On combinatorial Gray codes with distance 3

We suggest a construction of the cyclic binary combinatorial Gray codes with distance 3 and dimension $n=2^k-1$, where $k=3,4,\dots$. We give a method of construction of Hamiltonian cycles in the graphs of minimum distances of binary Hamming codes. For all admissible lengths $n\ge15$, we give nonlinear perfect binary codes whose graphs of minimum distances contain a Hamiltonian cycle.