Equivocation performance of linear wireless physical layer network coding

We consider linear mapping functions for wireless physical layer network coding (WPNC), 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 WPNC, particularly in terms of ensuring unambiguous decoding, and show that this requires that the coefficients of the linear mapping have an inverse within the coefficient set. We then evaluate the information lost on the channel in terms of the equivocation for arbitrary relative fade states, as well as singular fade states. This is then used to determine outage probability and outage capacity for LPNC systems.