Neighbour-transitive codes in Johnson graphs

Understand

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.