Efficient Integer Coefficient Search for Compute-and-Forward

TL;DR

This paper introduces efficient algorithms for selecting optimal integer coefficients in compute-and-forward schemes over complex channels, improving decoding performance and computational feasibility.

Contribution

It adapts existing real-valued algorithms to complex channels and extends them to MIMO systems, providing practical solutions for integer coefficient selection.

Findings

Algorithms outperform existing methods in simulation

Significant reduction in computational complexity

Achieve near-optimal decoding performance

Abstract

Integer coefficient selection is an important decoding step in the implementation of compute-and-forward (C-F) relaying scheme. Choosing the optimal integer coefficients in C-F has been shown to be a shortest vector problem (SVP) which is known to be NP hard in its general form. Exhaustive search of the integer coefficients is only feasible in complexity for small number of users while approximation algorithms such as Lenstra-Lenstra-Lovasz (LLL) lattice reduction algorithm only find a vector within an exponential factor of the shortest vector. An optimal deterministic algorithm was proposed for C-F by Sahraei and Gastpar specifically for the real valued channel case. In this paper, we adapt their idea to the complex valued channel and propose an efficient search algorithm to find the optimal integer coefficient vectors over the ring of Gaussian integers and the ring of Eisenstein integers. A second algorithm is then proposed that generalises our search algorithm to the Integer-Forcing MIMO C-F receiver. Performance and efficiency of the proposed algorithms are evaluated through simulations and theoretical analysis.