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This dissertation is devoted to the study of relative difference sets and circulant weighing matrices using algebraic tools. We are mainly interested in answering the existence problems for these objects: For what values of $v$ and $n$, does a circulant weighing matrix $CW(v,n)$ exist? For what values of $m$, $n$, $k$, $\lambda$ does a $(m, n, k, \lambda)$ difference set exist? The existence of such combinatorial structures is proved via explicit construction while non-existence proofs are based on algebraic and number theoretic methods and sometimes computer assistance. Circulant weighing matrices and relative difference sets exhibit some similarities. First of all, they can be characterized in terms of group ring equations. Hence, powerful algebraic methods can be adopted to extract information on the structure of these objects. Furthermore, their images under a character of a group satisfy the modulus equation $$X\bar{X}= n$$ where $X \in \Z[\zeta_v]$ and $n$ and $v$ are integers. In addition, these two combinatorial objects are closely related to each other as well as other well known combinatorial structures such as Hadamard designs, difference sets and sequences. Hence, methods used to study these combinatorial objects can be adapted to the study of the two combinatorial structures in question. Algebraic methods that are used in the study of these objects are the self-conjugacy approach, the multiplier approach, and the field descent method. We extend the usual concept of multipliers to group rings with cyclotomic integers as coefficients and develop new non-existence results. Seven open cases were solved using this approach. Solutions of the modulus equations are called Weil numbers. Knowledge of these Weil numbers helps to tackle the existence problems. We incorporate the field descent method in search for these Weil numbers and with the help of computer, we settle two other open cases. In total, we solved $18$ open cases of circulant weighing matrices of order less than $200$, as well as several infinite families of cases. Circulant weighing matrices are rare. In fact, all of our new results concerning circulant weighing matrices are non-existence results. Relative difference sets, on the other hand, have a higher chance for positive construction results. We employ binary as well as quaternary Golay sequences, Williamson matrices, and building sets to develop a recursive construction for $(2m, 2, 2m, m)$ relative difference sets. We introduce the term Golay transversal which serves as the ingredient for our construction. As a result, we obtain the most general known construction of normal $(2m, 2, 2m, m)$ relative difference sets, extending previously known results. We append the most updated version of Strassler's table concerning the existence status of circulant weighing matrices of order at most $200$ and weight at most $100$. Detailed references for each case are provided as well. For relative difference sets, a table of the existence status of Golay transversals in abelian groups were provided. Doctor of Philosophy (SPMS) |
