In this paper we construct distance-regular graphs admitting a vertex transitive action of the five sporadic simple groups discovered by E. Mathieu, the Mathieu groups \(M_{11}\), \(M_{12}\), \(M_{22}\), \(M_{23}\) and \(M_{24}\). From the binary code spanned by an adjacency matrix of the strongly regular graph with parameters (176,70,18,34) we obtain block designs having the full automorphism groups isomorphic to the Higman-Sims finite simple group. Moreover, from that code we obtain eight 2-designs having the full automorphism group isomorphic to \(M_{22}\), whose existence cannot be explained neither by the Assmus-Mattson theorem nor by a transitivity argument. Further, we discuss a possibility of permutation decoding of the codes spanned by adjacency matrices of the graphs constructed and find small PD-sets for some of the codes.

在本文中，我们构造距离正则图，该图承认 E. Mathieu 发现的五个零星单群的顶点传递作用，Mathieu 群\(M_{11}\)，\(M_{12}\)，\(M_ {22}\)、\(M_{23}\)和\(M_{24}\)。从带有参数（176,70,18,34）的强正则图的邻接矩阵跨越的二进制代码，我们获得了具有与 Higman-Sims 有限单群同构的完全自同构群的块设计。此外，从该代码中，我们获得了 8 个 2 设计，它们具有与\(M_{22}\)同构的完全自同构群，其存在既不能用 Assmus-Mattson 定理也不能用及物性论证来解释。此外，我们讨论了对由所构建图的邻接矩阵跨越的代码进行置换解码的可能性，并为一些代码找到小的 PD 集。