## Design Theory:This volume concludes the second edition of the standard text on design theory. Since the first edition there has been extensive development of the theory and this book has been thoroughly rewritten to reflect this. In particular, the growing importance of discrete mathematics to many parts of engineering and science have made designs a useful tool for applications, a fact that has been acknowledged here with the inclusion of an additional chapter on applications. The volume is suitable for advanced courses and for reference use, not only for researchers in discrete mathematics or finite algebra, but also for those working in computer and communications engineering and other mathematically oriented disciplines. Features include exercises and an extensive, updated bibliography of well over 1800 citations. |

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### Contents

Recursive constructions | 608 |

Difference families 468 | 611 |

2 Use of pairwise balanced designs | 617 |

3 Applications of divisible designs | 621 |

4 Applications of Hananis lemmas | 627 |

5 Block designs of block size three and four | 636 |

6 Solution of Kirkmans schoolgirl problem | 641 |

7 The basis of a closed set | 644 |

7 An existence theorem for resolvable block designs | 801 |

8 Some results for t 3 | 805 |

Characterisations of classical designs | 806 |

2 Characterisations of projective spaces | 808 |

3 Characterisation of affine spaces | 821 |

4 Locally projective linear spaces | 828 |

5 Good blocks | 833 |

6 Concluding remarks | 841 |

8 Block designs with block size five | 651 |

9 Divisible designs with small block sizes | 660 |

10 Steiner quadruple systems | 664 |

11 Embedding theorems for designs and partial designs | 673 |

12 Concluding remarks | 681 |

Transversal designs and nets | 690 |

2 Transversal designs with M 1 | 693 |

3 A construction of Wilson | 696 |

4 Six and more mutually orthogonal Latin squares | 703 |

5 The theorem of Chowla Erdös and Straus | 706 |

6 Further bounds for transversal designs and orthogonal arrays | 708 |

7 Completion theorems for Bruck nets | 713 |

8 Maximal nets with large deficiency | 725 |

9 Translation nets and maximal nets with small deficiency | 731 |

Examples and basic definitions 1 | 737 |

3 tdesigns Steiner systems and configurations 15 | 738 |

10 Completion results for u 1 | 749 |

11 Extending symmetric nets | 758 |

12 Complete mappings difference matrices and maximal nets | 761 |

13 Tarrys theorem | 772 |

14 Codes of Bruck nets | 778 |

Asymptotic existence theory | 781 |

2 The existence of Steiner systems with v in given residue classes | 783 |

3 The main theorem for Steiner systems S2 k v | 787 |

4 The eventual periodicity of closed sets | 790 |

5 The main theorem for X 1 | 793 |

6 The main theorem for X 1 | 796 |

Applications of designs | 852 |

2 Design of experiments | 856 |

3 Experiments with Latin squares and orthogonal arrays | 874 |

4 Application of designs in optics | 880 |

5 Codes and designs | 892 |

6 Discrete tomography | 926 |

7 Designs in data structures and computer algorithms | 930 |

8 Designs in hardware | 937 |

9 Difference sets rule matter and waves | 946 |

10 No waves no rules but security | 956 |

Appendix Tables | 971 |

2 Symmetric designs | 981 |

Further direct constructions 520 | 1001 |

Notation and symbols | 1005 |

1013 | |

1025 | |

1029 | |

Highly transitive groups 277 | 1037 |

1045 | |

1051 | |

1055 | |

1064 | |

1067 | |

1091 | |

1093 | |

### Common terms and phrases

1)-difference 2-design abelian affine design affine plane algebra algorithms applications assume automorphism group Beth block designs characterisation codeword coding theory Colbourn and Dinitz combinatorial consider construction contains Corollary corresponding cosets cyclic decoding defined Definition Delandtsheer denote design theory difference sets Discr disjoint equation error error-correcting codes exactly example existence Figure finite following result generalised geometry GF(q given graph group of order Hadamard designs Hadamard matrices Hadamard transform Hamming code Hanani Hence implies incidence matrix incidence structure intersect isomorphic Jungnickel Lemma linear spaces lines Math Mathon necessary conditions nets notation obtained orthogonal Latin squares parallel class parameters permutation point classes point set positive integer prime power problem projective plane projective space Proof Proposition proved quadruple Remark Section sequences Series Shrikhande Steiner systems Steiner triple systems subset subspace symmetric design Table Theorem transversal designs triple systems unique vector

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