Lattice Network Codes Based on Eisenstein Integers

In this paper, we investigate lattice network codes (LNCs) constructed from Eisenstein integer based lattices. 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. Next, the Gaussian reduction algorithm is generalized to be applicable to complex lattices over Eisenstein integers such that an optimal coefficient vector can be found in the two-transmitter single-relay system. Based on the UBE, design criteria for optimal LNCs with minimum decoding error probability are formulated and applied to construct both Gaussian integer and Eisenstein integer based good LNCs from rate-1/2 feed-forward convolutional codes by Complex Construction A. The constructed codes provide up to 7.65 dB nominal coding gains over Rayleigh fading channels. Furthermore, we introduce the construction of LNCs from linear codes by Complex Construction B. The nominal coding gains and error performance of the LNCs thus constructed are explicitly analyzed. Examples show that the LNCs constructed by Complex Construction B provide a better tradeoff between code rate and nominal coding gain.