Wireless bidirectional relaying and Latin squares

The design of modulation schemes for the physical layer network-coded two way relaying scenario is considered with the protocol which employs two phases: Multiple access (MA) Phase and Broadcast (BC) Phase. It was observed by Koike-Akino et al. that adaptively changing the network coding map used at the relay according to the channel conditions greatly reduces the impact of multiple access interference which occurs at the relay during the MA Phase and all these network coding maps should satisfy a requirement called the exclusive law. We show that every network coding map that satisfies the exclusive law is representable by a Latin Square and conversely, and this relationship can be used to get the network coding maps satisfying the exclusive law. Using the structural properties of the Latin Squares for a given set of parameters, the problem of finding all the required maps is reduced to finding a small set of maps for M-PSK constellations. This is achieved using the notions of isotopic and transposed Latin Squares. Furthermore, the channel conditions for which the bit-wise XOR will perform well is analytically obtained which holds for all values of M (for M any power of 2). We illustrate these results for the case where both the end users use QPSK constellation.