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

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