Compute-and-Forward: Error-Correcting Codes for Wireless Network Coding on the Physical Layer

By exploiting properties of the wireless medium, network coding may provide even larger throughput increases for wireless networks than possible for wired networks. In particular, simultaneously transmitted signals are seen as noisy linear combinations at the antennas of receivers. It is thus quite tempting to use the physical layer to directly perform network coding "on the air". However, classical communication schemes are ill-suited for carrying out these distributed computations and uncoded transmission accumulates noise with multiple transmissions. In previous work, we have developed a coding technique, computation codes, for reliable, efficient computation over noisy channels. We have also demonstrated how to use these codes to run a network code on finite-field and Gaussian multiple-access networks. In this note, we survey our recent work on reliable computation over channels and show how it can be applied to perform network coding.