Finite Simple Groups and Related Codes, Lattices and Vertex Operator Algebras

我们研究了有限单群的码、格和顶点算子代数。主要研究结果如下：完成了Fischer简单群F_<22>, F_<23>， F'_<24>的2-和3-基亚群的分类。对12维的偶模高斯格进行了分类。给出了z_6码极值的充分条件，并作为应用，构造了许多与Leech格相关的z_6码。并对长度为8的自对偶z_6码进行了分类。研究了用三元自对偶码构造非模格的方法。首次构造了维数为44,60和68的极值奇非模格。讨论了用z_8码和子代数V_<√<2>A_3>构造顶点算子代数。在Leech格中嵌入晶格（√<2>A_3）^8，给出了与子代数相关的Moonshine VOA V^*的完全分解。利用Dong-Mason给出的z_3 -轨道结构，给出了Moonshine VOA V^*与Leech格（√<2>A_2）^<12>嵌入所给出的子代数的完全分解。并给出了在V^*上某些Monster简单群3a元素的显式作用的定义。

We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :1.The classifications of the 2-and 3-radical subgroups of Fischer's simple groups F_<22>, F_<23>, F'_<24> have been completed.2.Even unimodular Gaussian lattices of dimension 12 have been classified.3.A sufficient condition for extremal Z_6-codes, and as an application, many Z_6-code related the Leech lattice had been constructed. Moreover the self-dual Z_6-codes of length 8 are classified.4.The construction method of unimodular lattices using ternary self-dual codes has been studied. Extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time.5.The constructions of vertex operator algebras using Z_8-codes and subalgebras V_<√<2>A_3> have been considered. We also give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_3)^8 into the Leech lattice.6.By using Z_3-orbifold construction given by Dong-Mason, we give a complete decomposition of the Moonshine VOA V^* associated with some subalgebra given by an embedding of the lattice (√<2>A_2)^<12> the Leech lattice. We also give the explicit actions of certain 3A-elements of the Monster simple group on V^* are defined.