Linear physical layer network coding for multihop wireless networks

We consider linear network coding functions that can be employed at the relays in wireless physical layer network coding, applied to a general multi-hop network topology. We introduce a general model of such a network, and discuss the algebraic basis of linear functions, deriving conditions for unambiguous decodability of the source data at the destination. We consider the use of integer rings, integer fields, binary extension fields and the ring of binary matrices as potential algebraic constructs, and show that the ring constructs provide more flexibility. We use the two-way relay channel and a network containing two sources and two relays to illustrate the concept and to demonstrate the effect of fading of the wireless channels. We show the capacity benefits of the more flexible rings.