Entry-Faithful 2-Neighbour Transitive Codes

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We consider a code to be a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of vertices, not in the code, at distance s from some codeword, but not distance less than s from any codeword. A 2-neighbour transitive code is a code which admits a group X of automorphisms which is transitive on the s-neighbours, for s=1,2, and transitive on the code itself. We give a classification of 2-neighbour transitive codes, with minimum distance δ≥ 5, for which X acts faithfully on the set of entries of the Hamming graph.