Hamiltonicity of Minimum Distance Graphs of 1-Perfect Codes

  • 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 n13, we construct   Hamiltonian nonlinear ternary 1-perfect  codes,   and for  all admissible lengths n21, 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, r1 there exist Hamiltonian nonlinear q-ary 1-perfect  codes of length n=(qm1)/(q1), m2. If q=2,3,4, then m2.  If q=2, then m3.

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

Published
2012-03-31
Article Number
P65