  This study deals with the further development of nuclear spin model of
scalable quantum register, which presents the one-dimensional chain of the
magnetic atoms with nuclear spins 1/2, substituting the basic atoms in the
plate of nuclear spin-free easy-axis 3D antiferromagnet. The decoherence rates
of one qubit state and entanglement state of two removed qubits and
longitudinal relaxation rates are caused by the interaction of nuclear
spins-qubits with virtual spin waves in antiferromagnet ground state were
calculated. It was considered also one qubit adiabatic decoherence, is caused
by the interaction of nuclear spin of quantum register with nuclear spins of
randomly distributed isotopes, substituting the basic nuclear spin-free
isotopes of antiferromagnet. We have considered finally encoded DFS
(Decoherence-Free Subspaces) logical qubits are constructed on clusters of the
four-physical qubits, given by the two states with zero total angular momentum.
