Group weighing matrices 1

2.1 Some Inductive Constructions of Group Weighing Matrices 12 3.1 Some Known Results on Abelian Groups Weighing Matrices 3.2 Some New Results on Abelian Groups Weighing Matrices 3.3 The

I would like to extend my warmest appreciation and profound gratitude to my supervisors, Assoc Professor Ma Siu Lun and Assoc Professor Leung Ka Hin for their excellent, unwavering and invaluable guidance in helping me complete my thesis at the National University of Singapore

Special thanks go out to Universiti Sains Malaysia, for their generosity in pro-viding me with the necessary financial aid, through the Academic Staff Training Scheme, without which I would not have been able to undertake my PhD studies

in Singapore

I am also forever indebted to my loving and supportive husband, Mr Tan Hooi Boon, family and friends for their encouragement and understanding throughout the course of my studies at the National University of Singapore

I would also like to express my deep gratitude to Assoc Professor Lang Mong Lung as well as Assoc Professor How Guan Aun (of Universiti Sains Malaysia) for their friendship, guidance and encouragement

2.1 Some Inductive Constructions of Group Weighing Matrices 12

3.1 Some Known Results on Abelian Groups Weighing Matrices

3.2 Some New Results on Abelian Groups Weighing Matrices

3.3 The Study of the Existence of Proper Circulant Weighing

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Chapter 4 Generalized Dihedral Group Weighing Matrices 48

4.1 Basic Properties of Generalized Dihedral Group Weighing

4.2 A Construction of Generalized Dihedral Group Weighing

4.3 Some Non-existent Results of Proper Generalized

5.1 Some Properties of Symmetric Abelian Group

5.2 Constructions of Symmetric Group Weighing

5.3 Exponent Bounds on Abelian Groups Admit

A weighing matrix of order n and weight ν2 is a square matrix M of order n

with entries from {0, ±1} such that M MT = ν2I for some integer ν where I is the

identity matrix of order n Let G = {g1, g2, , gn} be a group and A =Pn

i=1aigi ∈ Z[G] satisfies

(W1) A has 0, ±1 coefficients and

(W2) AA(−1) = ν2 where A(−1) =Pn

i=1aig−1i

If M = (bij) is a group matrix of G such that bij = ak if gigj−1 = gk, then M MT =

ν2I for some integer ν and A ∈ Z[G] is called a group weighing matrix denoted by

W (G, ν2)

For the case when G is abelian, group weighing matrices are essentially the same as perfect ternary arrays Chapter one is an introduction and the discussion

of some basic properties of group weighing matrices Some properties of perfect ternary arrays and character theory that will be needed in our further discussions are also given in this chapter

In Chapter two, we mainly study constructions and examples of proper W (G, ν2)

Some of the constructions are new

Chapter three discusses abelian group weighing matrices We study the structure

of W (G, p2t) where p is an odd prime and G is an abelian group having cyclic Sylow

p-subgroup Let G = hαi × H be an abelian group with o(α) = ps and p is an odd

prime that is relatively prime to the exponent of H We found that any W (G, p2f)

with f ≤ s − 1 is not proper

Apart from these results we also give a thorough study of the existence of proper circulant weighing matrices with weight 9 in chapter three

iv

Let DH = H ∪ θH be a group where H is a finite abelian group, o(θ) = 2 and

hθ = θh−1 for all h ∈ H The group DH is called a generalized dihedral group We

study generalized dihedral group weighing matrices in chapter four Some basic properties of generalized dihedral group weighing matrices and a construction of even weight generalized dihedral group weighing matrices are given If p is an odd prime and H is an abelian group with cyclic Sylow p-subgroup, then we found that

no proper W (DH, p2f) exist for f ≥ 1

The last chapter, that is chapter five, is on symmetric group weighing matrices Let A ∈ Z[G] be a W (G, ν2) It can be easily checked that the weighing matrix constructed by A is symmetric if and only if

(W3) A(−1) = A

Some new examples of symmetric group weighing matrices are found We have also obtained a few exponent bounds on abelian groups that admit symmetric W (G, ν2)

In particular, we prove that there is no symmetric W (G, p2) where G is an abelian

group of order 2pr, p is a prime and p ≥ 5