Combinatorics of Symmetric Designs
The aim of this book is to provide a unified exposition of the theory of symmetric designs with emphasis on recent developments. The authors cover the combinatorial aspects of the theory giving particular attention to the construction of symmetric designs and related objects. The last five chapters of the book are devoted to balanced generalized weighing matrices, decomposable symmetric designs, subdesigns of symmetric designs, non-embeddable quasi-residual designs, and Ryser designs. Most results in these chapters have never previously appeared in book form. The book concludes with a comprehensive bibliography of over 400 entries. Researchers in all areas of combinatorial designs, including coding theory and finite geometries, will find much of interest here. Detailed proofs and a large number of exercises make this book suitable as a text for an advanced course in combinatorial designs.
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2-design 2-subsets abelian group adjacency matrix afﬁne plane block matrix cardinality columns complement conference matrix constructed Corollary cyclic group deﬁned Deﬁnition denote design of index design with parameters difference sets disjoint distinct blocks eigenvalues element entries equal equation exactly exists a symmetric ﬁeld ﬁnite ﬁrst GF(q group G group of order group of symmetries group ring Hadamard 3-design implies incidence matrix incidence structure 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 classes partition permutation plane of order point set polynomial positive integer prime power projective plane Proof Proposition Prove quasi-symmetric regular Hadamard matrix replication number resolution class Ryser design S. S. Shrikhande Shrikhande strongly regular graphs subdesign subset subspaces Suppose symmetric design Theorem vector space vertices X)-design yields