On combinatorial Gray codes with distance 3

Abstract

We suggest a construction of the cyclic binary combinatorial Gray codes with distance 3 and dimension n = 2k – 1, where k = 3, 4, . . . We give a method of construction of Hamiltonian cycles in the graphs of minimum distances of binary Hamming codes. For all admissible lengths n ≥ 15, we give nonlinear perfect binary codes whose graphs of minimum distances contain a Hamiltonian cycle.