## Handbook of Combinatorial DesignsContinuing 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 Edition Meeting 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. |

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### Contents

3 | |

11 | |

25 | |

2 Triple Systems | 58 |

3 BIBDs with Small Block Size | 72 |

4 tDesigns with t 3 | 79 |

5 Steiner Systems | 102 |

6 Symmetric Designs | 110 |

26 Graphical Designs | 490 |

27 Grooming | 494 |

28 Hall Triple Systems | 496 |

29 Howell Designs | 499 |

30 Inﬁnite Designs | 504 |

Geometric Aspects | 506 |

32 Lotto Designs | 512 |

33 Low Density Parity Check Codes | 519 |

7 Resolvable and NearResolvable Designs | 124 |

1 Latin Squares | 135 |

2 Quasigroups | 152 |

3 Mutually Orthogonal Latin Squares MOLS | 160 |

4 Incomplete MOLS | 193 |

5 SelfOrthogonal Latin Squares SOLS | 211 |

6 Orthogonal Arrays of Index More Than One | 219 |

7 Orthogonal Arrays of Strength More Than Two | 224 |

1 PBDs and GDDs The Basics | 231 |

Recursive Constructions | 236 |

3 PBDClosure | 247 |

4 Group Divisible Designs | 255 |

5 PBDs Frames and Resolvability | 261 |

6 Pairwise Balanced Designs as Linear Spaces | 266 |

1 Hadamard Matrices and Hadamard Designs | 273 |

2 Orthogonal Designs | 280 |

3 DOptimal Matrices | 296 |

4 Bhaskar Rao Designs | 299 |

5 Generalized Hadamard Matrices | 301 |

6 Balanced Generalized Weighing Matrices and Conference Matrices | 306 |

7 Sequence Correlation | 313 |

8 Complementary Base and Turyn Sequences | 317 |

9 Optical Orthogonal Codes | 321 |

1 Association Schemes | 325 |

2 Balanced Ternary Designs | 330 |

3 Balanced Tournament Designs | 333 |

4 Bent Functions | 337 |

5 BlockTransitive Designs | 339 |

6 Complete Mappings and Sequencings of Finite Groups | 345 |

7 Configurations | 353 |

8 CorrelationImmune and Resilient Functions | 355 |

9 Costas Arrays | 357 |

10 Covering Arrays | 361 |

11 Coverings | 365 |

12 Cycle Decompositions | 373 |

13 Deﬁning Sets | 382 |

14 DeletionCorrecting Codes | 385 |

15 Derandomization | 389 |

16 Difference Families | 392 |

17 Difference Matrices | 411 |

18 Diﬀerence Sets | 419 |

19 Diﬀerence Triangle Sets | 436 |

20 Directed Designs | 441 |

21 Factorial Designs | 445 |

22 Frequency Squares and Hypercubes | 465 |

23 Generalized Quadrangles | 472 |

24 Graph Decompositions | 477 |

25 Graph Embeddings and Designs | 486 |

34 Magic Squares | 524 |

35 Mendelsohn Designs | 528 |

36 Nested Designs | 535 |

Comparing Block Designs | 540 |

38 Ordered Designs Perpendicular Arrays and Permutation Sets | 543 |

39 Orthogonal Main Effect Plans | 547 |

40 Packings | 550 |

41 Partial Geometries | 557 |

42 Partially Balanced Incomplete Block Designs | 562 |

43 Perfect Hash Families | 566 |

44 Permutation Codes and Arrays | 568 |

45 Permutation Polynomials | 572 |

46 Pooling Designs | 574 |

47 Quasi3 Designs | 576 |

48 QuasiSymmetric Designs | 578 |

49 r λdesigns | 582 |

50 Room Squares | 584 |

51 Scheduling a Tournament | 591 |

52 Secrecy and Authentication Codes | 606 |

53 Skolem and Langford Sequences | 612 |

54 Spherical Designs | 617 |

55 Starters | 622 |

56 Superimposed Codes and Combinatorial Group Testing | 629 |

57 Supersimple Designs | 633 |

58 Threshold and Ramp Schemes | 635 |

59 tmsNets | 639 |

60 Trades | 644 |

61 Turán Systems | 649 |

62 Tuscan Squares | 652 |

63 tWise Balanced Designs | 657 |

64 Whist Tournaments | 663 |

65 Youden Squares and Generalized Youden Designs | 668 |

1 Codes | 677 |

2 Finite Geometry | 702 |

3 Divisible Semiplanes | 729 |

4 Graphs and Multigraphs | 731 |

5 Factorizations of Graphs | 740 |

6 Computational Methods in Design Theory | 755 |

7 Linear Algebra and Designs | 783 |

8 Number Theory and Finite Fields | 791 |

9 Finite Groups and Designs | 819 |

10 Designs and Matroids | 847 |

11 Strongly Regular Graphs | 852 |

12 Directed Strongly Regular Graphs | 868 |

13 TwoGraphs | 875 |

Bibliography | 883 |

967 | |

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### Common terms and phrases

1-factorizations 2-graph aﬃne algorithm automorphism group Base blocks BIBD binary block design column combinatorial designs conference matrix conﬁguration Conjecture Construction contains coset Costas array cyclic deﬁned Deﬁnitions denoted difference family diﬀerence sets disjoint eﬀects elements equivalent exactly Example exists factor ﬁeld ﬁnite ﬁrst given group G group of order Hadamard matrices Hadamard matrix incidence matrix inﬁnite intersection isomorphic known Let G linear space modulo MOLS necessary conditions nonisomorphic obtained orbit orthogonal array pair pairwise parallel classes parameters partial geometry partition PBDs permutation plane of order points polynomial positive integers prime power projective plane quasi-symmetric quasigroup References Cited Remark resolvable Room square satisﬁes square of order square of side starter Steiner system Steiner triple system strongly regular graph subgroup subset symbols symmetric design t-design Table Theorem Theorem Let two-graph vector vertex vertices