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 M11\documentclass[12pt]{minimal}
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\begin{document}$$M_{11}$$\end{document}, M12\documentclass[12pt]{minimal}
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\begin{document}$$M_{12}$$\end{document}, M22\documentclass[12pt]{minimal}
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\begin{document}$$M_{22}$$\end{document}, M23\documentclass[12pt]{minimal}
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\begin{document}$$M_{23}$$\end{document} and M24\documentclass[12pt]{minimal}
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\begin{document}$$M_{24}$$\end{document}. 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 M22\documentclass[12pt]{minimal}
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\begin{document}$$M_{22}$$\end{document}, 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.
