BP+OSD: Hybrid Decoding for LDPC & Quantum Codes

• BP+OSD is a hybrid decoding framework that combines iterative belief propagation with an ordered statistics decoding stage to improve near-ML performance.
• It leverages accumulated reliability metrics from BP iterations to guide a combinatorial OSD search, yielding significant gains in FER, BER, and LER.
• The method efficiently bridges performance gaps in classical LDPC, polar, and quantum codes while keeping computational complexity tractable.

Belief Propagation with Ordered Statistics Decoding (BP+OSD) refers to a class of hybrid decoding algorithms that augment iterative belief propagation (BP)—or its close variants such as min-sum and neural message-passing—with an ordered statistics decoding (OSD) stage, typically invoked only upon decoding failure. This methodology is employed broadly in classical LDPC, polar, and quantum LDPC codes, where standard BP does not attain near-maximum-likelihood (ML) decoding performance, especially for short or medium block lengths, codes with substantial cycles, and highly degenerate quantum codes. BP+OSD leverages the soft reliability metrics from BP iterations to guide the combinatorial search in the OSD post-processing, yielding substantial gains in frame error rate (FER), bit error rate (BER), and logical error rate (LER), while maintaining practical computational complexity bounds.

1. Algorithmic Framework and Reliability Metrics

BP+OSD algorithms operate in two main stages: an initial BP phase followed (conditionally) by OSD reprocessing. BP is executed for a predetermined number of iterations $I_m$; reliability information from each iteration is accumulated for each variable node (bit or qubit), resulting in a composite metric. For classical codes, the accumulated log-likelihood ratio (LLR) metric is given by

$r_i = \sum_{k=0}^{I_m} \alpha^{I_m - k} L_i^{(k)},$

where $\alpha$ is a decay or weight factor set by simulation, and $L_i^{(k)}$ is the LLR at iteration $k$ with $L_i^{(0)}$ assigned as the channel reliability (e.g., $2y_i/\sigma^2$ for AWGN channels) (0710.5230). In probability-domain BP, the accumulative reliability metric takes the form

$$q_i = \begin{cases} \sum_{k=0}^{I_m} \alpha^{I_m - k} P_i^{(k)}, & \text{if } \hat{c}_i = 1 \ I_m + 1 - \sum_{k=0}^{I_m} \alpha^{I_m - k} P_i^{(k)}, & \text{if } \hat{c}_i = 0 \end{cases}$$

where $P_i^{(k)}$ is the probability output by BP at iteration $k$ (0710.5230). For quantum codes, BP yields quaternary reliability vectors $q_i = (q_i^I, q_i^X, q_i^Y, q_i^Z)$ and a hard-decision run-length $\ell_i$, summarized into sorting functions preserving $X/Z$ error correlations (Kung et al., 2023).

2. Ordered Statistics Decoding (OSD) Mechanisms

Upon BP failure (syndrome non-zero or no valid codeword found), OSD is invoked using the accumulated reliability metrics to rank and permute the columns of the parity-check or generator matrix. Reordered matrices facilitate formation of a least reliable basis (LRB) or most reliable independent basis (MRIB), and systematic conversion via Gaussian elimination. For an OSD of order $p$, all combinations of up to $p$ bit flips in the reliable set are tested to recover valid codewords (or Pauli error patterns in quantum codes). The error candidate is selected via a discrepancy test or weighted Hamming distance, such as

$D(y, \hat{e}) = \sum_{t:\hat{e}_t = 1} |y_t|$

or its generalizations (0710.5230, Mogilevsky et al., 2021).

To avoid exhaustive enumeration, advanced candidate selection methods assign a weight $w_i$ based on reliability rank and order candidates for reduced-complexity evaluation; only the most promising candidates (often as few as one when code minimum distance is high) undergo the full discrepancy test (0710.5230).

3. Performance and Complexity Trade-offs

BP+OSD is empirically shown to achieve substantial improvements relative to BP alone or other iterative schemes:

• For a (504,252) LDPC code, order-0 OSD gains approximately 0.5 dB in FER over BP at 20 iterations, and 0.12 dB over BP at 100 iterations; order-1 and order-2 OSD further improve performance by 0.35–0.5 dB (0710.5230).
• In classical short-block LDPC and polar codes, order-1 BP+OSD yields gains of 0.4–0.5 dB in E_b/N_0 at FER $10^{-3}$, with higher order or list decoding providing further gains, approaching ML performance (Mogilevsky et al., 2021, Urman et al., 2021).
• Quantum codes realize improved thresholds, e.g., 17.5%–17.7% for toric/XZZX and 15.42% for hexagonal planar color codes (Kung et al., 2023, Kung et al., 30 Dec 2024).

BP+OSD complexity comprises reliability sorting ($O(N\log_2 N)$), Gaussian elimination ($O(N^3)$), and candidate pattern testing (combinatorial in code dimension and OSD order). For small orders and when using optimization strategies, the complexity is tractable, and the average required iterations decrease over standalone BP (0710.5230, Zhang et al., 2023). Partial reprocessing in higher-order OSD restricts the candidate pool to least-reliable bits or pairs, retaining most of the performance benefit at reduced computational cost (Mogilevsky et al., 2021, Urman et al., 2021).

4. Generalizations and Adaptations

The BP+OSD approach has been extended to several domains:

• Probability-domain BP and min-sum variants allow accumulative reliability metrics without conversion to LLR, benefiting codes and channels with non-symmetric distributions or hardware-limited precision (0710.5230).
• For polar codes, BP+OSD leverages CRC-aided BP list decoding, using refined soft outputs for MRIB formation. Matrix triangulation and parallel Gaussian elimination are used for efficient candidate testing, achieving ML performance over the binary erasure channel with average complexity $O(N\log N)$ (Urman et al., 2021).
• Quantum codes benefit from quaternary reliability sorting—which preserves $X/Z$ correlations for stabilizer and non-CSS codes—and additional degenerate pruning conditions, such as highly reliable subset reduction, to eliminate unnecessary candidate testing when flips only alter the stabilizer coset (Kung et al., 2023, Kung et al., 30 Dec 2024).
• Neural BP-RNN diversity architectures create ensembles of specialized decoders (according to absorbing set support), optimizing soft outputs for OSD; these architectures are especially effective in the waterfall region for short LDPC codes (Rosseel et al., 2022).

5. Innovative Error Pattern Selection and Degeneracy Pruning

Recent BP+OSD variants focus on further reducing computational cost and improving error correction via:

• Selective candidate evaluation based on reliability scores, reducing arithmetic operations while maintaining optimality (0710.5230).
• Approximate degenerate OSD (ADOSD), where reliable subset reduction and degeneracy conditions allow high-order OSD to be collapsed to order-0 in most practical cases (Kung et al., 30 Dec 2024). For quantum codes, if flipping reliable bits only multiplies the error by a stabilizer, additional candidate evaluation is unnecessary.
• In quantum decoding, degeneracy cutting and syndrome-based preprocessing can further lower latency and complexity by identifying trivial errors locally before entering BP-OSD (Fan et al., 2 Sep 2025, Tsubouchi et al., 9 Oct 2025).

6. Practical Implications and Applications

BP+OSD has a broad range of practical applications:

• For short and medium blocklength LDPC codes, it bridges the suboptimality gap between iterative/BP decoding and ML decoding, offering near-ML performance suitable for applications ranging from 5G/uRLLC systems to deep-space communications (Zhang et al., 2023, 0710.5230).
• In polar and CRC-polar codes, BP+OSD enables low-latency, high-throughput decoding without the large permutation lists required by SCL (Mogilevsky et al., 2021, Urman et al., 2021).
• Quantum error correction for LDPC stabilizer codes (surface, color, BB, hypergraph codes) achieves higher thresholds, lower logical error rates, and scalability for fault-tolerant quantum computing, especially when degeneracy, syndrome measurement errors, and circuit-level noise are present (Kung et al., 2023, Kung et al., 30 Dec 2024, Liang et al., 11 Sep 2024).
• Extensions to machine learning message-passing (e.g., Astra (Maan et al., 13 Aug 2024)) and fully parallel post-processing schemes push BP+OSD beyond classical hardware limitations, reducing latency and scaling to larger code distances.

BP+OSD decoding thus provides an effective framework for bridging the performance gap to maximum-likelihood decoding with controlled complexity, robust error correction, and flexibility for both classical and quantum error-correcting code architectures.