Association schemes, codes, and difference sets

Abstract

This thesis consists of an introductory chapter and three independent chapters. In the first chapter, we give a brief description of the three independent chapters: abelian n-adic codes; generalized skew hadamard difference sets; and equivariant incidence structures. In the second chapter, we introduce n-adic codes, a generalization of the Duadic Codes studied by Pless and Rushanan, and we solve the corresponding existence problem. We introduce n-adic groups, canonical pplitters, and Margarita Codes to generalize the self-dual codes of Rushanan and Pless, and we solve the corresponding existence problem. In the third chapter, we consider the generalized skew hadamard difference set (GSHDS) existence problem. We introduce the combinatorial matrices AG,G1, where G1=(Z/exp(G)Z)* and G is a group, to reduce the existence problem to an integral equation. Using a special finite-dimensional algebra or Association Scheme, we show AG,G12 = |G|/pI for general G. With the aid of AG,G1, we show some necessary conditions for the existence of a GSHDS D in the group (Z/pZ) x (Z/p2Z)2α+1, we provide a proof of Johnsen's exponent bound, we provide a proof of Xiang's exponent bound, and we show a necessary existence condition for general G. In the fourth chapter, we study the incidence matrices Wt,k(v) of t-subsets of {1,...,v} vs. k-subsets of {1,...,v}. Also, given a group G acting on {1,...,v}, we define analogous incidence matrices Mt,k and M't,k of k-subsets' orbits vs. t-subsets' orbits. For general G, we show that Mt,k and M't,k have full rank over Q, we give a bound on the exponent of the Smith Group of Mt,k and M't,k, and we give a partial answer to the integral preimage problem for Mt,k and M't,k. We propose the Equivariant Sign Conjecture for the matrices Wt,k(v) using a special basis of the column module of Wt,k consisting of columns of Wt,k; we verify the Equivariant Sign Conjecture for small cases; and we reduce this conjecture to the case v=2k+t. For the case G=(Z/nZ), we conjecture that M't,k has a basis of the column module of M't,k that consists of columns of M't,k. We prove this conjecture for (t,k)=(2,3),(2,4), and we use these results to calculate the Smith Group of M2,4, M'2,4, M2,3, and M'2,3 for general n.