# Low Complexity Coefficient Selection Algorithms for Compute-and-Forward

## TL;DR

This paper introduces low complexity algorithms for optimal coefficient selection in compute-and-forward systems with complex channels, outperforming existing methods in performance and computational efficiency.

## Contribution

It presents the first low polynomial complexity algorithm for complex-valued channels in compute-and-forward, applicable to lattices over any algebraic integers.

## Key findings

- Algorithms outperform lattice reduction in complexity and performance.
- Numerical results show negligible performance loss with simple linear search.
- Proposed methods are suitable for complex channels and various algebraic integer lattices.

## Abstract

Compute-and-Forward (C&F) has been proposed as an efficient strategy to reduce the backhaul load for the distributed antenna systems. Finding the optimal coefficients in C&F has commonly been treated as a shortest vector problem (SVP), which is N-P hard. The point of our work and of Sahraei's recent work is that the C&F coefficient problem can be much simpler. Due to the special structure of C&F, some low polynomial complexity optimal algorithms have recently been developed. However these methods can be applied to real valued channels and integer based lattices only. In this paper, we consider the complex valued channel with complex integer based lattices. For the first time, we propose a low polynomial complexity algorithm to find the optimal solution for the complex scenario. Then we propose a simple linear search algorithm which is conceptually suboptimal, however numerical results show that the performance degradation is negligible compared to the optimal method. Both algorithms are suitable for lattices over any algebraic integers, and significantly outperform the lattice reduction algorithm. The complexity of both algorithms are investigated both theoretically and numerically. The results show that our proposed algorithms achieve better performance-complexity trade-offs compared to the existing algorithms.