  We present a general framework for studying the multilevel structure of
lattice network coding (LNC), which serves as the theoretical fundamental for
solving the ring-based LNC problem in practice, with greatly reduced decoding
complexity. Building on the framework developed, we propose a novel
lattice-based network coding solution, termed layered integer forcing (LIF),
which applies to any lattices having multilevel structure. The theoretic
foundations of the developed multilevel framework lead to a new general lattice
construction approach, the elementary divisor construction (EDC), which shows
its strength in improving the overall rate over multiple access channels (MAC)
with low computational cost. We prove that the EDC lattices subsume the
traditional complex construction approaches. Then a soft detector is developed
for lattice network relaying, based on the multilevel structure of EDC. This
makes it possible to employ iterative decoding in lattice network coding, and
simulation results show the large potential of using iterative multistage
decoding to approach the capacity.
