On Lattice-Partition-Based Physical-Layer Network Coding over GF(4)

This letter studies lattice-partition-based physical layer network coding (LNC) over the finite field GF(4), whose quaternary constellation has practical interests. The optimal dither method in terms of energy efficiency is first derived for LNC over GF(4), and it is shown to save 1/3 average transmission power. Next, LNC design from linear codes over GF(4) is investigated. Explicit connection between parameters of the linear code and of the corresponding LNC is established. The construction method is further extended to design LNCs from linear codes over a general finite field GF(4^m). As design examples, LNCs constructed from convolutional, BCH, and Reed-Solomon codes are presented and analyzed. It is illustrated that these LNCs can provide up to 9.5 dB coding gain.