ACKNOWLEDGEMENTS 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 providing 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. i CONTENTS Acknowledgements i Summary iv Chapter Introduction to Group Weighing Matrices 1.1 Weighing Matrices 1.2 Group Weighing Matrices 1.3 Perfect Ternary Sequences and Arrays 1.4 Character Theory Chapter Constructions of Group Weighing Matrices 12 2.1 Some Inductive Constructions of Group Weighing Matrices 12 2.2 Constructions Using Difference Sets 13 2.3 Constructions Using Divisible Difference Sets 17 2.4 Construction Using Hyperplane 20 2.5 Construction Using Finite Local Ring 23 Chapter Some Results on Abelian Group Weighing Matrices 27 3.1 Some Known Results on Abelian Groups Weighing Matrices with Odd Prime Power Weight 27 3.2 Some New Results on Abelian Groups Weighing Matrices with Odd Prime Power Weight 29 3.3 The Study of the Existence of Proper Circulant Weighing Matrices with Weight 36 ii Chapter Generalized Dihedral Group Weighing Matrices 48 4.1 Basic Properties of Generalized Dihedral Group Weighing Matrices 48 4.2 A Construction of Generalized Dihedral Group Weighing Matrices with Even Weight 51 4.3 Some Non-existent Results of Proper Generalized Dihedral Group Weighing Matrices Chapter Symmetric Abelian Group Weighing Matrices 52 55 5.1 Some Properties of Symmetric Abelian Group Weighing Matrices 55 5.2 Constructions of Symmetric Group Weighing Matrices 58 5.3 Exponent Bounds on Abelian Groups Admit Symmetric Group Weighing Matrices Bibliography 62 68 iii SUMMARY A weighing matrix of order n and weight ν is a square matrix M of order n with entries from {0, ±1} such that M M T = ν I for some integer ν where I is the identity matrix of order n. Let G = {g1 , g2 , . . . , gn } be a group and A = n i=1 gi ∈ Z[G] satisfies (W1) A has 0, ±1 coefficients and (W2) AA(−1) = ν where A(−1) = n i=1 gi−1 . If M = (bij ) is a group matrix of G such that bij = ak if gi gj−1 = gk , then M M T = ν I for some integer ν and A ∈ Z[G] is called a group weighing matrix denoted by W (G, ν ). 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, ν ). 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 = α × 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 − is not proper. Apart from these results we also give a thorough study of the existence of proper circulant weighing matrices with weight in chapter three. iv Let DH = H ∪ θH be a group where H is a finite abelian group, o(θ) = 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, ν ). 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, ν ). 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. v . encouragement. i CONTENTS Acknowledgements i Summary iv Chapter 1 Introduction to Group Weighing Matrices 1 1 .1 Weighing Matrices 1 1.2 Group Weighing Matrices 2 1. 3 Perfect Ternary Sequences and Arrays 6 1. 4 Character Theory 9 Chapter. Constructions of Group Weighing Matrices 12 2 .1 Some Inductive Constructions of Group Weighing Matrices 12 2.2 Constructions Using Difference Sets 13 2.3 Constructions Using Divisible Difference Sets 17 2.4. Dihedral Group Weighing Matrices with Even Weight 51 4.3 Some Non-existent Results of Proper Generalized Dihedral Group Weighing Matrices 52 Chapter 5 Symmetric Abelian Group Weighing Matrices 55 5.1