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

我们研究了与有限单群相关的码、格和顶点算子代数。主要结果如下：1.研究了长度为48的单偶码和秩为48的奇么模格。没有)极端的邻居。作为一个副产品，我们在Z/4Z.2上构造了一个新的极值编码。我们研究了一种假定的长度为72的极值二进制码。证明了如果存在一个5-(72，16，78)自正交设计，则其块点关联矩阵的行将生成一个长度为72.3的极双偶自对偶码。我们在Hall-Janko群下构造了长度为100不变的新的自对偶码。研究了二进制码C(G，n)，C(G，n)定义为给定群G的对合不动点集所成的码的对偶码，证明了任何长度为n的G-不变自正交码都包含在C(G，n)中。许多与零星简单群有关的自正交码，包括扩展的Golay码和上述用于Hall-Janko群的码，被获得为C(G，n)。构造了一些新的零星几乎单群下的自对偶码不变量。我们构造了具有未知重量枚举数的极值单偶自对偶[64，32，12]码。特别地，我们发现了一些影子具有最小重量12的码。通过考虑它们的双偶邻域，首次构造了覆盖半径为12的极值双偶自对偶码。我们研究了零星秩3图的最大团及其相关设计。从Witt系统和六进制码出发，给出了Hall-Janko图的一些重构。此外，我们还考虑了零星Suzuki图的最大团设计问题。我们利用16元数域上的酉群U_{3}(4)：4构造了一个新的3-(66，16，21)设计。利用这种设计，我们给出了零星Suzuki图的一种新的构造。

We have studied codes, lattices and vertex operator algebras related to finite simple groups. Main results are as follows :1. We studied singly even codes of length 48 and odd unimodular lattices of rank 48 which have (resp. do not have) extremal neighbors. As a byproduct, we have constructed a new extremal code over Z/4Z.2. We studied a putative extremal binary code of length 72. We showed that if there is a self-orthogonal 5-(72,16,78) design then the rows of its block-point incidence matrix generate an extremal doubly-even self-dual code of length 72.3. We constructed new self-dual codes of length 100 invariant under the Hall-Janko group. We studied the binary code C(G, n) defined as the dual code of the code spanned by the sets of fixed points of involutions of a given group G. We showed that any G-invariant self-orthogonal code of length n is contained in C(G, n). Many self-orthogonal codes related to sporadic simple groups, including the extended Golay code and the above code for the Hall-Janko group, are obtained as C(G, n). Some new self-dual codes invariant under sporadic almost simple groups are constructed.4. We constructed extremal singly even self-dual [64,32,12] codes with weight enumerators which were not known to be attainable. In particular, we find some codes whose shadows have minimum weight 12. By considering their doubly even neighbors, extremal doubly even self-dual [64,32,12] codes with covering radius 12 are constructed for the first time.5. We studied maximum cocliques of sporadic rank 3 graphs and related designs. We gave some reconstructions of the Hall-Janko graph from the Witt system and the hexacode. Moreover we considered the maximum coclique design of the sporadic Suzuki graph. We constructed a new 3-(66,16,21) design with the automorphism group U_{3}(4):4, the unitary group over the 16-element field. By using this design, we gave a new construction of the sporadic Suzuki graph.