Multipoint Code-Weight Sphere Decoding: Parallel Near-ML Decoding for Short-Blocklength Codes

TL;DR

This work addresses the challenge of achieving near-ML decoding for short-blocklength codes in URLLC by introducing a two-stage framework that first applies a low-complexity list decoder with CRC validation and, upon failure, launches Multipoint Code-Weight Sphere Decoding (MP-WSD). MP-WSD exploits code linearity by using a pre-computed code-weight sphere around multiple anchors, performing parallel, small-sphere searches within $\\mathcal{S}_r(oldsymbol{c})$ and selecting the best candidate; this yields near-ML performance with predictable, bounded latency. The approach provides an embarrassingly parallelizable architecture with reduced average complexity compared to high-order OSD or large-list SCL, and its gains are demonstrated across CRC-aided polar, CA-deep polar, and RM codes. The technique offers a practical path to high-throughput, reliable decoding for short packets in URLLC deployments, combining theoretical rigor with empirically verified efficiency.

Abstract

Ultra-reliable low-latency communications (URLLC) operate with short packets, where finite-blocklength effects make near-maximum-likelihood (near-ML) decoding desirable but often too costly. This paper proposes a two-stage near-ML decoding framework that applies to any linear block code. In the first stage, we run a low-complexity decoder to produce a candidate codeword and a cyclic redundancy check. When this stage succeeds, we terminate immediately. When it fails, we invoke a second-stage decoder, termed multipoint code-weight sphere decoding (MP-WSD). The central idea behind {MP-WSD} is to concentrate the ML search where it matters. We pre-compute a set of low-weight codewords and use them to generate structured local perturbations of the current estimate. Starting from the first-stage output, MP-WSD iteratively explores a small Euclidean sphere of candidate codewords formed by adding selected low-weight codewords, tightening the search region as better candidates are found. This design keeps the average complexity low: at high signal-to-noise ratio, the first stage succeeds with high probability and the second stage is rarely activated; when it is activated, the search remains localized. Simulation results show that the proposed decoder attains near-ML performance for short-blocklength, low-rate codes while maintaining low decoding latency.

Figures (7)

• Figure 1: Schematic of the proposed Parallelized WSD. The search starts from multiple initial candidates (red dots) derived from the list decoder, and each path independently converges to a local optimum using the pre-computed code-weight sphere (yellow area).
• Figure 2: BLER performance of CA-polar codes ($N=256, K=16$) with SCL and MP-WSD. Notably, SCL($L=16$) combined with MP-WSD achieves performance comparable to the higher-complexity SCL($L=32$).
• Figure 3: BLER performance of CA-DP codes ($N=128, K=16$) with varying number of initialization paths $|\mathcal{L}_{\text{init}}|$. MP-WSD with $|\mathcal{L}_{\text{init}}|=16$ significantly outperforms the baseline SCL-BPC($L=32$).
• Figure 4: BLER performance of $\mathcal{RM}(128, 29)$ codes. MP-WSD with a low-order OSD($k=2$) initialization approaches the reliability of the high-complexity OSD($k=4$).
• Figure 5: Average normalized complexity for CA-polar codes ($N=256, K=16$). The total complexity adaptively converges to the baseline SCL complexity as SNR increases.

Theorems & Definitions (2)

• Remark 1: Decoding Stability and Refinement
• Remark 2: Difference from Sphere Decoding