Multilevel coding schemes for compute-and-forward

We consider the design of coding schemes for the wireless two-way relaying channel when there is no channel state information at the transmitter. In the spirit of the compute and forward paradigm, we present a multilevel coding scheme that permits the recovery of a class of functions at the relay. We define such a class of functions and derive rates that are universally achievable over a set of channel gains when this class of functions is used at the relay. We develop our framework with general modulation formats in mind, but numerical results are presented for the case where each node transmits using the QPSK constellation. Numerical results with QPSK show that substantially higher rates are achievable with our proposed approach than those achievable by always using a fixed function or adapting the function at the relay but coding over GF(4).