Constellations Maximizing Minimal Distance for Physical-Layer Network Coding Multiway Relaying

This paper objects a design of constellations suitable for Physical-Layer Network Coding (PLNC) in a Multi-Way Relay Channel (M-WRC). We formulate the constellation design as a general constrained optimization problem maximizing minimal distance of network coding function decoding. The numerically obtained constellations are recognized to possess some regular structure. The optimal constellations for 2-WRC are power-balanced hexagonal constellations with particular type of indexing. As far as the optimization problem is numerically tractable, we declare optimality of power-scaled pre-rotated amplitude shift keying constellations for M-WRC with number of terminals > 2. All optimums decode non-linear modulo-sum network coding function and offer considerable performance gains in comparison to canonical schemes. This work also demonstrates that PLNC strategy tailored to M-WRC overcome PLNC 2-WRC approach.