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More About This Textbook
Overview
Design Theory, Second Edition presents some of the most important techniques used for constructing combinatorial designs. It augments the descriptions of the constructions with many figures to help students understand and enjoy this branch of mathematics.
This edition now offers a thorough development of the embedding of Latin squares and combinatorial designs. It also presents some pure mathematical ideas, including connections between universal algebra and graph designs.
The authors focus on several basic designs, including Steiner triple systems, Latin squares, and finite projective and affine planes. They produce these designs using flexible constructions and then add interesting properties that may be required, such as resolvability, embeddings, and orthogonality. The authors also construct more complicated structures, such as Steiner quadruple systems.
By providing both classical and state-of-the-art construction techniques, this book enables students to produce many other types of designs.
Editorial Reviews
From the Publisher
…it is remarkable how quickly the book propels the reader from the basics to the frontiers of design theory … Combined, these features make the book an excellent candidate for a design theory text. At the same time, even the seasoned researcher of triple systems will find this a useful resource.—Peter James Dukes (3-VCTR-MS; Victoria, BC), Mathematical Reviews, 2010
Booknews
A text for advanced undergraduate and graduate courses in combinatorial design theory, featuring explanations of basic designs such as Steiner and Kirkman triple systems, mutually orthogonal Latin squares, finite projective and affine planes, and Steiner quadruple systems. Covers both classic and modern construction techniques in combinatorial designs, progressing from simple to complex constructions, with exercises throughout each chapter. Annotation c. by Book News, Inc., Portland, Or.Product Details
Related Subjects
Table of Contents
Steiner Triple Systems
The Existence Problem
v ≡ 3 (mod 6): The Bose Construction
v ≡ 1 (mod 6): The Skolem Construction
v ≡ 5 (mod 6): The 6n + 5 Construction
Quasigroups with Holes and Steiner Triple Systems
The Wilson Construction
Cyclic Steiner Triple Systems
The 2n + 1 and 2n + 7 Constructions
λ-Fold Triple Systems
Triple Systems of Index λ > 1
The Existence of Indempotent Latin Squares
2-fold Triple Systems
λ= 3 and 6
λ-Fold Triple Systems in General
Quasigroup Identities and Graph Decompositions
Quasigroup Identities
Mendelsohn Triple Systems Revisited
Steiner Triple Systems Revisited
Maximum Packings and Minimum Coverings
The General Problem
Maximum Packings
Minimum Coverings
Kirkman Triple Systems
A Recursive Construction
Constructing Pairwise Balanced Designs
Mutually Orthogonal Latin Squares
Introduction
The Euler and MacNeish Conjectures
Disproof of the MacNeish Conjecture
Disproof of the Euler Conjecture
Orthogonal Latin Squares of Order n ≡ 2 (mod 4)
Affine and Projective Planes
Affine Planes
Projective Planes
Connections between Affine and Projective Planes
Connection between Affine Planes and Complete Sets of MOLS
Coordinating the Affine Plane
Intersections of Steiner Triple Systems
Teirlinck’s Algorithm
The General Intersection Problem
Embeddings
Embedding Latin Rectangles—Necessary Conditions
Edge-Coloring Bipartite Graphs
Embedding Latin Rectangles: Ryser’s Sufficient Conditions
Embedding Idempotent Commutative Latin Squares: Cruse’s Theorem
Embedding Partial Steiner Triple Systems
Steiner Quadruple Systems
Introduction
Constructions of Steiner Quadruple Systems
The Stern and Lenz Lemma
The (3v – 2u)-Construction
Appendix A: Cyclic Steiner Triple Systems
Appendix B: Answers to Selected Exercises
References
Index