Phase Precoded Compute-and-Forward with Partial Feedback

TL;DR

This paper introduces phase precoding with partial feedback for the compute-and-forward protocol, improving computation rates and demonstrating significant coding gains through simulations with lattice codes.

Contribution

It proposes a novel phase precoding scheme with partial feedback for CoF, jointly optimizing precoders and network equations to enhance performance.

Findings

Phase precoding increases computation rate over original CoF.

Partial feedback enables practical implementation of precoding.

Simulations show significant coding gains with lattice codes.

Abstract

In this work, we propose phase precoding for the compute-and-forward (CoF) protocol. We derive the phase precoded computation rate and show that it is greater than the original computation rate of CoF protocol without precoder. To maximize the phase precoded computation rate, we need to 'jointly' find the optimum phase precoding matrix and the corresponding network equation coefficients. This is 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 solve this problem, we introduce phase precoded CoF with partial feedback. It 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 will then be fedback to the transmitters. A "deep hole phase precoder" is presented as an example of such a scheme. We further simulate our scheme with a lattice code carved out of the Gosset lattice and show that significant coding gains can be obtained in terms of equation error performance.