Elusive Codes in Hamming Graphs

TL;DR

This paper investigates elusive code-group pairs in Hamming graphs, constructing an infinite family where the group acts transitively on neighbors, revealing new structural properties and open questions about their parameters.

Contribution

It introduces a new infinite family of elusive pairs with transitive group actions and analyzes their parameter constraints, advancing understanding of code symmetry in Hamming graphs.

Findings

Constructed an infinite family of elusive pairs with transitive group action

Found that the alphabet size divides the code length in these examples

Proved no elusive pairs exist for minimal parameter sets without this divisibility

Abstract

We consider a code to be a subset of the vertex set of a Hamming graph. We examine elusive pairs, code-group pairs where the code is not determined by knowledge of its set of neighbours. We construct a new infinite family of elusive pairs, where the group in question acts transitively on the set of neighbours of the code. In our examples, we find that the alphabet size always divides the length of the code, and prove that there is no elusive pair for the smallest set of parameters for which this is not the case. We also pose several questions regarding elusive pairs.