Binary Completely Transitive Codes

Given a code in a graph, the vertex set of the graph may be partitioned by the sets of vertices at each of the possible distances to the code. If the automorphism group of the code acts transitively on each cell of this `distance partition', then the code is called `completely transitive'. I will discuss recent work towards a classification of binary completely transitive codes, and briefly discuss related algebraic and combinatorial symmetry conditions.