Alphabet-affine 2-neighbour-transitive codes

Abstract

A code $\mathcal{C}$ is a subset of the vertex set of a Hamming graph $H(n,q)$, and $\mathcal{C}$ is $2$-neighbour-transitive if the automorphism group $G=\mathrm{Aut}(\mathcal{C})$ acts transitively on each of the sets $\mathcal{C}$, ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$, where ${\mathcal{C}}_1$ and ${\mathcal{C}}_2$ are the (non-empty) sets of vertices that are distances $1$ and $2$, respectively, (but no closer) to some element of $\mathcal{C}$. Suppose that $\mathcal{C}$ is a $2$-neighbour-transitive code with minimum distance at least $5$. For $q=2$, all `minimal' such $\mathcal{C}$ have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine $2$-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine $2$-transitive group must be a subgroup of $\mathrm{A}\hat{I}“{\mathrm{L}}_1(q)$ and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups.

18 pages