Distance-regular graphs obtained from the Mathieu groups

TL;DR

This paper constructs distance-regular graphs with transitive Mathieu group actions, explores their automorphism groups, and discusses permutation decoding methods for associated codes.

Contribution

It introduces new distance-regular graphs linked to Mathieu groups and analyzes their automorphism groups and decoding strategies.

Findings

Constructed graphs with Mathieu group symmetries.

Identified automorphism groups isomorphic to Higman-Sims group.

Proposed permutation decoding methods for related codes.

Abstract

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.