Lower Bound on the Error Rate of Genie-Aided Lattice Decoding

TL;DR

This paper analyzes a genie-aided lattice decoding method, establishing a lower bound on error rates and demonstrating potential gains over traditional decoders, with practical implementation insights for high-dimensional codes.

Contribution

It introduces a lower bound on lattice decoding error rates considering the covering sphere and demonstrates performance gains with a genie-aided approach.

Findings

Genie-aided decoder outperforms one-shot decoder at certain error rates.

Lower bounds on WER are derived using lattice Voronoi regions.

Practical decoder implementation with CRC-embedded polar codes is proposed.

Abstract

A genie-aided decoder for finite dimensional lattice codes is considered. The decoder may exhaustively search through all possible scaling factors $\alpha \in \mathbb{R}$. We show that this decoder can achieve lower word error rate (WER) than the one-shot decoder using $\alpha_{MMSE}$ as a scaling factor. A lower bound on the WER for the decoder is found by considering the covering sphere of the lattice Voronoi region. The proposed decoder and the bound are valid for both power-constrained lattice codes and lattices. If the genie is applied at the decoder, E8 lattice code has 0.5 dB gain and BW16 lattice code has 0.4 dB gain at WER of $10^{-4}$ compared with the one-shot decoder using $\alpha_{MMSE}$. A method for estimating the WER of the decoder is provided by considering the effective sphere of the lattice Voronoi region, which shows an accurate estimate for E8 and BW16 lattice codes. In the case of per-dimension power $P \rightarrow \infty$, an asymptotic expression of the bound is given in a closed form. A practical implementation of a simplified decoder is given by considering CRC-embedded $n=128$ polar code lattice.