Convex Cone Conditions on the Structure of Designs

Citation

Dukes, Peter James (2003) Convex Cone Conditions on the Structure of Designs. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/V7F7-FQ47. https://resolver.caltech.edu/CaltechETD:etd-07142002-210918

Abstract

Various known and original inequalities concerning the structure of combinatorial designs are established using polyhedral cones generated by incidence matrices. This work begins by giving definitions and elementary facts concerning t-designs. A connection with the incidence matrix W of t-subsets versus k-subsets of a finite set is mentioned. The opening chapter also discusses relevant facts about convex geometry (in particular, the Farkas Lemma) and presents an arsenal of binomial identities. The purpose of Chapter 2 is to study the cone generated by columns of W, viewed as an increasing union of cones with certain invariant automorphisms. The two subsequent chapters derive inequalities on block density and intersection patterns in t-designs. Chapter 5 outlines generalizations of W which correspond to hypergraph designs and poset designs. To conclude, an easy consequence of this theory for orthogonal arrays is used in a computing application which generalizes the method of two-point based sampling

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