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′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C'$$\end{document} that is equivalent to C under the full automorphism group of the Hamming graph such that C and C′\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C'$$\end{document} have the same set of s-neighbours. We show that the minimum distance of an s-elusive code is at most 2s+2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s+2$$\end{document}, and that an s-elusive code with minimum distance at least 2s+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2s+1$$\end{document} 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.