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

## DOI:

https://doi.org/10.26493/1855-3974.1240.515## Keywords:

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

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## Published

2017-09-30

## Issue

## Section

Articles

## License

Articles in this journal are published under Creative Commons Attribution 4.0 International License

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