  In this paper we construct distance-regular graphs admitting a 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 code
spanned by the 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. Further, we discuss a
possibility of permutation decoding of the codes spanned by the adjacency
matrices of the graphs constructed and find small PD-sets for some of the
codes.
