Symmetric designs are the principal objects of study in that vast area of constructive combinatorics called design theory. Combinatorics courses at the senior or beginning graduate level introduce such structures as difference sets, Latin squares, finite geometries, Hadamard matrices, and finally the subject that unifies them — block designs. The subject is full of beautiful little examples that come from geometry, number theory, group theory, coding theory, and linear algebra, such as the Fano plane, Kirkman’s Schoolgirl Problem, Hamming codes, and Euler’s Thirty-Six Officers. Students are captivated by the accessibility and puzzle-like playful aspect of these designs, which are merely special families of subsets of a finite set, arranged in interesting ways.
What lured me in was a combination of the examples and the language: Perfect Difference Sets, Mutually Orthogonal Latin Squares, Doubly Regular Round Robin Tournaments, Tactical Configurations — and above all, Symmetric Balanced Incomplete Block Designs. Marvelous words, aren’t they? When I learned what they were, I was hooked, and thus began a career-long fascination with these beautiful combinatorial objects, so full of symmetry and full of connections to so many other areas of mathematics. This brings us to the book under review, Ionin and Shrikhande’s Combinatorics of Symmetric Designs , which is a research monograph whose treatment of the title topic is both broad and deep, and which describes the most recent research developments in this area. But it is more than just a research monograph; more about that later.
Here is what this fine book is all about. A family of k-element subsets (called blocks) of a v-element set V such that each pair of distinct elements of V appears together in exactly λ of the blocks is called a (v,k, λ)-design. Such a design is symmetric if each element of V appears in exactly k blocks, and that there are exactly v blocks in all. Combinatorics of Symmetric Designs begins with an introduction to symmetric designs and the combinatorics of finite sets. Following this, the book proceeds to topics that might be touched on in some elementary combinatorics courses, such as finite geometries, Latin squares, linear codes, Hadamard matrices, and difference sets. Finally, the authors treat more advanced topics such as t-designs, strongly regular graphs, BGW matrices, decomposable designs and Ryser designs. In short, this book clearly belongs on the shelf of any serious researcher in the area of combinatorial designs.
I said that the book is more than a research monograph: in many places, it is fun to read! Like most such endeavors, Combinatorics of Symmetric Designs includes proofs of many standard theorems, such as the Bruck-Ryser-Chowla Theorem, Fisher’s Inequality and the Ryser-Woodall Theorem. It includes a large number of detailed worked-through examples. These range from the familiar Fano plane and Hamming codes to the (56, 11, 2) and (79, 13, 2) symmetric designs constructed using nontrivial properties of certain finite groups. I especially enjoyed reading through the examples and marveling at the ingenuity of the researchers who first constructed them.
Some theorems are stated but not proved, the proofs being beyond the scope of the book. (An indication about the tone of the book can be found in the authors’ characterization of a proof of Ray-Chaudhury and Wilson’s solution of the Kirkman Design Problem as being “far beyond the scope of this book!”) Unlike most research monographs, however, every chapter ends with exercises ranging from the routine to the very challenging, and every chapter ends with notes on sources, history, and references to the chapter’s topics. The exercises and notes are among the most attractive aspect of the book.
The authors state that the book could also be used as a text for a course in combinatorial designs, and provide a list of sections that would form such a course. The chapters and topics are well organized, the writing is clear, and the exercises are appropriate. The proofs are thorough, albeit occasionally so complex that one may be lost in a mass of detail; without proper background, the casual reader may find the going somewhat tough. As for the necessary background, the authors state that a standard course in linear algebra and the basics of combinatorics and modern algebra should be sufficient prerequisites for such a proposed course in designs. Such a course might be demanding, but if liberally supplemented with plenty of elementary examples, I think it could be done — and someday, I’d like to try.
Ezra Brown (firstname.lastname@example.org) is Alumni Distinguished Professor of Mathematics at Virginia Tech, with degrees from Rice and LSU. He is a number theorist by trade, but his first publication was about tournaments and Hadamard matrices. He is a fairly regular contributor to the MAA journals. He sings (everything from blues to opera), plays a tolerable jazz piano, and his wife Jo is teaching him to be a gardener. He occasionally bakes biscuits for his students.
1. Combinatorics of finite sets; 2. Introduction to designs; 3. Vector spaces over finite fields; 4. Hadamard matrices; 5. Resolvable designs; 6. Symmetric designs and t-designs; 7. Symmetric designs and regular graphs; 8. Block intersection structure of designs; 9. Difference sets; 10. Balanced generalized weighing matrices; 11. Decomposable symmetric designs; 12. Subdesigns of symmetric designs; 13. Non-embeddable quasi-residual designs; 14. Ryser designs; Appendix; Bibliography; Index.