Linear physical layer network coding based on rings

We consider linear mapping functions for physical layer network coding (PNC), using a generalised description of linear mapping, and considering alternative algebraic structures within which a linear mapping can be defined. We discuss the requirements on a linear mapping for PNC in terms of ensuring unambiguous decoding, and hence consider the requirements this imposes on the coefficients of the linear mapping. We then consider an alternative forms of linear mapping, based on coefficients from rings, and show that this generalizes the more common approach using fields. We then consider the performance of the mappings in the two-way relay channel using QPSK as an example, and evaluate the information loss in terms of the equivocation at the destination under the possible singular fade states of the source-relay channels.