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Algebraic design theory. (English) Zbl 1235.05001
Mathematical Surveys and Monographs 175. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4496-0/hbk). xi, 298 p. (2011).
This monograph is concerned with algebraic questions, objects and applications in combinatorial design theory. In particular, it develops a general theory for cocyclic pairwise combinatorial designs, with a focus on cocyclic Hadamard matrices. The algebraic themes include the applications of matrix algebra in design theory, the automorphism group and its regular subgroups, the composition of smaller designs to construct larger designs, and the connection between designs with regular group actions and solutions to group ring equations. The aspect of orthogonality is central in the authors’ unified approach. Necessary background material is provided in introductory chapters as well as many examples of classical and non-classical pairwise combinatorial designs that are of interest in diverse areas such as finite geometry, statistics, and electrical engineering. Chapters 3-16 and Chapter 20 develop the general abstract framework for pairwise combinatorial designs. Chapters 2, 17-19, 21-23 focus on practical case studies.
The comprehensive book is accessible and well written. It offers a valuable source to researchers of all levels, including undergraduate students, in the fields of algebra, matrix algebra, computational algebra, and combinatorics.

05-02 Research exposition (monographs, survey articles) pertaining to combinatorics
05Bxx Designs and configurations
05E18 Group actions on combinatorial structures
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures