Alphabet-affine 2-neighbour-transitive codes

TL;DR

This paper classifies 2-neighbour-transitive codes with minimum distance at least 5, showing the automorphism group must be a subgroup of AΓL_1(q) and providing infinite families of such codes using polynomial algebra and classical group representations.

Contribution

It proves that the automorphism group of these codes is contained in AΓL_1(q) and constructs infinite families of codes via polynomial algebra and classical group representations.

Findings

Automorphism group is a subgroup of AΓL_1(q).

Constructs infinite families of codes.

Provides classification for certain 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 $G={\rm Aut}({\mathcal C})$ 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 ${\rm A}\Gamma{\rm L}_1(q)$ 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.