Orbit Classification Of Blocks And Their Application

Combinatorics consists of combinatorial enumeration, combinatorial design, combinatorial matrix, combinatorial optimization, etc. As an important branch of combinatorics, combinatorial design theory has important applications in various fields, such as computer science, coding theory, cryptography, physics, chemistry and so on.The large set problem in the combinatorial design theory has a long history and a wide range of applications. It is unanimously recognized as the difficulties in design field due to its complex conditions. The related research work had been quite slow in making progress for a long period of time due to its sophistication. Being motivated by some new methods and techniques, the research in the large set problem has taken on a promising posture in recent years. Orbit of blocks is a commonly method to research combinatorial design and large set problem, so the study of orbit classification and find the representative elements of the orbit is important.In this dissertation, we investigate the problem of orbit classification of blocks. We research the problem of orbit classification of quadruples on an n-set X, under the action of cyclic group of order 4, order n and order n-1. This dissertation is divided into five chapters.In the first chapter, we introduce the research background and the general research situation. We introduce some basic concepts and some of the known research results.In the second chapter, we research the problem of orbit classification of quadruples on an n-set X, under the action of cyclic group of order 4. The orbit enumerations and orbit representative systems are obtained. We also give a example for application of orbit classification.In the third chapter, we research the problem of orbit classification of quadruples on an n-set X, under the action of cyclic group of order n. The orbit enumerations and orbit representative systems are obtained. We also give an example for application of orbit classification.In the fourth chapter, we research the problem of orbit classification of quadru-ples on an n-set X, under the action of cyclic group of order n-1. The orbit enumerations and orbit representative systems are obtained.In the fifth chapter, we summarize the results of this paper and give some prospects in this research field.