## A new class of designs and singly or doubly equivalent designs

##### Abstract

We are already familiar with ('v, k', [lambda])-difference sets and ('v, k', [lambda])-designs. In this thesis, we will introduce a new class of difference sets and designs: ('v, k', [[lambda]1, [lambda]2, ... , [lambda]' m']) difference sets and ('v, k', [[lambda] 1, [lambda]2, ... , [lambda]'m'])-designs. We will also introduce the concepts of singly equivalent designs and doubly equivalent designs. In Chapter 2, we will discuss some necessary conditions and nonexistence theorems. In Chapter 3, we will first discuss the necessary and sufficient conditions for a ('v, k', [[lambda]1, [lambda]2, ... , [lambda]'m'])-designs to be a singly equivalent design. We will prove that a [lambda]2-equivalent ('v, k ', [[lambda]1, [lambda]2];'t')-difference set is a (_, 't' + 1, 'k', [lambda] 2, [lambda]1)-DDS. We will show that a 0-equivalent (' n'2 - 1, 'n', [1,0])-design can be embedded into an affine plane. We will also prove that for a 0-equivalent ('v, k', [[lambda], 0]; 't')-design if ' t' >= 2 then the point 0 is missing from at least one parallel class. Define 'u' to be the number of parallel classes missing the point 0. We have obtained that up to isomorphism {0, 1} mod 4 is the only 0-equivalent ('v, k', [[lambda], 0]; 't')-design with 'u' = 0. We can attain a "standard" difference set from a 0-equivalent ('v, k', [[lambda], 0]; 't')-design with 'u' > 1. We will also prove that we can add some points to a base set of a 0-equivalent ('v, k', [[lambda], 0]; ' t')-design with 'u' = 1 and get a set which generates a ('t' + 1)-equivalent ('v, k' + 't' + 1, [[lambda] + 2, 't' + 1]; 't')-design or a ('v, k' + 't' + 1, [lambda] + 2)-design. In Chapter 4, we will discuss doubly equivalent designs as well as the notion of super classes. We will describe the structure of super classes and discuss properties of doubly equivalent designs. We will extend results of singly equivalent design to doubly equivalent designs. In Chapter 5, we will give an example to construct a ('v, k', [[lambda]1, [lambda]2, [lambda]3])-design. Some more general necessary conditions and existence the rems than those in Elliott and Butson [12]. Ryser [35]. Wei, Gao and Yang [37] and/or Theorem 2.44 in Chapter 2 will be given. We will also show how to construct difference sets from ('v, k', [[lambda], 0])-difference sets. Many other results will be also given. We will include the tables of singly equivalent difference sets obtained by computers as appendixes. We will also include a 'C' ++ program to search ('v, k', [[lambda]1, [lambda] 2])-difference sets.