2014/04/03 by Daniel R. Hawtin, Hawtin, Daniel R.
Computer Science · Engineering · #Cellular Automata and Applications #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1404.0950
openalex publication_date 2014/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A code is a subset of the vertex set of a Hamming graph. The set of s-neighbours of a code is the set of all vertices at Hamming distance s from their nearest codeword. A code C is s-elusive if there exists a distinct code C' that is equivalent to C under the full automorphism group of the Hamming graph such that C and C' have the same set of s-neighbours. It is proved here that the minimum distance of an s-elusive code is at most 2s+2, and that an s-elusive code with minimum distance at least 2s+1 gives rise to a q-ary t-design with certain parameters. This leads to the construction of: an infinite family of 1-elusive and completely transitive codes, an infinite family of 2-elusive codes, and a single example of a 3-elusive code. Answers to several open questions on elusive codes are also provided.