2013/11/01 by Robert A. Liebler, Cheryl E. Praeger, Liebler, Robert A. +1
Computer Science · Mathematics · #05C25 #20B25 #94B60 #Coding theory and cryptography #Combinatorics (math.CO) #Cooperative Communication and Network Coding #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.1311.0113
openalex publication_date 2013/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Γsuch that the subgroup G of graph automorphisms leaving Γinvariant is transitive on both the set Γof `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 on V and is intransitive or imprimitive on the underlying v-set V. In the remaining case where G lies in Sym(V) and G is primitive on V, we prove that, provided distinct codewords are at distance at least 3 in J(v,k), 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.