Existence and Construction of Difference Families and Their Applications to Combinatorial Codes in Multiple-Access Communications

My interest in combinatorics began six years ago when I was studying in the laboratory of Professor Sanpei Kageyama at Hiroshima University as an undergraduate student. I started my study for combinatorics with reading the famous book " Introduction to Combinatorial Theory " written by R.C. Bose and B. Manvel. After one and a half years, I decided to enter Nagoya University and join the laboratory of Professor Masakazu Jimbo in order to study more deeply. I was inspired not only by his expert knowledge for combinatorial designs and their applications but also by his personality. This thesis presents results from my four and a half years research on difference families and combinatorial codes at the This thesis consists of six chapters. In the first chapter, a general introduction to combina-torial design theory, particularly, difference families and their application to combinatorial codes, and a survey of current new results are given. The next four chapters contain the au-thor's results in [88, 89, 90, 91, 92]. These chapters provide various new results on difference families including several known results via purely combinatorial techniques, the theory of finite fields, and number theory. Some of these papers are written jointly with other mathematicians , who are Professor Masakazu Jimbo In the final chapter, some concluding remarks and open problems related to results in this thesis are provided. This thesis demands of a little preparatory knowledge in combinatorial design theory but most concepts are thoroughly defined in each chapter of this thesis or self-explanatory. I would like to express my deepest appreciation to my advisor Professor Masakazu Jimbo. He has supervised my study for five and a half years and his continuous encouragement, patience, and excellent guidance had led me to successfully complete this thesis. I am most grateful to his invitation me to the academic world. I would also like to thank Professor Sanpei Kageyama. He had introduced me to the field of combinatorics in my undergraduate course. I am grateful for his kind advices not only on my study but also on my private life. I am especially indebted to Professor Ryoh Fuji-Hara, Professor Mieko Yamada, and Professor Junya Satoh for helping me to study combinatorics, number theory, and their applications to communication theory. I would like to give special thanks to Professor Marco Buratti and i Dr. Anita Pasotti for their knowledge and fruitful discussions about difference families, the main topic …