- Develops a deep understanding of the methods through both formal descriptions of the constructions and the accompanying illustrations
- Focuses on construction methods to provide students with the expertise to produce nonstandard experimental designs when needed
- Includes important results in combinatorial designs, such as the existence of orthogonal Latin squares, balanced incomplete and pairwise balanced designs, affine and projective planes, and quadruple systems
- Contains extensive new material on embeddings, directed designs, universal algebraic representations of designs, and intersection properties of designs

**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.

**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 6*n *+ 5 Construction

Quasigroups with Holes and Steiner Triple Systems

The Wilson Construction

Cyclic Steiner Triple Systems

The 2*n* + 1 and 2*n *+ 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 (3*v* – 2*u*)-Construction

**Appendix A: Cyclic Steiner Triple Systems **

**Appendix B: Answers to Selected Exercises **

**References **

**Index**

…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