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Alexander Mikhailovich Romanov
Keywords:
Hamiltonian cycle, minimum distance graph, Hamming code, nonlinear code, q-ary 1-perfect code.
Abstract
A 1-perfect code Cnq is called Hamiltonian if its minimum distance graph G(Cnq) 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≠2. If q=2, then m≠3.
Author Biography
Alexander Mikhailovich Romanov, Sobolev Institute of Mathematics of the Siberian Branch of the Russian Academy of Sciences
Department of Theoretical Cybernetics
Senior Scientific Researcher