Low Complexity Coefficient Selection Algorithms for Compute-and-Forward

Compute-and-forward (C&F) has been proposed as an efficient strategy to reduce the backhaul load for distributed antenna systems. Finding the optimal coefficients in C&F has commonly been treated as a shortest vector problem, which is NP-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, and however, numerical results show that the performance degradation is negligible compared with 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 is investigated both theoretically and numerically. The results show that our proposed algorithms achieve better performance-complexity tradeoffs compared with the existing algorithms.