设计理论

Preface

1.BIBDs

1.1 Definition and Fundamental Properties of BIBDs

1.2 Isomorphisms and Automorphisms

1.3 Constructions of New BIBDs from Old Ones

1.4 Exercises

2.Symmetric BIBDs

2.1 Definition and Fundamental Properties

2.2 Bruck-Ryser-Chowla Theorem

2.3 Finite Projective Planes as Symmetric BIBDs

2.4 Difference Sets and Symmetric BIBDs

2.5 Hadamard Matrices and Symmetric BIBDs

2.6 Derived and Residual BIBDs

2.7 Exercises

3.Resolvable BIBDs

3.1 Definitions and Examples

3.2 Finite Affine Planes

3.3 Properties of Resolvable BIBDs

3.4 Exercises

4.Orthogonal Latin Squares

4.1 Orthogonal Latin Squares

4.2 Mutually Orthogonal Latin Squares

4.3 Singular Direct Product of Latin Squares

4.4 Sum Composition of Latin Squares

4.5 Orthogonal Arrays

4.6 Transversal Designs

4.7 Exercises

5.Pairwise Balanced Designs；Group Divisible Designs

5.1 Pairwise Balanced Designs

5.2 Group Divisible Designs

5.3 Closedness of Some Sets of Positive Integers

5.4 Exercises

6.Construction of Some Families of BIBDs

6.1 Steiner Triple Systems

6.2 Cyclic Steiner Triple Systems

6.3 Kirkman Triple Systems

6.4 Triple Systems

6.5 Biplanes

6.6 Exercises

7.t-Designs

7.1 Definition and Fundamental Properties of t-Designs

7.2 Restriction and Extension

7.3 Extendable SBIBDs and Hadamard 3-Designs

7.4 Finite Inversive Planes

7.5 Exercises

8.Steiner Systems

8.1 Steiner Systems

8.2 Some Designs from Hadamard 2-Designs and 3-Designs

8.3 Steiner Systems S(4；11，5) and S(5；12，6)

8.4 Binary Codes

8.5 Binary Golay Codes and Steiner Systems S(4；23，7) and S(5；24,8)

8.6 Exercises

9.Association Schemes and PBIBDs

9.1 Association Schemes

9.2 PBIBDs

9.3 Association Schemes (Continued)

9.4 Exercises

References

This book deals with the basic subjects of design theory.It begins with balanced incomplete block designs，various constructions of which are described in ample detail.In particular，finite projective and affine planes,difference sets and Hadamard matrices，as tools to construct balanced incomplete block designs，are included.Orthogonal latin squares are also treated in detail.Zhu's simpler proof of the falsity of Euler's conjecture is included.The construction of some classes of balanced incomplete block designs，such as Steiner triple systems and Kirkman triple systems，are also given.
T-designs and partially balanced incomplete block designs (together with association schemes)，as generalizations of balanced incomplete block designs，are included.Some coding theory related to Steiner triple systems are clearly explained.
The book is written in a lucid style and is algebraic in nature.It can be used as a text or a reference book for graduate students and researchers in combinatorics and applied mathematics.It is also suitable for self-study.

《设计理论》由高等教育出版社与新加坡世界科技出版社（WSP）合作出版，全球发行。
设计理论是组合数学的一个重要分支，《设计理论》是根据作者在南开大学组合中心为研究生讲课的讲义，润色、补充而成，是一本设计理论的引论性书籍，涵盖最基本的古典设计理论。内容包括：Symmetric BIBDs、Resolvable BIBDs、Orthogonal Latin Squares等。
《设计理论》适合组合数学、计算机科学等相关专业的学生和教师使用参考。