Applications Of Weil's Theorem In Combinatorial Designs

Weil's theorem on character sums is an important theorem in number theory. This theorem can be written in two versions; one is on additive character sums and the other is on multiplicative character sums. In this dissertation, some applications of Weil's theorem in combinatorial designs are provided.By using Weil's theorem on multiplicative character sums, Wilson's bound on difference families for general k is improved in Chapter 2. For k = 4,5,6, the existence of (q,k,l) difference families is solved. For k = 7, partial results are also given. In Chapter 3, Lu's bound on spectrum Q(k, A) of coset difference array with k = 2λ + 1 is improved; and the spectrum Q(3,1) is determined. In addition, the degenerate case when k = λ + 1 is also discussed. In Chapter 4, the existence of APAV(q, k) in GF(q) with q ≡ 3 (mod 4) is investigated. In particular, the existence of an APAV(q, k) with a prime power q ≡ 3 (mod 4) and k = 7, 9,11,13,15 is solved. In Chapter 5, the existence of V(m, t) for general m is investigated; and the spectrum for V(m,t)'s with m = 7 is determined. For variant V(m,t)'s, such as V²(m,t), V⁴ (m,t) and V_λ(m,t) are also investigated. Some special sum graphs and difference graphs, based on abelian groups, are discussed in Chapter 6. In addition to W.-C. W. Li's result on character sum estimates, Weil's theorem on additive character sums is used to show that these are indeed Ramanujan graphs.