Book description
Through both formal descriptions and many illustrations, this text presents some of the most important techniques used for constructing combinatorial designs. This edition contains extensive new material on embeddings, directed designs, universal algebraic representations of designs, and intersection properties of designs. It focuses on construction methods to provide readers with the expertise to produce nonstandard experimental designs when needed. The text also 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.
Table of contents
- 1 Steiner Triple Systems
- 1.1 The existence problem
- 1.2 The Bose Construction
- 1.3 The Skolem Construction
- 1.4 The 6n + 5 Construction
- 1.5 Quasigroups with holes and Steiner triple systems : The 6n + 5 Construction
- 1.5.1 Constructing quasigroups with holes
- 1.5.2 Constructing Steiner triple systems using quasigroupswith holes
- 1.6 The Wilson Construction
- 1.7 Cyclic Steiner triple systems
- 1.8 The 2n + 1 and 2n + 7 Constructions
- 2 λ-Fold Triple Systems
- 2.1 Triple systems of index λ > 1 Triple systems of index λ > 1
- 2.2 The existence of idempotent latin squares
- 2.3 2-Fold triple systems
- 2.3.1 Constructing2-foldtriple systems
- 2.4 Mendelsohn triple systems
- 2.5 λ = 3 and 6
- 2.6 λ-Fold triple systems in general
- 3 Quasigroup Identities and Graph Decompositions
- 3.1 Quasigroup identities
- 3.2 Mendelsohntriple systems revisited
- 3.3 Steiner triple systems revisited
- 4 Maximum Packings and Minimum Coverings
- 4.1 The general problem
- 4.2 Maximum packings
- 4.3 Minimum coverings
- 5 Kirkman Triple Systems
- 5.1 A recursive construction
- 5.2 Constructing pairwise balanced designs
- 6 Mutually Orthogonal Latin Squares
- 6.1 Introduction
- 6.2 The Euler and MacNeish Conjectures
- 6.3 Disproof of the MacNeish Conjecture
- 6.4 Disproof of the Euler Conjecture
- 6.5 Orthogonal latin squares of order n ≡ 2(mod 4)
- 7 Affine and Projective Planes
- 7.1 Affine planes
- 7.2 Projective planes
- 7.3 Connections between affine and projective planes
- 7.4 Connection between affine planes and complete sets of MOLS
- 7.5 Coordinatizing the affine plane
- 8 Intersections of Steiner Triple System
- 8.1 Teirlinck’s Algorithm
- 8.2 Thegeneral intersectionproblem
- 9 Embeddings
- 9.1 Embedding latin rectangles‐necessary conditions
- 9.2 Edge‐coloring bipartite graphs
- 9.3 Embedding latin rectangles: Ryser’s Sufficient Conditions
- 9.4 Embedding idempotent commutative latin squares: Cruse’s Theorem
- 9.5 Embedding partial Steiner triple systems
- 10 Steiler Quadruple Systems
- 10.1 Introduction
- 10.2 Constructions of Steiner Quadruple Systems
- 10.3 The Stern and Lenz Lemma
- 10.4 The (3v - 2u)‐Construction
- Appendices
- Appendices A: Cyclic Steiner Triple Systems
- Appendices B: Answers to Selected Exercises
- References
Product information
- Title: Design Theory, 2nd Edition
- Author(s):
- Release date: March 2017
- Publisher(s): Chapman and Hall/CRC
- ISBN: 9781351606455
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