## Abstract

Let X be a subgroup of the full automorphism group of the Hamming graph H(m, q), and C a subset of the vertices of the Hamming graph. We say that C is an (X, 2)-neighbour-transitive code if X is transitive on C, as well as C_{1} and C_{2}, the sets of vertices which are distance 1 and 2 from the code. It has been shown that, given an (X, 2)-neighbour-transitive code C, there exists a subgroup of X with a 2-transitive action on the alphabet; this action is thus almost-simple or affine. This paper completes the classification of (X, 2)-neighbour-transitive codes, with minimum distance at least 5, where the subgroup of X stabilising some entry has an almost-simple action on the alphabet in the stabilised entry. The main result of this paper states that the class of (X, 2) neighbour-transitive codes with an almost-simple action on the alphabet and minimum distance at least 3 consists of one infinite family of well known codes.

Original language | English |
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Pages (from-to) | 345-357 |

Number of pages | 13 |

Journal | Ars Mathematica Contemporanea |

Volume | 14 |

Issue number | 2 |

Early online date | 30 Sept 2017 |

Publication status | Published - 2018 |

## Keywords

- 2-neighbour-transitive
- Alphabet-almost-simple
- Automorphism groups
- Completely transitive
- Hamming graph