Alphabet-Almost-Simple 2-Neighbour Transitive Codes

TL;DR

This paper classifies a special class of symmetric codes in Hamming graphs, showing that the only alphabet-almost-simple (X,2)-neighbour transitive code with minimum distance at least 3 is a specific repetition code in H(3,q).

Contribution

It provides a classification of alphabet-almost-simple (X,2)-neighbour transitive codes with minimum distance at least 3, identifying the unique such code as a repetition code in H(3,q).

Findings

Only the repetition code in H(3,q) qualifies under the specified conditions.

The classification focuses on codes with a non-trivial kernel action on the alphabet.

The main result restricts the structure of such highly symmetric 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 \emph{$(X,2)$-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. This paper begins the classification of $(X,2)$-neighbour transitive codes where the action of $X$ on the entries of the Hamming graph has a non-trivial kernel. There exists a subgroup of $X$ with a $2$-transitive action on the alphabet; this action is thus almost-simple or affine. If this $2$-transitive action is almost simple we say $C$ is \emph{alphabet-almost-simple}. The main result in this paper states that the only alphabet-almost-simple $(X,2)$-neighbour transitive code with minimum distance $\delta\geq 3$ is the repetition code in $H(3,q)$, where $q\geq 5$.