Belief Propagation List (BPL) Decoder

• The Belief Propagation List (BPL) Decoder is an iterative method that deploys multiple BP decoders on distinct permuted factor graphs to enhance error-correcting performance.
• Its strategy of using diverse graph permutations boosts decoding diversity, reduces error floors, and approaches maximum-likelihood bounds for polar, LDPC, and quantum codes.
• Integration of CRC checks, soft-decision updates, and postprocessing like OSD enables high-throughput, low-latency decoding optimized for hardware implementations.

A Belief Propagation List (BPL) decoder is a modern iterative decoding architecture that seeks to approach or match the error-correcting performance of maximum-likelihood (ML) or CRC-aided successive cancellation list (CA-SCL) decoding for polar codes and related linear block codes. A BPL decoder instantiates multiple belief-propagation (BP) decoders, each operating on a uniquely permuted version of the code's factor graph or parity-check structure, then aggregates candidate codeword decisions into a list and selects the optimal output using a suitable metric or outer code constraint. BPL decoders exploit the inherent parallelism of BP, the diversity induced by graph permutations, and (in advanced forms) soft-decision processing or post-list reprocessing, enabling high-throughput, low-latency, and near-ML performance, particularly for moderate blocklengths and hardware-oriented deployments.

1. Core Principles and Algorithmic Foundations

The foundational concept of Belief Propagation List decoding is the parallel deployment of $L$ independent BP decoders, each mapping the channel observation $\mathbf{y}$ onto an $n$-stage permuted factor graph $\mathcal{G}_\pi$. For polar codes, this leverages the property (Arıkan's theorem) that stage-order permutations of the factor graph $G_N$ preserve the underlying code but induce distinct cycle structures (Elkelesh et al., 2018, Elkelesh et al., 2018, Geiselhart et al., 2020). Each BP instance iteratively propagates log-likelihood ratios (LLRs) using min-sum or sum-product updates along the permuted graph:

• The "boxplus" or $\mathrm{atanh}$ sum at check nodes,
• The appropriate variable node update (with temporary hard decisions as needed),
• Enforcement of frozen constraints or outer CRC constraints on designated nodes.

Upon convergence or maximum iteration, each BP decoder outputs a candidate codeword $\hat{\mathbf{x}}^{(l)}$, and the set $$\{\hat{\mathbf{x}}^{(1)}, \ldots, \hat{\mathbf{x}}^{(L)}\}$$ forms the BPL candidate list. Final selection is performed by evaluation of a metric (e.g., Euclidean distance metric $\|\mathbf{y} - \hat{\mathbf{x}}\|^2$), and, in CRC-aided settings, filtering by CRC validity (Geiselhart et al., 2020, Elkelesh et al., 2018).

For LDPC and quantum codes, analogous BPL mechanisms instantiate BP on alternative parity-check bases (e.g., through redundant subtree checks or multiple bases), with list-aggregation rules possibly incorporating candidate reliability and frequency scoring (Rabeti et al., 4 Nov 2025, Bocharova et al., 2017).

2. Permuted Factor Graphs and List Diversity

Stage-order permutations and associated factor-graph diversity are central to BPL's error performance. Each permutation $\pi \in S_n$ (for length $\mathbf{y}$0 polar codes) defines a unique FG topology with different cycle configurations and fixed-point properties. Since BP decoding can fail due to graph-induced trapping sets or specific error pattern alignments, running multiple decoders across $\mathbf{y}$1 diverse permutations raises the probability that at least one BP instance converges to the transmitted codeword.

Permutation selection strategies include:

• Uniform random sampling of $\mathbf{y}$2 possible stage orders (Elkelesh et al., 2018, Elkelesh et al., 2018, Ren et al., 2022),
• Systematic selections based on maximizing pairwise Hamming distance between permutations,
• Greedy sequential generation (SG) based on conditional independence and error-coverage statistics (Ren et al., 2022),
• Algebraic affine permutations for subcoded polar or CRC-polar codes (Geiselhart et al., 2022).

The resulting BPL error-rate is theoretically proven to approach the ML bound exponentially in $\mathbf{y}$3 under reasonable independence assumptions (Elkelesh et al., 2018, Elkelesh et al., 2018, Geiselhart et al., 2020).

3. Outer Constraints: CRC-Aid, Soft CRC, and Subcodes

Aggregation performance improves markedly when an outer code or constraint is applied:

• CRC-aided BPL (CA-BPL) uses CRC checks to prune invalid candidates; early stopping upon CRC success further reduces average list size and latency (Geiselhart et al., 2020, Elkelesh et al., 2018, Mogilevsky et al., 2021).
• Advanced variants integrate the CRC as a soft-in/soft-out code within BP by applying BCJR or SPA decoding over the CRC trellis or parity-check matrix, feeding back extrinsic CRC LLRs into the polar graph after each BP sweep. This enables genuine extrinsic error correction beyond mere detection (Geiselhart et al., 2020).
• For polar subcodes, permutations correspond to nontrivial affine transforms, yielding a larger effective permutation space and improved diversity, especially for dynamic-frozen or CRC-constraint codes (Geiselhart et al., 2022).

The combination of iteratively-updated BP, CRC validation, and metric-based selection allows BPL decoders to approach or match the performance of CA-SCL for moderate list sizes and iterations (Geiselhart et al., 2020, Geiselhart et al., 2022, Raviv et al., 2023).

4. List Management, Candidate Selection, and Postprocessing

The BPL workflow includes:

• Parallel or sequential BP decodings across the selected permutation set.
• For each BP instance, optional early stopping on CRC success, or forced exit after $\mathbf{y}$4 iterations.
• Aggregation of candidates, pruning by CRC outcome and path metric (typically the minimum Euclidean distance, or in certain constructions, weighted Hamming distance to the received vector or LLR-based reliability metrics (Geiselhart et al., 2022, Rabeti et al., 4 Nov 2025)).
• In enhanced designs, such as BP-OSD (Mogilevsky et al., 2021), ordered statistics decoding (OSD) reprocessing is applied within each BP candidate: a reliability ordering is computed, the most reliable basis extracted, and all possible patterns up to weight $\mathbf{y}$5 are tested to form a local superlist, selecting the candidate closest to $\mathbf{y}$6. This can provide $\mathbf{y}$7--$\mathbf{y}$8 dB additional gain with only moderate increase in complexity.

For quantum LDPC codes, list aggregation is supplemented by frequency-weighted scoring (FWS)—choosing the candidate that appears most frequently across the multiple base decodings relative to its Hamming weight (Rabeti et al., 4 Nov 2025).

5. Hardware Considerations and Latency/Complexity

BPL decoders are highly suitable for high-throughput hardware owing to the inherent parallelism of BP and the independence of list branches:

• Each iteration per decoder entails $\mathbf{y}$9 operations for polar codes (Elkelesh et al., 2018, Ren et al., 2022, Geiselhart et al., 2020).
• The aggregate effort is $n$0, but with proper early stopping, average iterations can be significantly lower than the worst case.
• Recent hardware implementations realize on-the-fly permutation generation via networks of basic subroutings rather than a full crossbar, yielding compact, low-area, and high-throughput designs. For example, a length-1024, rate-1/2 BPL decoder with $n$1 achieves $n$2 Gbps and area efficiency $n$3 Gbps/mm$n$4 at SNR $n$5 dB, outperforming BP flip and SCL decoders by factors of $n$6 and $n$7, respectively (Ren et al., 2022).
• Sequential BPL strategies can reuse a single BP core for all permutations, with permutation latency negligible compared to the BP decoding time for $n$8 (Ren et al., 2022).

At high SNR, early stopping becomes efficient—many branches terminate after a few iterations, and complements such as a "gate" BPL agent or early-exit designs further reduce average latency (Raviv et al., 2023, Geiselhart et al., 2022).

6. Extended BPL Variants and Related Techniques

Several advanced BPL forms and related frameworks have been developed:

• Weighted BPL (ensemble of weighted BP decoders, each trained or specialized to subsets of the input space, possibly indexed by the CRC remainder), with learned weights and CRC gating to improve ensemble diversity (Raviv et al., 2023).
• BP-LED (Belief Propagation List-Erasure Decoding), primarily for LDPC codes, where after BP failure, a small set of unreliable bits is erased and an ML-based list decoder for the induced erasure channel is used. Each candidate codeword is compared by distance to the original channel output (Bocharova et al., 2017).
• Neural-BPL: Neural BP decoders can utilize list-decimation strategies, where the least reliable bits are forcibly set (decimated) to both values, expanding the candidate set along an "NBP-list" tree. Subsequent learned decimation steps refined by a neural network prune the list size, improving block error rates for short codes (Buchberger et al., 2020).
• Quantum BPL: Multiple bases BP decoders leveraging redundant checks (maximum cycle-free subgraphs) in QLDPC codes, with frequency-based list aggregation or standard least-metric scoring, achieve lower logical error rates at linear (in $n$9) complexity (Rabeti et al., 4 Nov 2025).

7. Performance, Trade-offs, and Theoretical Bounds

Empirical and theoretical evaluations demonstrate the following:

• For (1024,512) polar codes, BPL with $\mathcal{G}_\pi$0–$\mathcal{G}_\pi$1 approaches within $\mathcal{G}_\pi$2–$\mathcal{G}_\pi$3 dB of the ML bound and matches SCL-32 decoding at moderate iteration counts ($\mathcal{G}_\pi$4–$\mathcal{G}_\pi$5) (Elkelesh et al., 2018, Elkelesh et al., 2018, Ren et al., 2022).
• With CRC-aid and list sizes $\mathcal{G}_\pi$6–$\mathcal{G}_\pi$7, CA-BPL is within $\mathcal{G}_\pi$8–$\mathcal{G}_\pi$9 dB of CA-SCL on standardized 5G polar codes, with hardware latency reductions of $G_N$0–$G_N$1, especially at high SNR (Geiselhart et al., 2020, Geiselhart et al., 2022).
• Gains saturate with increasing $G_N$2 beyond $G_N$3; further improvements are realized by adding weighted message updates, BP-based frozen bit reordering, or OSD postprocessing.
• For LDPC and quantum codes, list-based BP (including BPL-LED and MBBP-LD) closes a significant fraction of the gap to near-ML decoding with linear-time scaling (Bocharova et al., 2017, Rabeti et al., 4 Nov 2025).

A typical performance comparison (for a (1024,512) polar code, FER $G_N$4):

Decoder	$G_N$5 (dB)
SC	4.75
BP (single)	4.55
BPL ($G_N$6)	4.25
BPL ($G_N$7)	4.15
SCL ($G_N$8)	4.10
ML Bound	4.05

(Elkelesh et al., 2018, Geiselhart et al., 2020, Geiselhart et al., 2022, Ren et al., 2022)

The exponential approach of BPL FER to the ML bound with $G_N$9 has been theoretically established, and practical results confirm that BPL is a viable, efficient, and high-performance decoding solution for polar codes and beyond.

---

References:

(Elkelesh et al., 2018, Elkelesh et al., 2018, Geiselhart et al., 2020, Geiselhart et al., 2022, Raviv et al., 2023, Ren et al., 2022, Mogilevsky et al., 2021, Bocharova et al., 2017, Buchberger et al., 2020, Rabeti et al., 4 Nov 2025)