四阶循环群作用下四元组的轨道分类

Orbit Classification of Quadruples Under the Action of Cyclic Group of Order Four

组合数学中的组合设计及其大集问题的研究有着悠久的历史,其在计算机科学、编码论等领域都有广泛的应用.由于条件复杂,组合设计大集问题被公认为组合设计中的难点,长期以来进展缓慢.近年来,在新的方法和理论推动下,组合设计大集问题的研究呈现了很好的态势.研究组合设计大集的首要问题是将区组进行轨道分类.本文研究了n元集X中四元组(即4子集)集合在X上对称群S_n的四阶循环子群作用下的轨道分类问题,得到了四元组的轨道计数以及轨道代表系的一种取法,并给出了轨道分类的一些应用.

Combinatorial designs and their large sets have a long history and a wide range of applications in computer science and coding theory, etc. The large set problem 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 classification is the first step for constructing large sets. In this paper, we research the problem of orbit classification of quadruples on an n-set X, under the action of cyclic group of order four, which is a subgroup of symmetric group Sn on set X. The orbit enumeration and an orbit representative system are obtained. We also give some applications of this orbit classification.