### Alphabet-almost-simple 2-neighbour-transitive codes

#### 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*(*if*X*, 2)-neighbour-transitive code*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.#### Keywords

2-neighbour-transitive, alphabet-almost-simple, automorphism groups, Hamming graph, completely transitive

DOI: https://doi.org/10.26493/1855-3974.1240.515

ISSN: 1855-3974

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