Neighbour-transitive codes in Johnson graphs

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