Alphabet-almost-simple 2-neighbour-transitive codes

Let X be a subgroup of the full automorphism group of the Hamming graph H(m, q), and C a subset of the vertices of the Hamming graph. We say that C is an (X, 2)-neighbour-transitive code if X is transitive on C, as well as C₁ and C₂, the sets of vertices which are distance 1 and 2 from the code. It has been shown that, given an (X, 2)-neighbour-transitive code C, there exists a subgroup of X with a 2-transitive action on the alphabet; this action is thus almost-simple or affine. This paper completes the classification of (X, 2)-neighbour-transitive codes, with minimum distance at least 5, where the subgroup of X stabilising some entry has an almost-simple action on the alphabet in the stabilised entry. The main result of this paper states that the class of (X, 2) neighbour-transitive codes with an almost-simple action on the alphabet and minimum distance at least 3 consists of one infinite family of well known codes.