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.