MAP decoder for physical-layer network coding using lattice sphere decoding

We focus in this work on optimal maximum a posteriori (MAP) decoding for lattice-based physical-layer network coding operating in Gaussian multiple access relay channels. We consider a general lattice design that makes our results hold for any lattice coding schemes of any dimensions including the Compute-and-Forward framework. By examining the MAP decoding rule, we first derive an analytical bound on the codeword error probability taking into consideration decoding errors at the relay. Besides, we derive a novel MAP decoding metric using which we develop a novel, practical and easy-to-implement MAP decoding algorithm based on lattice sphere decoding. We further provide numerical results that demonstrate the effectiveness of our algorithm and show its outperformance over existing suboptimal decoders.