On the classification of binary completely transitive codes with almost-simple top-group

Abstract

A code C in the Hamming metric, that is, is a subset of the vertex set VΓ of the Hamming graph Γ=H(m,q), gives rise to a natural distance partition{C,C1,…,Cρ}, where ρ is the covering radius of C. Such a code C is called completely transitive if the automorphism group Aut(C) acts transitively on each of the sets C, C1, …, Cρ. A code C is called 2-neighbour-transitive if ρ⩾2 and Aut(C) acts transitively on each of C, C1 and C2.Let C be a completely transitive code in a binary (q=2) Hamming graph having full automorphism group Aut(C) and minimum distance δ⩾5. Then it is known that Aut(C) induces a 2-homogeneous action on the coordinates of the vertices of the Hamming graph. The main result of this paper classifies those C for which this induced 2-homogeneous action is not an affine, linear or symplectic group. We find that there are 13 such codes, 4 of which are non-linear codes. Though most of the codes are well-known, we obtain several new results. First, a non-linear completely transitive code that does not explicitly appear in the existing literature is constructed, as well as a related non-linear code that is 2-neighbour-transitive but not completely transitive. Moreover, new proofs of the complete transitivity of several codes are given. Additionally, we consider the question of the existence of distance-regular graphs related to the completely transitive codes appearing in our main result.