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

我们研究了与有限单群有关的码、格和顶点算子代数。主要研究结果如下：1.本文研究了长为48的单偶码和秩为48的奇幺模格。没有）极端的邻居。作为副产品，我们构造了一个新的极值码在Z/4Z.2。我们研究了一个长度为72的假定极值二进制码。证明了如果存在自正交5-（72，16，78）设计，则其区组点关联矩阵的行生成长度为72.3的极值双偶自对偶码。我们在Hall-Janko群下构造了新的长度为100不变的自对偶码。研究了二元码C（G，n），C（G，n）定义为给定群G的对合的不动点集所生成的码的对偶码。证明了任意长度为n的G-不变自正交码都包含在C（G，n）中.许多与散在单群有关的自正交码，包括推广的Golay码和上述的Hall-Janko群的自正交码，都可以用C（G，n）表示.构造了一些新的在散在几乎单群下不变的自对偶码.我们构造了极值单偶自对偶[64，32，12]码，其权元是不可达到的.特别是，我们发现一些代码的阴影有最小的重量12。通过考虑它们的双偶邻居，首次构造了覆盖半径为12的极值双偶自对偶[64，32，12]码.研究了3阶偶发图的最大角点及相关设计。给出了由Witt系统和六角码重构的Hall-Janko图。此外，我们还考虑了偶发Suzuki图的最大斜团设计。我们构造了一个新的3-（66，16，21）设计，其自同构群为16元域上的酉群U_{3}（4）：4。利用这种设计，我们给出了一个新的零星Suzuki图的构造。

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.