Eisenstein integer based multi-dimensional coded modulation for physical-layer network coding over F4 in the two-way relay channels

A novel multi-dimensional coded modulation based on Eisenstein integer Z[w] is proposed for physical-layer network coding (PNC) over the finite field F₄ in the two-way relay channel (TWRC). The duo-binary turbo code is used to design the Construction A-based Z[w]-lattice for the proposed PNC over the lattice partition F₄≅ Z[w]/2Z[w]. In order to improve the spectral efficiency, Ungerboeck´s mapping by set partitioning (MSP) is adopted for transmission of coded and uncoded layers of the designed Construction A-based Z[w]-lattice. A novel soft-demapper and a new ML decoder with size-reduced search space are proposed to decode the superimposed lattices at the relay. A new mapping selection criterion which maximizes the mutual information between the received signal and the network coded combination is introduced for the proposed soft-demapper. Numerical results show that the proposed PNC over IF₄ is superior to compute-and-forward using integer-forcing and channel decoder [5, 10].