2020/12/15 by Bailey, Robert F., Hawtin, Daniel R.
#05E18 #20B25 #94B27 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2012.08436
A code C in the Hamming metric, that is, is a subset of the vertex set V\varGamma of the Hamming graph \varGamma=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 \rmAut(C) acts transitively on each of the sets C, C1, …, Cρ. A code C is called 2-neighbour-transitive if ρ≥ 2 and \rmAut(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 \rmAut(C) and minimum distance δ≥ 5. Then it is known that \rmAut(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 new non-linear completely transitive code 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 answer the question of the existence of distance-regular graphs related to the completely transitive codes appearing in our main result.