Lattices over Eisenstein Integers for Compute-and-Forward

TL;DR

This paper extends the compute-and-forward protocol to use Eisenstein integer lattice codes, demonstrating improved performance in wireless networks without channel state information at transmitters.

Contribution

It introduces a new class of lattice codes over Eisenstein integers for compute-and-forward, proving their existence and superior performance over traditional integer lattices.

Findings

Eisenstein integer lattices can be constructed with good covering and Poltyrev limit-achieving properties.

Nested lattice codebooks over Eisenstein integers outperform integer lattice codes in outage and error correction.

No additional computational complexity is required for the improved lattice codes.

Abstract

In this paper, we consider the use of lattice codes over Eisenstein integers for implementing a compute-and-forward protocol in wireless networks when channel state information is not available at the transmitter. We extend the compute-and-forward paradigm of Nazer and Gastpar to decoding Eisenstein integer combinations of transmitted messages at relays by proving the existence of a sequence of pairs of nested lattices over Eisenstein integers in which the coarse lattice is good for covering and the fine lattice can achieve the Poltyrev limit. Using this result, we show that both the outage performance and error-correcting performance of nested lattice codebooks over Eisenstein integers surpasses lattice codebooks over integers considered by Nazer and Gastpar with no additional computational complexity.