Abstract
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.
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Acknowledgements
The authors would like to thank Sven Reichard for pointing out that the graphs \(\Gamma _3^1\) and \(\Gamma _5^2\) constructed in this paper are not arising from orthogonal arrays, and to the anonymous referee for helpful comments and suggestions.
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This work has been fully supported by Croatian Science Foundation under the project 6732.
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Appendix
Appendix
In this Appendix we give short description of the procedure which was used for obtaining the results described in this paper. The procedures follow the method described in Section 3.
Procedures:
ActionGcosets(class) - function that defines transitive permutation representation of the group G;
BaseBlock(x) - function that calculates the orbit of the subgroups under the action;
SelfPaired(x), MutuallyPaired(x) - functions that check whether the orbits are self-paired or mutually paired;
DesignGraph(x) - function that takes the base block, builds 1-designs which incidence matrices are adjacency matrices of regular graphs
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Crnković, D., Mostarac, N. & Švob, A. Distance-regular graphs and new block designs obtained from the Mathieu groups. AAECC 35, 177–194 (2024). https://doi.org/10.1007/s00200-022-00542-x
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DOI: https://doi.org/10.1007/s00200-022-00542-x