Neighbour-transitive codes in Johnson graphs

TL;DR

This paper introduces and classifies neighbour-transitive codes in Johnson graphs, revealing their structure under various symmetry group actions and constructing numerous examples, with some cases still unresolved.

Contribution

It defines neighbour-transitive codes in Johnson graphs, classifies their symmetry groups, and constructs many examples, advancing understanding of their algebraic and combinatorial properties.

Findings

Classification of neighbour-transitive codes with intransitive or imprimitive symmetry groups.

Proof that primitive symmetry groups imply 2-transitivity under certain conditions.

Construction of rich families of neighbour-transitive codes from 2-transitive groups.

Abstract

The Johnson graph J(v,k) has, as vertices, the k-subsets of a v-set V, and as edges the pairs of k-subsets with intersection of size k-1. We introduce the notion of a neighbour-transitive code in J(v,k). This is a vertex subset \Gamma such that the subgroup G of graph automorphisms leaving \Gamma invariant is transitive on both the set \Gamma of `codewords' and also the set of `neighbours' of \Gamma, which are the non-codewords joined by an edge to some codeword. We classify all examples where the group G is a subgroup of the symmetric group on V and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G lies in Sym(V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3 in J(v,k), then G is 2-transitive on V. We examine many of the infinite families of finite 2-transitive permutation groups and construct surprisingly rich families of examples of neighbour-transitive codes. A major unresolved case remains.