Construction of leaves and excesses when κ = 3,4

Abstract

A packing design, or a PD (υ, κ, λ) is a family of κ-subsets (called blocks), of a υ-set S, such that every 2-subset (called a pair), of S is contained in at most λ blocks. The packing number P (υ, κ, λ) is the number of blocks in a PD (υ, κ, λ). -- The edges in the multigraph λΚυ not contained in the packing form the leave of the PD (υ, κ, λ), denoted by leave (υ, κ, λ). Generally we consider maximum packings (packings with maximum number of blocks) unless stated otherwise. -- A covering design, or a CD (υ, κ, λ) is a family of κ-subsets (called blocks), of a υ-set S, such that every 2-subset (called a pair), of S is contained in at least λ blocks. The covering number C (υ, κ, λ) is the number of blocks in a CD (υ, κ, λ). -- The extrone edges added to the multigraph λΚυ in the covering form the excess of the CD (υ, κ, λ), denoted by excess (υ, κ, λ). Generally we consider minimum coverings (coverings with minimum number of blocks) unless stated otherwise. -- In this thesis we give the direct constructions of the leaves and excesses for κ = 3, 4. Some of them are from existing papers, some are the author's original work. This is the first time to put all the leaves and excesses for κ = 4 and all λs together (with only few possible exceptions).