Hamiltonicity of minimum distance graphs of 1-perfect codes (Q426843)

Summary: A 1-perfect code \(\mathcal{C}_{q}^{n}\) is called Hamiltonian if its minimum distance graph \(G(\mathcal{C}_{q}^{n})\) contains a Hamiltonian cycle. In this paper, for all admissible lengths \(n \geq 13\), we construct Hamiltonian nonlinear ternary 1-perfect codes, and for all admissible lengths \(n \geq 21\), 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 \(q = p^r\), where \(p\) is prime, \(r \geq 1\) there exist Hamiltonian nonlinear \(q\)-ary 1-perfect codes of length \(n = (q ^{m} -1) / (q-1)\), \(m \geq 2\). If \(q =2, 3, 4\), then \( m \neq 2\). If \(q =2\), then \( m \neq 3\).