2024/11/13 by Daniel R. Hawtin, Hawtin, Daniel R.
Computer Science · Engineering · #05E18 (Primary) 20B25 #94B25 (Secondary) #Advanced Wireless Communication Techniques #Coding theory and cryptography #Combinatorics (math.CO) #Error Correcting Code Techniques #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2411.08351
openalex publication_date 2024/11/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A code \mathcal C is a subset of the vertex set of a Hamming graph H(n,q), and \mathcal C is 2-neighbour-transitive if the automorphism group G=\rm Aut(\mathcal C) acts transitively on each of the sets \mathcal C, \mathcal C1 and \mathcal C2, where \mathcal C1 and \mathcal C2 are the (non-empty) sets of vertices that are distances 1 and 2, respectively, (but no closer) to some element of \mathcal C. Suppose that \mathcal C is a 2-neighbour-transitive code with minimum distance at least 5. For q=2, all `minimal' such \mathcal C have been classified. Moreover, it has previously been shown that a subgroup of the automorphism group of the code induces an affine 2-transitive group action on the alphabet of the Hamming graph. The main results of this paper are to show that this affine 2-transitive group must be a subgroup of \rm AΓ\rm L1(q) and to provide a number of infinite families of examples of such codes. These examples are described via polynomial algebras related to representations of certain classical groups.