Inverse compute-and-forward: Extracting messages from simultaneously transmitted equations

We consider the transmission of independent messages over a Gaussian relay network with interfering links. Using the compute-and-forward framework, relays can efficiently decode equations of the transmitted messages. The relays can then send their collected equations to the destination, which solves for its desired messages. Here, we study a special case of the inverse compute-and-forward problem: transmitting the equations to a single destination over a multiple-access channel. We observe that if the underlying messages have unequal rates, the set of possible values of an equation is constrained by the value of the other equations. We use this fact to improve the rate region for downloading equations. Interestingly, the rate region achieved over relay networks with interfering links using a combination of compute-and-forward and inverse compute-and-forward is larger than the best rate region achievable in the absence of interfering links. This verifies that interference may be used to beneficially “mix” messages over a wireless network.