Introduction to Combinatorial Designs, Second Edition

Introduction to Combinatorial Designs, Second Edition

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Features

  • Covers classical designs such as Latin squares, balanced incomplete block designs, and finite projective and affine planes
  • Introduces modern extensions of design theory, including one-factorizations, Room squares, tournament designs, and nested designs
  • Features applications in several areas, including cryptography, computer science, experimental design, and communications theory
  • Includes instructive examples and theorems with every topic
  • Provides exercises in each section, select answers in the back of the book, and more complete solutions on the author’s website
  • Contains references to classical literature to put results in a historical perspective
  • Summary

    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

    Table of Contents

    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