Alphabet-Almost-Simple 2-Neighbour Transitive Codes
Neil I. Gillespie, Daniel R. Hawtin

TL;DR
This paper classifies a special class of symmetric codes in Hamming graphs, showing that the only alphabet-almost-simple (X,2)-neighbour transitive code with minimum distance at least 3 is a specific repetition code in H(3,q).
Contribution
It provides a classification of alphabet-almost-simple (X,2)-neighbour transitive codes with minimum distance at least 3, identifying the unique such code as a repetition code in H(3,q).
Findings
Only the repetition code in H(3,q) qualifies under the specified conditions.
The classification focuses on codes with a non-trivial kernel action on the alphabet.
The main result restricts the structure of such highly symmetric codes.
Abstract
Let be a subgroup of the full automorphism group of the Hamming graph , and a subset of the vertices of the Hamming graph. We say that is an \emph{-neighbour transitive code} if is transitive on , as well as and , the sets of vertices which are distance and from the code. This paper begins the classification of -neighbour transitive codes where the action of on the entries of the Hamming graph has a non-trivial kernel. There exists a subgroup of with a -transitive action on the alphabet; this action is thus almost-simple or affine. If this -transitive action is almost simple we say is \emph{alphabet-almost-simple}. The main result in this paper states that the only alphabet-almost-simple -neighbour transitive code with minimum distance is the repetition code in , where .
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Taxonomy
TopicsCoding theory and cryptography · Cooperative Communication and Network Coding · Finite Group Theory Research
