Finite Dimensional Lattice Codes with Self Error-Detection and Retry Decoding

TL;DR

This work tackles the challenge of finite-dimensional lattice codes where $R<C$ yet ML-like decoding can fail; it proposes a retry decoding framework that adjusts decoding coefficients instead of immediately retransmitting. It introduces CRC-embedded (LBC-embedded) lattice codes to enable physical-layer error detection, enabling controlled retries for both single-user transmission and compute-forward relaying, with shaping requirements for CF to detect linear-combinations without individual messages. A semi-analytic CRC-length optimization balances detection capability against SNR penalties, and offline candidate generation keeps online decoding practical. Numerical results with $E_8$, $BW_{16}$, and Construction D polar lattices show gains up to about $1.31$ dB (128) and $1.08$ dB (256) in CF relaying at $EER=10^{-5}$, demonstrating meaningful reductions in error rate and latency relative to one-shot decoding.

Abstract

Lattice codes with optimal decoding coefficient are capacity-achieving when dimension $N \rightarrow \infty$. In communications systems, finite dimensional lattice codes are considered, where the optimal decoding coefficients may still fail decoding even when $R< C$. This paper presents a new retry decoding scheme for finite dimensional lattice-based transmissions. When decoding errors are detected, the receiver is allowed to adjust the value of decoding coefficients and retry decoding, instead of requesting a re-transmission immediately which causes high latency. This scheme is considered for both point-to-point single user transmission and compute-forward (CF) relaying with power unconstrained relays, by which a lower word error rate (WER) is achieved than conventional one-shot decoding with optimal coefficients. A lattice/lattice code construction, called CRC-embedded lattice/lattice code, is presented to provide physical layer error detection to enable retry decoding. For CF relaying, a shaping lattice design is given so that the decoder is able to detect errors from CF linear combinations without requiring individual users' messages. The numerical results show gains of up to 1.31 dB and 1.08 dB at error probability $10^{-5}$ for a 2-user CF relay using 128- and 256-dimensional lattice codes with optimized CRC length and 2 decoding trials in total.

Figures (12)

• Figure 1: Relationship among lattice Voronoi region, covering sphere (dashed line) and effective sphere (dotted-dashed line).
• Figure 2: System model of multiple access network compute-forward with $L$ users and $J$ relays.
• Figure 3: $P(\alpha)$, $P(\alpha | e_1)$ and $P(\alpha | e_2)$ curve for $E_8$ lattice code with hypercube shaping and code rate $R= 2$. $SNR= 17$dB so that $1- P(\alpha_{MMSE})\approx 10^{-3}$. The $\alpha_{MMSE}$ and search results $\alpha_{1, 1} \cdots \alpha_{3, 4}$ are marked at the corresponding curves.
• Figure 4: $\mathcal{D}(\mathbf{x})$ of $A_2$ lattice and $\mathbf{x}= [\sqrt{3}, 0]^T$ with a decodable $\mathbf{y}_1$ and a non-decodable $\mathbf{y}_2$.
• Figure 5: Example of rotated $\mathcal{D}_c(\mathbf{x})- \mathbf{x}$ in 2 dimensions. The vertex of the decodable region is $(-\sqrt{n P_{\mathbf x}},0)$, where $P_{\mathbf x}$ is the message power. $r_c$ is the covering radius.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• Definition 4
• Definition 5
• Definition 6
• Theorem 1
• proof
• Definition 7