$s$-Elusive Codes in Hamming Graphs

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.