TABLE OF CONTENTS:

CHAPTER 1

Introduction and preliminary definitions

CHAPTER 2

The existence of certain designs

2.1 BTDs from one-factorizations

2.2 BTDs with block size four

2.2.1 The case $\lambda=2$

2.2.2 The case $\lambda=3$

2.2.3 The case $\lambda=4$

2.2.4 The case $\lambda=5$

2.2.5 The case $\lambda=6$

2.2.6 The case $\lambda=7$

2.2.7 The case $\lambda\geqslant 8$

CHAPTER 3

The intersection problem for BTDs

3.1 BTDs with a hole

3.1.1 $\rho_2$ and $v$ are the same parity

3.1.2 $\rho_2$ and $v$ are not the same parity

3.2 The intersection problem for $(v,3,2)$ BIBDs

3.2.1 The case $v=6$

3.2.2 The case $v=7$

3.2.3 The case $v=9$

3.2.4 The case $v=10$

3.2.5 The case $v=12$

3.2.6 The case $v=13$

3.2.7 The case $v=15$

3.2.8 The case $v=16$

3.2.9 The case $v=18$

3.2.10 The case $v=19$

3.3 The intersection problem for $(v;3;3,2)$ BTDs

3.3.1 The case $v=7$

3.3.2 The case $v=9$

3.3.3 The case $v=10$

3.3.4 The case $v=12$

3.3.5 The case $v=13$

3.3.6 The case $v=15$

3.4 The intersection problem for $(v;4;3,2)$ BTDs

3.4.1 The case $v=9$

3.4.2 The case $v=12$

3.4.3 The case $v=15$

3.4.4 The case $v=18$

3.5 The intersection problem for $(v;3;3,3)$ BTDs

3.5.1 Construction of designs

3.5.2 Pairs of designs

3.5.3 The small cases

3.6 The intersection problem for $(v,2)$ DTSs

3.6.1 The case $v=4$

3.6.2 The case $v=6$

3.6.3 The case $v=7$

3.6.4 The case $v=10$

3.6.5 Conclusion

CHAPTER 4

The fine structure of BTDs

4.1 Introduction and definitions

4.2 Recursive construction

4.3 Small cases for recursion

CHAPTER 5

Various super-simple designs with block size four

5.1 Introduction and definitions

5.2 Completely reducible super-simple $(v,4,2)$ BIBDs

5.3 Super-simple $(v,4,2)$ BIBDs

5.4 Super-simple $(v;1;4,2)$ BTDs

5.5 Super-simple $(v;2;4,2)$ BTDs

5.6 Super-simple $(v,4,4)$ BIBDs

5.7 Nonexistence of super-simple GDD$(4,4,3;15)$

CHAPTER 6

On smallest covering sets for block designs

6.1 Introduction and preliminaries

6.2 The case $k=t+1$

6.3 Some small cases

CHAPTER 7

Smallest defining sets for the $36$ non-isomorphic two-fold triple systems of order nine

7.1 Introduction

7.2 Technique

APPENDICES

Appendix A: Some BTDs and frame-BTDs with block size four

Appendix B: 64 simple $(v,3,2)$ BIBDs

Appendix C: 15 simple $(v;\rho_2;3,2)$ BTDs; $\rho_2=3$, $4$

Appendix D: Decomposition of a certain design into 3-factors

Appendix E: 143 $(v;3;3,3)$ BTDs with different types

Appendix F: 27 3-factorizations of $3K_6$ with different types

Appendix G: 42 3-factorizations of a certain graph with different types

Appendix H: Four 3-factorizations of a certain graph with different types

Appendix I: 12 3-factorizations of a certain graph with different types

Appendix J: Some super-simple designs with small $v$

Appendix K: 24 super-simple BTDs and frame-BTDs

Appendix L: The 36 non-isomorphic $(9,3,2)$ BIBDs with their smallest defining sets

REFERENCES