## Combinatorics of Symmetric DesignsProviding a unified exposition of the theory of symmetric designs with emphasis on recent developments, this volume covers the combinatorial aspects of the theory, giving particular attention to the construction of symmetric designs and related objects. The last five chapters are devoted to balanced generalized weighing matrices, decomposable symmetric designs, subdesigns of symmetric designs, non-embeddable quasi-residual designs, and Ryser designs. The book concludes with a comprehensive bibliography of over 400 entries. Detailed proofs and a large number of exercises make it suitable as a text for an advanced course in combinatorial designs. |

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### Contents

Exercises ll | 11 |

Exercises | 53 |

Hadamard matrices | 114 |

Resolvable designs | 154 |

Symmetric designs anddesigns | 186 |

Symmetric designs and regular graphs | 212 |

Block intersection structure of designs | 247 |

Difference sets | 289 |

Decomposable symmetric designs | 368 |

Subdesigns of symmetric designs | 407 |

Nonembeddable quasiresidual designs | 429 |

Ryser designs | 447 |

Appendix | 488 |

495 | |

515 | |

Balanced generalized weighing matrices | 323 |

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### Common terms and phrases

2-design 2-subsets abelian group adjacency matrix affine plane block matrix cardinality columns complement conference matrix constructed Corollary cyclic group define Definition denote design of index design with parameters diagonal difference sets disjoint distinct blocks eigenvalues element entries equal equation exactly exists a symmetric GF(q group G group of order group of symmetries group ring Hadamard 3-design implies incidence matrix incidence structure infinite families intersection numbers isomorphic Kronecker product Latin square Lemma Let G Let H Let q matrix of order non-embeddable nonzero number of blocks obtain order q parallel class partition permutation plane of order point set polynomial positive integer prime power projective plane Proof Proposition Prove quasi-residual design quasi-symmetric regular Hadamard matrix replication number resolution class Ryser design S. S. Shrikhande satisfying strongly regular graphs subdesign subset subspaces Suppose symmetric design Theorem type-l vector space vertices X)-design yields