Lattices over Eisenstein Integers for Compute-and-Forward

We consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We prove the existence of a sequence of infinite-dimensional nested lattices over Eisenstein integers where the coarse lattice is simultaneously good for quantization and additive white Gaussian noise (AWGN) channel coding and the fine lattice is good for AWGN channel coding. Using this, we show that the information rates achievable with nested lattice codebooks over Eisenstein integers can be higher than those achievable with nested lattices over integers considered by Nazer and Gastpar in [1] for some set of channel realizations. We also propose a practical coding scheme based on the concatenation of a non-binary low density parity check code with a modulation scheme derived from the ring of Eisenstein integers.