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 \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 $q\geq 5$.