Classification of a family of completely transitive codes

TL;DR

This paper investigates a specific subclass of highly symmetric error-correcting codes in Hamming graphs, focusing on those with automorphism groups acting transitively on each distance partition, advancing towards their classification.

Contribution

It initiates the classification of completely transitive codes with faithful automorphism group actions, extending the understanding of their structure and symmetry.

Findings

Identified conditions for automorphism groups to be faithful on entries.

Established initial classifications for a subfamily of completely transitive codes.

Abstract

The completely regular codes in Hamming graphs have a high degree of combinatorial symmetry and have attracted a lot of interest since their introduction in 1973 by Delsarte. This paper studies the subfamily of completely transitive codes, those in which an automorphism group is transitive on each part of the distance partition. This family is a natural generalisation of the binary completely transitive codes introduced by Sole in 1990. We take the first step towards a classification of these codes, determining those for which the automorphism group is faithful on entries.