Entry-Faithful $2$-Neighbour Transitive Codes

TL;DR

This paper classifies 2-neighbour transitive codes in Hamming graphs with minimum distance at least 5, where the automorphism group acts faithfully on the entries, advancing understanding of symmetrical error-correcting codes.

Contribution

It provides a classification of 2-neighbour transitive codes with certain minimum distance and faithful group action, a new result in the theory of symmetric codes.

Findings

Classification of 2-neighbour transitive codes with δ ≥ 5

Identification of conditions for faithful automorphism group action

Extension of known results in symmetric error-correcting codes

Abstract

We consider a code to be a subset of the vertex set of a Hamming graph. The set of $s$-neighbours of a code is the set of vertices, not in the code, at distance $s$ from some codeword, but not distance less than $s$ from any codeword. A $2$-neighbour transitive code is a code which admits a group $X$ of automorphisms which is transitive on the $s$-neighbours, for $s=1,2$, and transitive on the code itself. We give a classification of $2$-neighbour transitive codes, with minimum distance $\delta\geq 5$, for which $X$ acts faithfully on the set of entries of the Hamming graph.