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

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.

💡 Research Summary

The paper addresses the pressing need for near‑maximum‑likelihood (near‑ML) decoding in ultra‑reliable low‑latency communications (URLLC), where short packets and finite‑blocklength effects make conventional high‑complexity decoders impractical. The authors propose a universal two‑stage decoding framework that can be applied to any binary linear block code.

Stage 1 – Low‑Complexity List Decoding.
A conventional low‑complexity list decoder (e.g., successive‑cancellation list (SCL) or ordered‑statistics decoding (OSD)) generates a small list of candidate codewords (L_{\text{init}}={\hat{c}_1,\dots,\hat{c}_L}). If a cyclic redundancy check (CRC) is employed, the most reliable candidate is checked; a successful CRC terminates the process immediately, dramatically reducing average latency. If the CRC fails (or no CRC is present), the decoder proceeds to Stage 2.

Stage 2 – Multipoint Code‑Weight Sphere Decoding (MP‑WSD).
The core contribution is MP‑WSD, a parallelized sphere‑search algorithm that exploits the geometric uniformity of linear codes. The authors pre‑compute a compact set of low‑weight codewords, denoted (S_r(0)), which constitute a Hamming sphere of radius index (r) around the all‑zero codeword. Because any codeword’s neighborhood is a coset of this sphere, the same set can be translated to any anchor (\hat{c}^{(0)}_k) via XOR, yielding the local search space (S_r(\hat{c}^{(0)}_k)=\hat{c}^{(0)}_k\oplus S_r(0)).

Each anchor initiates an independent trajectory. At iteration (i) of trajectory (k), the decoder evaluates a filtered subset (M_k\subset S_r(0)) containing the top‑(m) patterns according to a correlation‑based gain metric
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