Codes and designs in Johnson graphs with high symmetry

Abstract: The Johnson graph J(v,k) has, as vertices, all k-subsets of a v-set V, with two k-subsets adjacent if and only if they share k − 1 common elements of V. Subsets of vertices of J(v,k) can be interpreted as the blocks of an incidence structure, or as the codewords of a code, and automorphisms of J(v,k) leaving the subset invariant are then automorphisms of the corresponding incidence structure or code. This approach leads to interesting new designs and codes. For example, numerous actions of the Mathieu sporadic simple groups give rise to examples of Delandtsheer designs (which are both flag-transitive and anti-flag transitive), and codes with large minimum distance (and hence strong error-correcting properties). The paper surveys recent progress, explores links between designs and codes in Johnson graphs which have a high degree of symmetry, and discusses several open questions.