Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Jiajie Xue, Brian M. Kurkoski

TL;DR

The paper addresses finite-dimensional lattice codes over AWGN where the standard MMSE scaling $\alpha_{MMSE}$ can be suboptimal. It introduces a genie-aided exhaustive search decoder that scans all $\alpha\in\mathbb R$ and employs a correctness check, together with a covering-sphere based lower bound and an effective-sphere based WER estimator. Key contributions include a computable bound $P_{e,Dec}>P_{e,cover}$, an accurate estimator $P_{e,effc}$ for sphere-like lattices, and a closed-form asymptotic expression as $P\to\infty$, along with demonstrated gains of $\approx 0.5$ dB (E8) and $\approx 0.4$ dB (BW16) at $P_e=10^{-4}$, plus a practical CRC-embedded $(n=128)$ polar lattice implementation. These results quantify finite-dimensional decoding gains and provide practical pathways to implement improved lattice decoders in power-constrained settings.

Abstract

A genie-aided decoder for finite dimensional lattice codes is considered. The decoder may exhaustively search through all possible scaling factors $α\in \mathbb{R}$. We show that this decoder can achieve lower word error rate (WER) than the one-shot decoder using $α_{MMSE}$ 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 $α_{MMSE}$. 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.

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

TL;DR

The paper addresses finite-dimensional lattice codes over AWGN where the standard MMSE scaling can be suboptimal. It introduces a genie-aided exhaustive search decoder that scans all and employs a correctness check, together with a covering-sphere based lower bound and an effective-sphere based WER estimator. Key contributions include a computable bound , an accurate estimator for sphere-like lattices, and a closed-form asymptotic expression as , along with demonstrated gains of dB (E8) and dB (BW16) at , plus a practical CRC-embedded polar lattice implementation. These results quantify finite-dimensional decoding gains and provide practical pathways to implement improved lattice decoders in power-constrained settings.

Abstract

A genie-aided decoder for finite dimensional lattice codes is considered. The decoder may exhaustively search through all possible scaling factors . We show that this decoder can achieve lower word error rate (WER) than the one-shot decoder using 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 compared with the one-shot decoder using . 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 , 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 polar code lattice.
Paper Structure (12 sections, 1 theorem, 18 equations, 7 figures)

This paper contains 12 sections, 1 theorem, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lattices and lattice codes
  4. Encoding/decoding scheme
  5. Universal error bound for lattice codes
  6. Lower bound on error rate
  7. Decoding strategy
  8. Lower bound on error rate using covering sphere
  9. Estimate error rate using effective sphere
  10. Asymptotic analysis
  11. Numerical Evaluation
  12. Conclusion

Key Result

Theorem 1

Let non-zero $\mathbf x$ be a lattice point of an $n \geq 2$ dimensional lattice $\Lambda$ having covering radius $r_c$ and per-dimensional power $P_{\mathbf{x}}= \|\mathbf{x}\|^2/n$. With the restriction of $r_c^2 < n P_{\mathbf x}$, the probability of word error for the genie-aided decoder on the where if $n$ is odd: and if $n$ is even: with $t= f^2(z) / (2 \sigma^2)$ and $f(z)= \left| \frac{

Figures (7)

  • Figure 1: Relationship among lattice Voronoi region, covering sphere and effective sphere.
  • Figure 2: Encoding/decoding scheme and channel model.
  • Figure 3: Example of decodable region $\mathcal{D}(\mathbf{x})$ using A2 lattice and $\mathbf{x}= (\sqrt{3}, 3)$.
  • Figure 4: Example of rotated $\mathcal{D}_c'$ in 2 dimensions. The vertex of the decodable region is $(-\sqrt{n P_{\mathbf x}},0)$.
  • Figure 5: Asymptotic result of the error bound when $P_{\mathbf x} \rightarrow \infty$ given dimension $n= 8,\ 16$ and $r= 5.4512,\ 6.5552$ respectively. The noise variance $\sigma^2= 1$.
  • ...and 2 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof