Neighbour-transitive codes in Johnson graphs

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 Γ such that the subgroup G of graph automorphisms leaving Γ invariant is transitive on both the set Γ of ‘codewords’ and also the set of ‘neighbours’ of Γ, 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 Sym (V) and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G ≤ Sym (V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3, 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. Key-words: codes in graphs, Johnson graph, 2-transitive permutation group, neighbour-transitive. Mathematics Subject Classification (2010): 05C25, 20B25, 94B60.