Neighbour transitivity on codes in Hamming graphs

TL;DR

This paper studies neighbour transitive codes in Hamming graphs, characterizing their automorphism groups and constructing examples where neighbours do not uniquely determine the code.

Contribution

It provides conditions for neighbour transitive groups to fix codes setwise and constructs an infinite family of codes with specific properties.

Findings

Neighbour transitive groups can sometimes fix codes setwise.

An infinite family of neighbour transitive codes with minimum distance 4 is constructed.

Neighbours do not always uniquely determine the code.

Abstract

We consider a \emph{code} to be a subset of the vertex set of a \emph{Hamming graph}. In this setting a \emph{neighbour} of the code is a vertex which differs in exactly one entry from some codeword. This paper examines codes with the property that some group of automorphisms acts transitively on the \emph{set of neighbours} of the code. We call these codes \emph{neighbour transitive}. We obtain sufficient conditions for a neighbour transitive group to fix the code setwise. Moreover, we construct an infinite family of neighbour transitive codes, with \emph{minimum distance} $\delta=4$, where this is not the case. That is to say, knowledge of even the complete set of code neighbours does not determine the code.