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 proper 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. © 2014 Springer Science+Business Media New York.