Continuing in the bestselling, informative tradition of the first edition, the Handbook of Combinatorial Designs, Second Edition remains the only resource to contain all of the most important results and tables in the field of combinatorial design. This handbook covers the constructions, properties, and applications of designs as well as existence results.Over 30% longer than the first edition, the book builds upon the groundwork of its predecessor while retaining the original contributors' expertise. The first part contains a brief introduction and history of the subject. The following parts focus on four main classes of combinatorial designs: balanced incomplete block designs, orthogonal arrays and Latin squares, pairwise balanced designs, and Hadamard and orthogonal designs. Closely connected to the preceding sections, the next part surveys 65 additional classes of designs, such as balanced ternary, factorial, graphical, Howell, quasi-symmetric, and spherical. The final part presents mathematical and computational background related to design theory.New to the Second EditionAn introductory part that provides a general overview and a historical perspective of the areaNew chapters on the history of design theory, various codes, bent functions, and numerous types of designsFully updated tables, including BIBDs, MOLS, PBDs, and Hadamard matricesNearly 2,200 references in a single bibliographic sectionMeeting the need for up-to-date and accessible tabular and reference information, this handbook provides the tools to understand combinatorial design theory and applications that span the entire discipline. The author maintains a website with more information.
Author(s): Charles J. Colbourn, Jeffrey H. Dinitz
Series: Discrete Mathematics and Its Applications
Edition: 2
Publisher: Chapman and Hall/CRC
Year: 2006
Language: English
Pages: 1016
Cover......Page 1
Title......Page 6
Copyright......Page 7
Preface......Page 8
Contributors......Page 14
Contents......Page 20
Part I: Introduction......Page 28
1 Opening the Door......Page 30
2 Design Theory: Antiquity to 1950......Page 38
Part II: Block Designs......Page 50
1 2-(v, k, Lambda) Designs of Small Order......Page 52
2 Triple Systems......Page 85
3 BIBDs with Small Block Size......Page 99
4 t-Designs with t Greater-Than Or Equal To 3......Page 106
5 Steiner Systems......Page 129
6 Symmetric Designs......Page 137
7 Resolvable and Near-Resolvable Designs......Page 151
Part III: Latin Squares......Page 160
1 Latin Squares......Page 162
2 Quasigroups......Page 179
3 Mutually Orthogonal Latin Squares (MOLS)......Page 187
4 Incomplete MOLS......Page 220
5 Self-Orthogonal Latin Squares (SOLS)......Page 238
6 Orthogonal Arrays of Index More Than One......Page 246
7 Orthogonal Arrays of Strength More Than Two......Page 251
Part IV: Pairwise Balanced Designs......Page 256
1 PBDs and GDDs: The Basics......Page 258
2 PBDs: Recursive Constructions......Page 263
3 PBD-Closure......Page 274
4 Group Divisible Designs......Page 282
5 PBDs, Frames, and Resolvability......Page 288
6 Pairwise Balanced Designs as Linear Spaces......Page 293
Part V: Hadamard Matrices and Related Designs......Page 298
1 Hadamard Matrices and Hadamard Designs......Page 300
2 Orthogonal Designs......Page 307
3 D-Optimal Matrices......Page 323
4 Bhaskar Rao Designs......Page 326
5 Generalized Hadamard Matrices......Page 328
6 Balanced Generalized Weighing Matrices and Conference Matrices......Page 333
7 Sequence Correlation......Page 340
8 Complementary, Base, and Turyn Sequences......Page 344
9 Optical Orthogonal Codes......Page 348
Part VI: Other Combinatorial Designs......Page 350
1 Association Schemes......Page 352
2 Balanced Ternary Designs......Page 357
3 Balanced Tournament Designs......Page 360
4 Bent Functions......Page 364
5 Block-Transitive Designs......Page 366
6 Complete Mappings and Sequencings of Finite Groups......Page 372
7 Configurations......Page 380
8 Correlation-immune and Resilient Functions......Page 382
9 Costas Arrays......Page 384
10 Covering Arrays......Page 388
11 Coverings......Page 392
12 Cycle Decompositions......Page 400
13 Defining Sets......Page 409
14 Deletion-correcting Codes......Page 412
15 Derandomization......Page 416
16 Difference Families......Page 419
17 Difference Matrices......Page 438
18 Difference Sets......Page 446
19 Difference Triangle Sets......Page 463
20 Directed Designs......Page 468
21 Factorial Designs......Page 472
22 Frequency Squares and Hypercubes......Page 492
23 Generalized Quadrangles......Page 499
24 Graph Decompositions......Page 504
25 Graph Embeddings and Designs......Page 513
26 Graphical Designs......Page 517
27 Grooming......Page 521
28 Hall Triple Systems......Page 523
29 Howell Designs......Page 526
30 Infinite Designs......Page 531
31 Linear Spaces: Geometric Aspects......Page 533
32 Lotto Designs......Page 539
33 Low Density Parity Check Codes......Page 546
34 Magic Squares......Page 551
35 Mendelsohn Designs......Page 555
36 Nested Designs......Page 562
37 Optimality and Effciency: Comparing Block Designs......Page 567
38 Ordered Designs, Perpendicular Arrays, and Permutation Sets......Page 570
39 Orthogonal Main Effect Plans......Page 574
40 Packings......Page 577
41 Partial Geometries......Page 584
42 Partially Balanced Incomplete Block Designs......Page 589
43 Perfect Hash Families......Page 593
44 Permutation Codes and Arrays......Page 595
45 Permutation Polynomials......Page 599
46 Pooling Designs......Page 601
47 Quasi-3 Designs......Page 603
48 Quasi-Symmetric Designs......Page 605
50 Room Squares......Page 611
51 Scheduling a Tournament......Page 618
52 Secrecy and Authentication Codes......Page 633
53 Skolem and Langford Sequences......Page 639
54 Spherical Designs......Page 644
55 Starters......Page 649
56 Superimposed Codes and Combinatorial Group Testing......Page 656
57 Supersimple Designs......Page 660
58 Threshold and Ramp Schemes......Page 662
59 (t,m,s)-Nets......Page 666
60 Trades......Page 671
61 Turán Systems......Page 676
62 Tuscan Squares......Page 679
63 t-Wise Balanced Designs......Page 684
64 Whist Tournaments......Page 690
65 Youden Squares and Generalized Youden Designs......Page 695
Part VII: Related Mathematics......Page 702
1 Codes......Page 704
2 Finite Geometry......Page 729
3 Divisible Semiplanes......Page 756
4 Graphs and Multigraphs......Page 758
5 Factorizations of Graphs......Page 767
6 Computational Methods in Design Theory......Page 782
7 Linear Algebra and Designs......Page 810
8 Number Theory and Finite Fields......Page 818
9 Finite Groups and Designs......Page 846
10 Designs and Matroids......Page 874
11 Strongly Regular Graphs......Page 879
12 Directed Strongly Regular Graphs......Page 895
13 Two-Graphs......Page 902
Bibliography......Page 910
Index......Page 994
