Hexagonal Constellations for Adaptive Physical-Layer Network Coding 2-Way Relaying

We focus on a constellation design for adaptive physical-layer network coding strategy in a wireless 2-way relay channel. It is well known that 4QAM constellations require extended-cardinality network coding adaptation to avoid all singular channel parameters at the Multiple-Access (MA) stage. The cardinality extension is undesirable since it introduces redundancy decreasing the data rates at the broadcast stage. In this paper, we target a design of constellations avoiding all the singularities without the cardinality extension. We show that such a constellation is 4-ary constellation taken from hexagonal lattice (4HEX) which keeps comparable error performance at the MA stage as 4QAM, however without the cardinality extension. The similar properties has been found also by unconventional 3HEX and 7HEX constellations.