Lattice network codes based on Eisenstein integers

In this paper, we investigate lattice network codes (LNCs) constructed from lattices over the ring of Eisenstein integers. Quantization and encoding algorithms over Eisenstein integers are first introduced. Then, a union bound estimation (UBE) of the decoding error probability is derived when the shaping region of the LNC is a product of regular hexagons. We show that the UBE is in the same form as the one for hypercube shaped LNCs, such as in the Gaussian integer case. We also demonstrate that in the Eisenstein integer case, the nominal coding gain and the shaping gain of a baseline LNC are, respectively, 0.625 dB and 0.167 dB, in contrast to the Gaussian integer case, where both gains are 0 dB. This is consistent with the simulation results comparing the performance of decoding error probability of baseline LNCs.