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Design Theory, Second Edition, 2nd Edition

Book Description

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.

Table of Contents

  1. 1 Steiner Triple Systems
  2. 1.1 The existence problem
  3. 1.2 The Bose Construction
  4. 1.3 The Skolem Construction
  5. 1.4 The 6n + 5 Construction + 5 Construction
  6. 1.5 Quasigroups with holes and Steiner triple systems : The 6n + 5 Construction + 5 Construction
  7. 1.5.1 Constructing quasigroups with holes
  8. 1.5.2 Constructing Steiner triple systems using quasigroupswith holes
  9. 1.6 The Wilson Construction
  10. 1.7 Cyclic Steiner triple systems
  11. 1.8 The 2n + 1 and 2 + 1 and 2n + 7 Constructions + 7 Constructions
  12. 2 λ-Fold Triple Systems λ-Fold Triple Systems
  13. 2.1 Triple systems of index λ > 1 Triple systems of index λ > 1
  14. 2.2 The existence of idempotent latin squares
  15. 2.3 2-Fold triple systems
  16. 2.3.1 Constructing2-foldtriple systems
  17. 2.4 Mendelsohn triple systems
  18. 2.5 λ = 3 and 6 λ = 3 and 6
  19. 2.6 λ-Fold triple systems in general λ-Fold triple systems in general
  20. 3 Quasigroup Identities and Graph Decompositions
  21. 3.1 Quasigroup identities
  22. 3.2 Mendelsohntriple systems revisited
  23. 3.3 Steiner triple systems revisited
  24. 4 Maximum Packings and Minimum Coverings
  25. 4.1 The general problem
  26. 4.2 Maximum packings
  27. 4.3 Minimum coverings
  28. 5 Kirkman Triple Systems
  29. 5.1 A recursive construction
  30. 5.2 Constructing pairwise balanced designs
  31. 6 Mutually Orthogonal Latin Squares
  32. 6.1 Introduction
  33. 6.2 The Euler and MacNeish Conjectures
  34. 6.3 Disproof of the MacNeish Conjecture
  35. 6.4 Disproof of the Euler Conjecture
  36. 6.5 Orthogonal latin squares of order n ≡ 2(mod 4) ≡ 2(mod 4)
  37. 7 Affine and Projective Planes
  38. 7.1 Affine planes
  39. 7.2 Projective planes
  40. 7.3 Connections between affine and projective planes
  41. 7.4 Connection between affine planes and complete sets of MOLS
  42. 7.5 Coordinatizing the affine plane
  43. 8 Intersections of Steiner Triple System
  44. 8.1 Teirlinck’s Algorithm Teirlinck’s Algorithm
  45. 8.2 Thegeneral intersectionproblem
  46. 9 Embeddings
  47. 9.1 Embedding latin rectangles‐necessary conditions Embedding latin rectangles‐necessary conditions
  48. 9.2 Edge‐coloring bipartite graphs Edge‐coloring bipartite graphs
  49. 9.3 Embedding latin rectangles: Ryser’s Sufficient Conditions Embedding latin rectangles: Ryser’s Sufficient Conditions
  50. 9.4 Embedding idempotent commutative latin squares: Cruse’s Theorem Embedding idempotent commutative latin squares: Cruse’s Theorem
  51. 9.5 Embedding partial Steiner triple systems
  52. 10 Steiler Quadruple Systems
  53. 10.1 Introduction
  54. 10.2 Constructions of Steiner Quadruple Systems
  55. 10.3 The Stern and Lenz Lemma
  56. 10.4 The (3v - 2u)‐Construction)‐Construction
  57. Appendices
  58. Appendices A: Cyclic Steiner Triple Systems
  59. Appendices B: Answers to Selected Exercises
  60. References