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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.
Auburn University, Alabama, USA Monmouth University, Middletown, New Jersey, USA
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 lambda-Fold Triple Systems Triple Systems of Index lambda > 1 The Existence of Indempotent Latin Squares 2-fold Triple Systems lambda= 3 and 6 lambda-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
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- ID: 9781420082968
2nd Revised edition
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