Introduction to Combinatorial Designs, Second Edition

This new edition presents a comprehensive look at combinatorial designs. It covers classical designs such as Latin squares, balanced incomplete block designs, and finite projective and affine planes as well as more contemporary designs that include one-factorizations, Room squares, tournament designs, and nested designs. The book features applications in cryptography, computer science, experimental design, communications theory, and more. With every topic, it includes instructive examples and theorems. The text also provides exercises in each section, select answers in the back of the book, and more complete solutions on the author's website.

Combinatorial theory is one of the fastest growing areas of modern mathematics. Focusing on a major part of this subject, Introduction to Combinatorial Designs, Second Edition provides a solid foundation in the classical areas of design theory as well as in more contemporary designs based on applications in a variety of fields. After an overview of basic concepts, the text introduces balanced designs and finite geometries. The author then delves into balanced incomplete block designs, covering difference methods, residual and derived designs, and resolvability. Following a chapter on the existence theorem of Bruck, Ryser, and Chowla, the book discusses Latin squares, one-factorizations, triple systems, Hadamard matrices, and Room squares. It concludes with a number of statistical applications of designs. Reflecting recent results in design theory and outlining several applications, this new edition of a standard text presents a comprehensive look at the combinatorial theory of experimental design. Suitable for a one-semester course or for self-study, it will prepare readers for further exploration in the field. To access supplemental materials for this volume, visit the author’s website at http://www.math.siu.edu/Wallis/designs

Basic Concepts Combinatorial Designs Some Examples of Designs Block Designs Systems of Distinct Representatives Balanced Designs Pairwise Balanced Designs Balanced Incomplete Block Designs Another Proof of Fisher’s Inequality t-Designs Finite Geometries Finite Affine Planes Finite Fields Construction of Finite Affine Geometries Finite Projective Geometries Some Properties of Finite Geometries Ovals in Projective Planes The Desargues Configuration Difference Sets and Difference Methods Difference Sets Construction of Difference Sets Properties of Difference Sets General Difference Methods Singer Difference Sets More about Block Designs Residual and Derived Designs Resolvability The Main Existence Theorem Sums of Squares The Bruck–Ryser–Chowla Theorem Another Proof Latin Squares Latin Squares and Subsquares Orthogonality Idempotent Latin Squares Transversal Designs More about Orthogonality Spouse-Avoiding Mixed Doubles Tournaments Three Orthogonal Latin Squares Bachelor Squares One-Factorizations Basic Ideas The Variability of One-Factorizations Starters Applications of One-Factorizations An Application to Finite Projective Planes Tournament Applications of One-Factorizations Tournaments Balanced for Carryover Steiner Triple Systems Construction of Triple Systems Subsystems Simple Triple Systems Cyclic Triple Systems Large Sets and Related Designs Kirkman Triple Systems and Generalizations Kirkman Triple Systems Kirkman Packings and Coverings Hadamard Matrices Basic Ideas Hadamard Matrices and Block Designs Further Hadamard Matrix Constructions Regular Hadamard Matrices Equivalence Room Squares Definitions Starter Constructions Subsquare Constructions The Existence Theorem Howell Rotations Further Applications of Design Theory Statistical Applications Information and Cryptography Golf Designs References ANSWERS AND SOLUTIONS INDEX