Phase Precoding for the Compute-and-Forward Protocol

The compute-and-forward (CoF) is a relaying protocol, which uses algebraic structured codes to harness the interference and remove the noise in wireless networks. We propose the use of phase precoders at the transmitters of a network, where relays apply CoF strategy. Firstly, we define the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol. To maximize the phase precoded computation rate, we need to solve a mixed integer programming problem where the optimum precoders should be obtained at the transmitters and the network equation coefficients have to be computed at the relays. To practically solve this problem, we introduce the phase precoded CoF with limited feedback. This is a quantized precoding system, where the relay jointly computes both a quasi-optimal precoder from a finite codebook and the corresponding network equations. The index of the obtained phase precoder within the codebook is then fedback to the transmitters. Secondly, we show that the proposed precoding scheme increases the degrees-of-freedom (DoF) of CoF protocol. This overcomes the limitations on the DoF of the CoF protocol, recently presented by Niesen and Whiting. Using tools from Diophantine approximation and algebraic geometry, we prove the existence of a phase precoder that achieves the maximum DoF when the number of transmitters tends to infinity. Finally, we conduct computer simulations to verify the effectiveness of the proposed phase precoding technique. We further give a new low-complexity method for finding network equations. The gain in computation rate and the higher DoF expected from theoretical results are confirmed by computer simulations with lattice codes.