Hardcover ISBN:  9780821844960 
Product Code:  SURV/175 
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Electronic ISBN:  9781470414023 
Product Code:  SURV/175.E 
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Book DetailsMathematical Surveys and MonographsVolume: 175; 2011; 298 ppMSC: Primary 05; 16; 20; Secondary 15;
Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs—new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.
ReadershipGraduate students and research mathematicians interested in algebra or design theory.

Table of Contents

Chapters

1. Overview

2. Many kinds of pairwise combinatorial designs

3. A primer for algebraic design theory

4. Orthogonality

5. Modeling $\Lambda $equivalence

6. The Grammian

7. Transposability

8. New designs from old

9. Automorphism groups

10. Group development and regular actions on arrays

11. Origins of cocyclic development

12. Group extensions and cocycles

13. Cocyclic pairwise combinatorial designs

14. Centrally regular actions

15. Cocyclic associates

16. Special classes of cocyclic designs

17. The Paley matrices

18. A large family of cocyclic Hadamard matrices

19. Substitution schemes for cocyclic Hadamard matrices

20. Calculating cocyclic development rules

21. Cocyclic Hadamard matrices indexed by elementary abelian groups

22. Cocyclic concordant systems of orthogonal designs

23. Asymptotic existence of cocyclic Hadamard matrices


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Combinatorial design theory is a source of simply stated, concrete, yet difficult discrete problems, with the Hadamard conjecture being a prime example. It has become clear that many of these problems are essentially algebraic in nature. This book provides a unified vision of the algebraic themes which have developed so far in design theory. These include the applications in design theory of matrix algebra, the automorphism group and its regular subgroups, the composition of smaller designs to make larger designs, and the connection between designs with regular group actions and solutions to group ring equations. Everything is explained at an elementary level in terms of orthogonality sets and pairwise combinatorial designs—new and simple combinatorial notions which cover many of the commonly studied designs. Particular attention is paid to how the main themes apply in the important new context of cocyclic development. Indeed, this book contains a comprehensive account of cocyclic Hadamard matrices. The book was written to inspire researchers, ranging from the expert to the beginning student, in algebra or design theory, to investigate the fundamental algebraic problems posed by combinatorial design theory.
Graduate students and research mathematicians interested in algebra or design theory.

Chapters

1. Overview

2. Many kinds of pairwise combinatorial designs

3. A primer for algebraic design theory

4. Orthogonality

5. Modeling $\Lambda $equivalence

6. The Grammian

7. Transposability

8. New designs from old

9. Automorphism groups

10. Group development and regular actions on arrays

11. Origins of cocyclic development

12. Group extensions and cocycles

13. Cocyclic pairwise combinatorial designs

14. Centrally regular actions

15. Cocyclic associates

16. Special classes of cocyclic designs

17. The Paley matrices

18. A large family of cocyclic Hadamard matrices

19. Substitution schemes for cocyclic Hadamard matrices

20. Calculating cocyclic development rules

21. Cocyclic Hadamard matrices indexed by elementary abelian groups

22. Cocyclic concordant systems of orthogonal designs

23. Asymptotic existence of cocyclic Hadamard matrices