Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes

A 1-perfect code Cn q is called Hamiltonian if its minimum distance graph G(Cn q ) contains a Hamiltonian cycle. In this paper, for all admissible lengths n ≥ 13, we construct Hamiltonian nonlinear ternary 1-perfect codes, and for all admissible lengths n ≥ 21, we construct Hamiltonian nonlinear quaternary 1-perfect codes. The existence of Hamiltonian nonlinear q-ary 1-perfect codes of length N = qn + 1 is reduced to the question of the existence of such codes of length n. Consequently, for q = pr, where p is prime, r ≥ 1 there exist Hamiltonian nonlinear q-ary 1-perfect codes of length n = (qm − 1)/(q − 1), m ≥ 2. If q = 2, 3, 4, then m 6= 2. If q = 2, then m 6= 3.