b b d a b q p b b d q p p r b p a r a q q s p q b q p p d q p p r b p a p b br s a r r d b s s r c s c b s s r p c c s d c d d q r d s d r r a d r r a s r c r s s b q b a a c p b b d a b a p b b d q p p r b p a q q s p q b q p p r q p p r b p a p b b d s a r r d b s s r c r c b s s r p c c s d s d d q r d s d r r a c r r a s r c r s s b d b q b a a c p b b d a p a p b b d q p p r b q q s p q b q p p r a p p r b p a p b b d q a r r d b s s r c r s b s s r p c c s d s c d q r d s d r r a c d r a s r c r s s b d r a b q b a a c p b b d b p a p b b d q p p r q s p q b q p p r a q p r b p a p b b d q p r r d b s s r c r s a s s r p c c s d s c b q r d s d r r a c d d a s r c r s s b d r r d a b q b a a c p b b r b p a p b b d q p p s p q b q p p r a q q r b p a p b b d q p p r d b s s r c r s a r s r p c c s d s c b s r d s d r r a c d d q s r c r s s b d r r a b d a b q b a a c p b p r b p a p b b d q p p q b q p p r a q q s b p a p b b d q p p rd b s s r c r s a r r r p c c s d s c b s s d s d r r a c d d q r r c r s s b d r r a s b b d a b q b a a c p p p r b p a p b b d q q b q p p r a q q s p p a p b b d q p p r bb s s r c r s a r r d p c c s d s c b s s r s d r r a c d d q r d c r s s b d r r a s r p b b d a b q b a a c q p p r b p a p b b d b q p p r a q q s p q a p b b d q p p r b p s s r c r s a r r d b c c s d s c b s s r p d r r a c d d q r d s r s s b d r r a s r c c p b b d a b q b a a d q p p r b p a p b b q p p r a q q s p q b p b b d q p p r b p as r c r s a r r d b s c s d s c b s s r p c r r a c d d q r d s d s s b d r r a s r c r a c p b b d a b q b a b d q p p r b p a p b p p r a q q s p q b q b b d q p p r b p a p s d s c b s s r p c c r c r s a r r d b s s s b d r r a s r c r s r a c d d q r d s d r b b d q p p r b p a p a a c p b b d a b q b b d q p p r b p a p b p r a q q s p q b q p d s c b s s r p c c s c r s a r r d b s s r b d r r a s r c r s s a c d d q r d s d r r p b b d q p p r b p a b a a c p b b d a b q d q p p r b p a p b b r a q q s p q b q p p s c b s s r p c c s d r s a r r d b s s r c d r r a s r c r s s b c d d q r d s d r r a a p b b d q p p r b p q b a a c p b b d a b q p p r b p a p b b d a q q s p q b q p p r c b s s r p c c s d s s a r r d b s s r c r r r a s r c r s s b d d d q r d s d r r a c p a p b b d q p p r b b q b a a c p b b d a p p r b p a p b b d q q q s p q b q p p r ab s s r p c c s d s c a r r d b s s r c r s r a s r c r s s b d r d q r d s d r r a c d b p a p b b d q p p r a b q b a a c p b b d p r b p a p b b d q p q s p q b q p p r a q s s r p c c s d s c b r r d b s s r c r s a a s r c r s s b d r r q r d s d r r a c d d r b p a p b b d q p p d a b q b a a c p b b r b p a p b b d q p p s p q b q p p r a q q s r p c c s d s c b s r d b s s r c r s a r s r c r s s b d r r a r d s d r r a c d d q p r b p a p b b d q p b d a b q b a a c p b b p a p b b d q p p r p q b q p p r a q q sr p c c s d s c b s s d b s s r c r s a r r r c r s s b d r r a s d s d r r a c d d q r p p r b p a p b b d q b b d a b q b a a c p p a p b b d q p p r b q b q p p r a q q s p p c c s d s c b s s r b s s r c r s a r r d c r s s b d r r a s r s d r r a c d d q r d q p p r b p a p b b d p b b d a b q b a a c a p b b d q p p r b p b q p p r a q q s p q c c s d s c b s s r p s s r c r s a r r d b r s s b d r r a s r c d r r a c d d q r d s d q p p r b p a p b b c p b b d a b q b a a p b b d q p p r b p a q p p r a q q s p q b c s d s c b s s r p c s r c r s a r r d b s s s b d r r a s r c r r r a c d d q r d s d b d q p p r b p a p b a c p b b d a b q b a b b d q p p r b p a p p p r a q q s p q b q s d s c b s s r p c c c r c d p c c s q d d s b d r r a s r c r s c p r s s b c s d s c a c p b b d a b q b a q s b a a c q a p a q a a c p b b d a b q b b b d q p p r b p a pd s c b s s r p c c s r c d p c c s q d d c b d r r a s r c r s s p r s s b c s d s c c c p b b d a b q b a a s b a a c q a p a q q b a a c p b b d a b q p b b d q p p r b p a s c b s s r p c c s d c d p c c s q d d c r d r r a s r c r s s b r s s b c s d s c c p p b b d a b q b a a c b a a c q a p a q q s q b a a c p b b d a b a p b b d q p p r b p c b s s r p c c s d s d p c c s q d d c r c r r a s r c r s s b d s s b c s d s c c p r b b d a b q b a a c p a a c q a p a q q s b b q b a a c p b b d a p a p b b d q p p r bb s s r p c c s d s c p c c s q d d c r c d r a s r c r s s b d r s b c s d s c c p r s b d a b q b a a c p b a c q a p a q q s b a a b q b a a c p b b d b p a p b b d q p p r s s r p c c s d s c b c c s q d d c r c d p a s r c r s s b d r r b c s d s c c p r s s d a b q b a a c p b b c q a p a q q s b a a d a b q b a a c p b b r b p a p b b d q p p s r p c c s d s c b s c s q d d c r c d p c s r c r s s b d r r a c s d s c c p r s s b a b q b a a c p b b d q a p a q q s b a a c b d a b q b a a c p b p r b p a p b b d q p r p c c s d s c b s s s q d d c r c d p c c r c r s s b d r r a s s d s c c p r s s b c b q b a a c p b b d a a p a q q s b a a c q b b d a b q b a a c p p p r b p a p b b d q p c c s d s c b s s r q d d c r c d p c c s c r s s b d r r a s r d s c c p r s s b c s q b a a c p b b d a b p a q q s b a a c q a p b b d a b q b a a c q p p r b p a p b b d c c s d s c b s s r p d d c r c d p c c s q r s s b d r r a s r c s c c p r s s b c s d b a a c p b b d a b q a q q s b a a c q a p c p b b d a b q b a a d q p p r b p a p b b c s d s c b s s r p c d c r c d p c c s q d s s b d r r a s r c r c c p r s s b c s d s a a c p b b d a b q b q q s b a a c q a p a a c p b b d a b q b a b d q p p r b p a p b c r c d p c c s q d d s d s c b s s r p c c c p r s s b c s d s c s b d r r a s r c r s q s b a a c q a p a q a c p b b d a b q b a b b d q p p r b p a p a a c p b b d a b q br c d p c c s q d d c d s c b s s r p c c s p r s s b c s d s c c b d r r a s r c r s s s b a a c q a p a q q c p b b d a b q b a a p b b d q p p r b p a b a a c p b b d a b q c d p c c s q d d c r s c b s s r p c c s d r s s b c s d s c c p d r r a s r c r s s b b a a c q a p a q q s p b b d a b q b a a c a p b b d q p p r b p q b a a c p b b d a b d p c c s q d d c r c c b s s r p c c s d s s s b c s d s c c p r r r a s r c r s s b d a a c q a p a q q s b b b d a b q b a a c p p a p b b d q p p r b b q b a a c p b b d a p c c s q d d c r c d b s s r p c c s d s c s b c s d s c c p r s r a s r c r s s b d r a c q a p a q q s b a b d a b q b a a c p b b p a p b b d q p p r a b q b a a c p b b dc c s q d d c r c d p s s r p c c s d s c b b c s d s c c p r s s a s r c r s s b d r r c q a p a q q s b a a d a b q b a a c p b b r b p a p b b d q p p d a b q b a a c p b bc s q d d c r c d p c s r p c c s d s c b s c s d s c c p r s s b s r c r s s b d r r a q a p a q q s b a a c a b q b a a c p b b d p r b p a p b b d q p b d a b q b a a c p bs q d d c r c d p c c r p c c s d s c b s s s d s c c p r s s b c r c r s s b d r r a s a p a q q s b a a c q b 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a r r q a r a b q a p b q q d d p q d q a p q p r d s r s c r r p p s r b d c r d r s d d b b r d s c r r p p s r p r d s r c d s s q q c s q s r c sa s s b b q a b a p b s b q r r a a p q a q b a r a p a a r r q a r a b q a q b q q d d p q d q a p q p r p r d s r s c r r p p s d b d c r d r s d d b b r c r r p p s r p r d s r s d s s q q c s q s r c s c s s b b q a b a p b s b a r r a a p q a q b a r a q a a r r q a r a b q a q p q q d d p q d q a p q p b s r p r d s r s c r r p p r d b d c r d r s d d b b r r p p s r p r d s r s c s s q q c s q s r c s c ds b b q a b a p b s b a s r a a p q a q b a r a q r a r r q a r a b q a q p a q d d p q d q a p q p b q p s r p r d s r s c r r p b r d b d c r d r s d d b r p p s r p r d s r s c r s q q c s q s r c s c d s b b q a b a p b s b a s s a a p q a q b a r a q r r r r q a r a b q a q p a a d d p q d q a p q p b q q p p s r p r d s r s c r r b b r d b d c r d r s d d p p s r p r d s r s c r r q q c s q s r c s c d s sb q a b a p b s b a s s b a p q a q b a r a q r r a r q a r a b q a q p a a r d p q d q a p q p b q q d r p p s r p r d s r s c r d b b r d b d c r d r s d p s r p r d s r s c r r p q c s q s r c s c d s s qq a b a p b s b a s s b b p q a q b a r a q r r a a q a r a b q a q p a a r r p q d q a p q p b q q d d r r p p s r p r d s r s c d d b b r d b d c r d r s s r p r d s r s c r r p p c s q s r c s c d s s q qa b a p b s b a s s b b q q a q b a r a q r r a a p a r a b q a q p a a r r q q d q a p q p b q q d d p c r r p p s r p r d s r s s d d b b r d b d c r d r r p r d s r s c r r p p s s q s r c s c d s s q q cb a p b s b a s s b b q a a q b a r a q r r a a p q r a b q a q p a a r r q a d q a p q p b q q d d p q s c r r p p s r p r d s r r s d d b b r d b d c r d p r d s r s c r r p p s r q s r c s c d s s q q c s a p b s b a s s b b q a b q b a r a q r r a a p q a a b q a q p a a r r q a r q a p q p b q q d d p q d r s c r r p p s r p r d s d r s d d b b r d b d c r r d s r s c r r p p s r p s r c s c d s s q q c s qp b s b a s s b b q a b a b a r a q r r a a p q a q b q a q p a a r r q a r a a p q p b q q d d p q d q s r s c r r p p s r p r d r d r s d d b b r d b d c d s r s c r r p p s r p r r c s c d s s q q c s q sb s b a s s b b q a b a p p c p b c c p p a b p b q q a q p a a r r q a r a b a b a q b b s s a b s b p r d r s d d b b r d b d c d c d r c c a a d c a c s c r d r s d d b b r d b d d s r s c r r p p s r p r s b a s s b b q a b a p b c p b c c p p a b p b q p a q p a a r r q a r a b q b a q b b s s a b s b p a d r s d d b b r d b d c r c d r c c a a d c a c s d d c r d r s d d b b r d b r d s r s c r r p p s r pb a s s b b q a b a p b s p b c c p p a b p b q p c q p a a r r q a r a b q a a q b b s s a b s b p a b r s d d b b r d b d c r d d r c c a a d c a c s d c b d c r d r s d d b b r d p r d s r s c r r p p s r a s s b b q a b a p b s b b c c p p a b p b q p c p p a a r r q a r a b q a q q b b s s a b s b p a b a s d d b b r d b d c r d r r c c a a d c a c s d c d d b d c r d r s d d b b r r p r d s r s c r r p p s s s b b q a b a p b s b a c c p p a b p b q p c p b a a r r q a r a b q a q p b b s s a b s b p a b a q d d b b r d b d c r d r s c c a a d c a c s d c d r r d b d c r d r s d d b b s r p r d s r s c r r p p s b b q a b a p b s b a s c p p a b p b q p c p b c a r r q a r a b q a q p a b s s a b s b p a b a q b d b b r d b d c r d r s d c a a d c a c s d c d r c b r d b d c r d r s d d b p s r p r d s r s c r r pb b q a b a p b s b a s s p p a b p b q p c p b c c r r q a r a b q a q p a a s s a b s b p a b a q b b b b r d b d c r d r s d d a a d c a c s d c d r c c b b r d b d c r d r s d d p p s r p r d s r s c r rb q a b a p b s b a s s b p a b p b q p c p b c c p r q a r a b q a q p a a r s a b s b p a b a q b b s b r d b d c r d r s d d b a d c a c s d c d r c c a d b b r d b d c r d r s d r p p s r p r d s r s c r q a b a p b s b a s s b b a b p b q p c p b c c p p q a r a b q a q p a a r r a b s b p a b a q b b s s r d b d c r d r s d d b b d c a c s d c d r c c a a d d b b r d b d c r d r s r r p p s r p r d s r s ca b a p b s b a s s b b q b p b q p c p b c c p p a a r a b q a q p a a r r q b s b p a b a q b b s s a d b d c r d r s d d b b r c a c s d c d r c c a a d s d d b b r d b d c r d r c r r p p s r p r d s r sb a p b s b a s s b b q a p b q p c p b c c p p a b r a b q a q p a a r r q a s b p a b a q b b s s a b b d c r d r s d d b b r d a c s d c d r c c a a d c r s d d b b r d b d c r d s c r r p p s r p r d s ra p b s b a s s b b q a b b q p c p b c c p p a b p a b q a q p a a r r q a r b p a b a q b b s s a b s d c r d r s d d b b r d b c s d c d r c c a a d c a d r s d d b b r d b d c r r s c r r p p s r p r d sp b s b a s s b b q a b a q p c p b c c p p a b p b b q a q p a a r r q a r a p a b a q b b s s a b s b c r d r s d d b b r d b d s d c d r c c a a d c a c r d r s d d b b r d b d c s r s c r r p p s r p r dp c p b c c p p a b p b q b s b a s s b b q a b a p a b a q b b s s a b s b p q a q p a a r r q a r a b d c d r c c a a d c a c s r d r s d d b b r d b d c d s r s c r r p p s r p r c r d r s d d b b r d b dc p b c c p p a b p b q p s b a s s b b q a b a p b b a q b b s s a b s b p a a q p a a r r q a r a b q c d r c c a a d c a c s d d r s d d b b r d b d c r r d s r s c r r p p s r pd c r d r s d d b b r d bp b c c p p a b p b q p c b a s s b b q a b a p b s a q b b s s a b s b p a b q p a a r r q a r a b q a d r c c a a d c a c s d c r s d d b b r d b d c r d p r d s r s c r r p p s r b d c r d r s d d b b r db c c p p a b p b q p c p a s s b b q a b a p b s b q b b s s a b s b p a b a p a a r r q a r a b q a q r c c a a d c a c s d c d s d d b b r d b d c r d r r p r d s r s c r r p p s d b d c r d r s d d b b r c c p p a b p b q p c p b s s b b q a b a p b s b a b b s s a b s b p a b a q a a r r q a r a b q a q p c c a a d c a c s d c d r d d b b r d b d c r d r s s r p r d s r s c r r p p r d b d c r d r s d d b bc p p a b p b q p c p b c s b b q a b a p b s b a s b s s a b s b p a b a q b a r r q a r a b q a q p a c a a d c a c s d c d r c d b b r d b d c r d r s d p s r p r d s r s c r r p b r d b d c r d r s d d bp p a b p b q p c p b c c b b q a b a p b s b a s s s s a b s b p a b a q b b r r q a r a b q a q p a a a a d c a c s d c d r c c b b r d b d c r d r s d d p p s r p r d s r s c r r b b r d b d c r d r s d dp a b p b q p c p b c c p b q a b a p b s b a s s b s a b s b p a b a q b b s r q a r a b q a q p a a r a d c a c s d c d r c c a b r d b d c r d r s d d b r p p s r p r d s r s c r d b b r d b d c r d r s da b p b q p c p b c c p p q a b a p b s b a s s b b a b s b p a b a q b b s s q a r a b q a q p a a r r d c a c s d c d r c c a a r d b d c r d r s d d b b r r p p s r p r d s r s c d d b b r d b d c r d r s
b p b q p c p b c c p p a a b a p b s b a s s b b q b s b p a b a q b b s s a a r a b q a q p a a r r q c a c s d c d r c c a a d d b d c r d r s d d b b r c r r p p s r p r d s r s s d d b b r d b d c r d r
p b q p c p b c c p p a b b a p b s b a s s b b q a s b p a b a q b b s s a b r a b q a q p a a r r q a a c s d c d r c c a a d c b d c r d r s d d b b r d s c r r p p s r p r d s r r s d d b b r d b d c r d
b q p c p b c c p p a b p a p b s b a s s b b q a b b p a b a q b b s s a b s a b q a q p a a r r q a r c s d c d r c c a a d c a d c r d r s d d b b r d b r s c r r p p s r p r d s d r s d d b b r d b d c r
q p c p b c c p p a b p b p b s b a s s b b q a b a p a b a q b b s s a b s b b q a q p a a r r q a r a s d c d r c c a a d c a c c r d r s d d b b r d b d s r s c r r p p s r p r d r d r s d d b b r d b d c
OD(104;26,26,26,26)
73
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q p p a d r p b a a p d c a q a a p s r a q a a p s r a b p p a d r p b a r d p a a b a r d p a a b p c s a p p p p a d c p q p p a s r p b p p a s r p b p p a d r p q p p a d r p q a a p d c a b a a p s r a q a a p s r a q p p a d r p b a r d p a a b a r d p a a b p c s a p p b q p p a d c p b p p a s r p b p p a s r p p a d r p q p p a d r p q p a p d c a b a a p s r a q a a p s r a q a p a d r p b p r d p a a b a r d p a a b a c s a p p b p p q p p a d c p b p p a s r p b p p a s r a d r p q p p a d r p q p p p d c a b a a p s r a q a a p s r a q a a a d r p b p p d p a a b a r d p a a b a r s a p p b p c c p q p p a d r p b p p a s r p b p p a s d r p q p p a d r p q p p a d c a b a a p s r a q a a p s r a q a a p d r p b p p a p a a b a r d a p p b p c s p a a b a r d s r p b p p a d c p q p p a s r p b p p a r p q p p a d c a b a a p d r p q p p a d r a q a a p s r p b p p a d r a q a a p s a a b a r d p p p b p c s a a a b a r d p a s r p b p p a d c p q p p a s r p b p p p q p p a d r a b a a p d c p q p p a d r a q a a p s r p b p p a d r a q a a p s r a b a r d p a p b p c s a p a b a r d p a p a s r p b p p a d c p q p p a s r p b p q p p a d r p b a a p d c a q p p a d r p q a a p s r a b p p a d r p q a a p s r ab a r d p a a b p c s a p p b a r d p a a p p a s r p b p p a d c p q p p a s r p b p p a d r p q a a p d c a b p p a d r p q a a p s r a q p p a d r p b a a p s r a qa r d p a a b p c s a p p b a r d p a a b b p p a s r p q p p a d c p b p p a s r p p a d r p q p a p d c a b a p a d r p q p a p s r a q a p a d r p b p a p s r a q a r d p a a b a c s a p p b p r d p a a b a p b p p a s r p q p p a d c p b p p a s r a d r p q p p p d c a b a a a d r p q p p p s r a q a a a d r p b p p p s r a q a ad p a a b a r s a p p b p c d p a a b a r r p b p p a s c p q p p a d r p b p p a s d r p q p p a d c a b a a p d r p q p p a s r a q a a p d r p b p p a s r a q a a pa p p b p c s p a a b a r d p a a b a r d s r p b p p a s r p b p p a d c p q p p a c a b a a p d r p q p p a d r p q p p a d r p b p p a d r a q a a p s r a q a a p s p p b p c s a a a b a r d p a a b a r d p a s r p b p p a s r p b p p a d c p q p p a b a a p d c p q p p a d r p q p p a d r p b p p a d r a q a a p s r a q a a p s r p b p c s a p a b a r d p a a b a r d p a p a s r p b p p a s r p b p p a d c p q p b a a p d c a q p p a d r p q p p a d r p b p p a d r p q a a p s r a q a a p s r ab p c s a p p b a r d p a a b a r d p a a p p a s r p b p p a s r p b p p a d c p q a a p d c a b p p a d r p q p p a d r p q p p a d r p b a a p s r a q a a p s r a q p c s a p p b a r d p a a b a r d p a a b b p p a s r p b p p a s r p q p p a d c p a p d c a b a p a d r p q p p a d r p q p p a d r p b p a p s r a q a a p s r a q a c s a p p b p r d p a a b a r d p a a b a p b p p a s r p b p p a s r p q p p a d c p d c a b a a a d r p q p p a d r p q p p a d r p b p p p s r a q a a p s r a q a a s a p p b p c d p a a b a r d p a a b a r r p b p p a s r p b p p a s c p q p p a d d c a b a a p d r p q p p a d r p q p p a d r p b p p a s r a q a a p s r a q a a p p a a q a r s p a a q a r s a p p b p r d c a b a a p s c a b a a p s r p q p p a s d c p q p p a s r p b p p a s r p b p p a c p q p p a s c p q p p a s r a q a a p da a q a r s p a a q a r s p p p b p r d a a b a a p s c a b a a p s c p q p p a s r a d c p q p p a s r p b p p a s r p b p p p q p p a s c p q p p a s c a q a a p d ra q a r s p a a q a r s p a p b p r d a p b a a p s c a b a a p s c a q p p a s r p p a d c p q p p a s r p b p p a s r p b p q p p a s c p q p p a s c p q a a p d r aq a r s p a a q a r s p a a b p r d a p p a a p s c a b a a p s c a b p p a s r p q p p a d c p q p p a s r p b p p a s r p b p p a s c p q p p a s c p q a a p d r a qa r s p a a q a r s p a a q p r d a p p b a p s c a b a a p s c a b a p a s r p q p q p p a d c p b p p a s r p b p p a s r p p a s c p q p p a s c p q p a p d r a q a r s p a a q a r s p a a q a r d a p p b p p s c a b a a p s c a b a a a s r p q p p p q p p a d c p b p p a s r p b p p a s r a s c p q p p a s c p q p p p d r a q a a s p a a q a r s p a a q a r d a p p b p r s c a b a a p s c a b a a p s r p q p p a c p q p p a d r p b p p a s r p b p p a s s c p q p p a s c p q p p a d r a q a a p p a a q a r s a p p b p r d p a a q a r s c a b a a p s r p q p p a s c a b a a p s s r p b p p a d c p q p p a s r p b p p a c p q p p a s r a q a a p d c p q p p a sa a q a r s p p p b p r d a a a q a r s p a b a a p s c p q p p a s r a b a a p s c a s r p b p p a d c p q p p a s r p b p p p q p p a s c a q a a p d r p q p p a s ca q a r s p a p b p r d a p a q a r s p a b a a p s c a q p p a s r p b a a p s c a p a s r p b p p a d c p q p p a s r p b p q p p a s c p q a a p d r a q p p a s c pq a r s p a a b p r d a p p q a r s p a a a a p s c a b p p a s r p q a a p s c a b p p a s r p b p p a d c p q p p a s r p b p p a s c p q a a p d r a q p p a s c p qa r s p a a q p r d a p p b a r s p a a q a p s c a b a p a s r p q p a p s c a b a b p p a s r p q p p a d c p b p p a s r p p a s c p q p a p d r a q a p a s c p q p r s p a a q a r d a p p b p r s p a a q a p s c a b a a a s r p q p p p s c a b a a p b p p a s r p q p p a d c p b p p a s r a s c p q p p p d r a q a a a s c p q p p s p a a q a r d a p p b p r s p a a q a r s c a b a a p s r p q p p a s c a b a a p r p b p p a s c p q p p a d r p b p p a s s c p q p p a d r a q a a p s c p q p p a a p p b p r d p a a q a r s p a a q a r s r p q p p a s c a b a a p s c a b a a p s s r p b p p a s r p b p p a d c p q p p a r a q a a p d c p q p p a s c p q p p a s p p b p r d a a a q a r s p a a q a r s p p q p p a s r a b a a p s c a b a a p s c a s r p b p p a s r p b p p a d c p q p p a q a a p d r p q p p a s c p q p p a s c p b p r d a p a q a r s p a a q a r s p a q p p a s r p b a a p s c a b a a p s c a p a s r p b p p a s r p b p p a d c p q p q a a p d r a q p p a s c p q p p a s c p b p r d a p p q a r s p a a q a r s p a a p p a s r p q a a p s c a b a a p s c a b p p a s r p b p p a s r p b p p a d c p q a a p d r a q p p a s c p q p p a s c p q p r d a p p b a r s p a a q a r s p a a q p a s r p q p a p s c a b a a p s c a b a b p p a s r p b p p a s r p q p p a d c p a p d r a q a p a s c p q p p a s c p q p r d a p p b p r s p a a q a r s p a a q a a s r p q p p p s c a b a a p s c a b a a p b p p a s r p b p p a s r p q p p a d c p d r a q a a a s c p q p p a s c p q p p d a p p b p r s p a a q a r s p a a q a r s r p q p p a s c a b a a p s c a b a a p r p b p p a s r p b p p a s c p q p p a d d r a q a a p s c p q p p a s c p q p p a p a a b a c s p a a b a c s a p p q p r s c p b p p a d c p b p p a d c a q a a p s r a b a a p d r a b a a p d c p b p p a s d c p q p p a s r p b p p a s r p b p p a a a b a c s p a a b a c s p p p q p r s a p b p p a d c p b p p a d c a q a a p s c a b a a p d r a b a a p d r p b p p a s c a d c p q p p a s r p b p p a s r p b p p a b a c s p a a b a c s p a p q p r s a p b p p a d c p b p p a d c p q a a p s c a b a a p d r a b a a p d r a b p p a s c p p a d c p q p p a s r p b p p a s r p b p b a c s p a a b a c s p a a q p r s a p p p p a d c p b p p a d c p b a a p s c a q a a p d r a b a a p d r a b p p a s c p b p p a d c p q p p a s r p b p p a s r p ba c s p a a b a c s p a a b p r s a p p q p a d c p b p p a d c p b p a p s c a q a a p d r a b a a p d r a b a p a s c p b p q p p a d c p b p p a s r p b p p a s r p c s p a a b a c s p a a b a r s a p p q p a d c p b p p a d c p b p p p s c a q a a p d r a b a a p d r a b a a a s c p b p p p q p p a d c p b p p a s r p b p p a s r s p a a b a c s p a a b a c s a p p q p r d c p b p p a d c p b p p a s c a q a a p d r a b a a p d r a b a a p s c p b p p a c p q p p a d r p b p p a s r p b p p a s p a a b a c s a p p q p r s p a a b a c s c p b p p a d c a q a a p s c p b p p a d r a b a a p d c p b p p a s r a b a a p d s r p b p p a d c p q p p a s r p b p p aa a b a c s p p p q p r s a a a b a c s p p b p p a d c a q a a p s c p b p p a d c a b a a p d r p b p p a s c a b a a p d r a s r p b p p a d c p q p p a s r p b p pa b a c s p a p q p r s a p a b a c s p a b p p a d c p q a a p s c a b p p a d c p b a a p d r a b p p a s c p b a a p d r a p a s r p b p p a d c p q p p a s r p b pb a c s p a a q p r s a p p b a c s p a a p p a d c p b a a p s c a q p p a d c p b a a p d r a b p p a s c p b a a p d r a b p p a s r p b p p a d c p q p p a s r p ba c s p a a b p r s a p p q a c s p a a b p a d c p b p a p s c a q a p a d c p b p a p d r a b a p a s c p b p a p d r a b a b p p a s r p q p p a d c p b p p a s r p c s p a a b a r s a p p q p c s p a a b a a d c p b p p p s c a q a a a d c p b p p p d r a b a a a s c p b p p p d r a b a a p b p p a s r p q p p a d c p b p p a s r s p a a b a c s a p p q p r s p a a b a c d c p b p p a s c a q a a p d c p b p p a d r a b a a p s c p b p p a d r a b a a p r p b p p a s c p q p p a d r p b p p a sa p p q p r s p a a b a c s p a a b a c s c a q a a p s c p b p p a d c p b p p a d c p b p p a s r a b a a p d r a b a a p d s r p b p p a s r p b p p a d c p q p p ap p q p r s a a a b a c s p a a b a c s p a q a a p s c p b p p a d c p b p p a d c p b p p a s c a b a a p d r a b a a p d r a s r p b p p a s r p b p p a d c p q p pp q p r s a p a b a c s p a a b a c s p a q a a p s c a b p p a d c p b p p a d c p b p p a s c p b a a p d r a b a a p d r a p a s r p b p p a s r p b p p a d c p q pq p r s a p p b a c s p a a b a c s p a a a a p s c a q p p a d c p b p p a d c p b p p a s c p b a a p d r a b a a p d r a b p p a s r p b p p a s r p b p p a d c p q
p r s a p p q a c s p a a b a c s p a a b a p s c a q a p a d c p b p p a d c p b p p a s c p b p a p d r a b a a p d r a b a b p p a s r p b p p a s r p q p p a d c p
r s a p p q p c s p a a b a c s p a a b a p s c a q a a a d c p b p p a d c p b p p a s c p b p p p d r a b a a p d r a b a a p b p p a s r p b p p a s r p q p p a d c
s a p p q p r s p a a b a c s p a a b a c s c a q a a p d c p b p p a d c p b p p a s c p b p p a d r a b a a p d r a b a a p r p b p p a s r p b p p a s c p q p p a d
OD(84;48,12,12,12)
74