Multiple-Bases BP List Decoder (MBBP-LD)

• The paper’s main contribution is introducing an ensemble BP decoder that leverages multiple distinct code representations to overcome pseudocodeword limitations and approach maximum-likelihood performance.
• The methodology exploits redundant parity-check matrices via cyclic shifts, automorphism groups, and graph permutations across classical, quantum, and polar code families.
• The approach enables parallel decoding with scalable computational complexity, reducing latency while maintaining high decoding reliability in diverse communication contexts.

The Multiple-Bases Belief-Propagation List Decoder (MBBP-LD) is an advanced ensemble decoding methodology that augments the belief-propagation (BP) paradigm through the exploitation of multiple code “bases”, leveraging structural symmetries, automorphism groups, or redundant parity-check ensembles to obtain near-maximum-likelihood performance for short or moderate blocklength codes. MBBP-LD operates by running multiple independent BP decoders, each corresponding to a distinct code representation (basis), in parallel, with the subsequent selection or fusion of candidate codewords via a list-based ML metric or post-processing rule appropriate to the application domain and code type. Its architecture is applicable across classical and quantum LDPC codes, polar subcodes, and dense cyclic codes, and has manifested marked performance gains in both classical and quantum communications contexts (0905.0079, Elkelesh et al., 2018, Geiselhart et al., 2022, Ren et al., 2022, Krieg et al., 2024, Rabeti et al., 4 Nov 2025, Geiselhart et al., 2020).

1. Fundamental Principles and Motivation

The core principle underlying MBBP-LD is exploiting the redundant representation of block codes—cyclic shifts, dual codewords, or automorphisms—to synthesize distinct parity-check matrices or factor-graph embeddings of the same code. Classical BP on a single code graph may suffer from convergence to suboptimal pseudocodewords due to short cycles or unfavorable local structures. By parallelizing BP over multiple such distinct bases, each with its own distinctive pseudocodeword landscape, MBBP-LD delivers a diversity gain: certain error patterns or trapping sets that are uncorrectable under one basis may be efficiently resolved in another (0905.0079, Elkelesh et al., 2018, Krieg et al., 2024, Ren et al., 2022).

Once each basis yields a (possibly different) soft candidate output, MBBP-LD aggregates all candidates into a list, and selects the most likely codeword according to a channel-based metric—typically minimum Euclidean distance in the AWGN case or a model-specific log-likelihood criterion (0905.0079, Elkelesh et al., 2018, Krieg et al., 2024).

2. Construction of Bases and Code Structure Exploitation

Classical Cyclic and LDPC Codes: For cyclic codes, each base is constructed by cyclically shifting a minimum-weight dual codeword, assembling full-rank parity-check matrices with distinct cycle structures (0905.0079). For LDPC codes, bases are formed by selecting collections of low-weight dual-code codewords, ensuring every resultant parity-check matrix remains sparse and of full rank (Krieg et al., 2024).

Quantum LDPC Codes: In quantum CSS codes, tree-based procedures extract maximal cycle-free subgraphs from the code's Tanner graph, yielding multiple redundant parity-check matrices that enhance BP performance when used as bases in an MBBP-LD ensemble (Rabeti et al., 4 Nov 2025).

Polar and RM Codes: In polar codes, affine or stage-wise permutations of the factor-graph (or bit index shuffling under LTA(n) automorphisms) produce distinct graph topologies—each corresponding to a different BP basis (Elkelesh et al., 2018, Ren et al., 2022, Geiselhart et al., 2022). Reed–Muller codes similarly leverage large automorphism groups to develop bases for parallel BP or SCL decoding (Geiselhart et al., 2020).

The following table summarizes base construction strategies for major code families:

Code Family	Basis Construction Method	Reference
Cyclic/LDPC	Cyclic shifts of dual codewords	(0905.0079)
General LDPC	Low-weight dual subsets	(Krieg et al., 2024)
Quantum LDPC (CSS/BB)	Tree-based maximal subgraphs	(Rabeti et al., 4 Nov 2025)
Polar/RM	Permutation of factor graph/affine group	(Elkelesh et al., 2018, Geiselhart et al., 2022, Geiselhart et al., 2020)

3. Algorithmic Structure and Message-Passing Details

Each BP decoder performs standard sum-product or min-sum message passing on its assigned parity-check or factor-graph representation. The main algorithmic steps, canonically defined for classical LDPC codes, are as follows (Krieg et al., 2024, Elkelesh et al., 2018):

1. Initialization: For each basis $j=1,...,L$, initialize the variable-to-check messages with the channel log-likelihood ratios (LLRs).
2. Iterative Updates: For $t=1$ to $N_{\text{it}}$, update check-to-variable and variable-to-check messages using

$$L_{c \to v}^{(t)} = 2\tanh^{-1} \Big( \prod_{v' \in N(c) \setminus v} \tanh\big(\tfrac{L_{v' \to c}^{(t-1)}}{2}\big) \Big)$$

$$L_{v \to c}^{(t)} = L_{\text{ch},v} + \sum_{c' \in N(v) \setminus c} L_{c' \to v}^{(t)}$$

1. Hard Decisions and List Construction: After $N_{\text{it}}$ iterations, each branch produces an output $\hat c_j$; only those passing the syndrome check (i.e., valid codewords under $\mathbf{H}_j$) enter the candidate list $\mathcal L$.
2. ML-in-the-list Selection: The final codeword is selected as

$$\hat c = \arg\min_{c \in \mathcal L} \| y - (1-2c) \|^2$$

for AWGN/BPSK channels, or another metric as appropriate (0905.0079, Krieg et al., 2024, Elkelesh et al., 2018).

The permutation-based extension in polar codes and RM codes involves factor-graph isomorphs, where message-passing proceeds identically but with permuted connectivity, maximizing the diversity in pseudocodeword structure (Elkelesh et al., 2018, Ren et al., 2022, Geiselhart et al., 2022).

4. List Decoding, Candidate Generation, and Post-Processing

If the number of valid codeword candidates is less than the desired list size $t=1$0, MBBP-LD can employ controlled bit-flip expansions: selected least-reliable positions in the current candidate are flipped, and BP is rerun (possibly on one or more bases) to generate additional candidates. Subsequent pruning retains at most the $t=1$1 most-likely codewords. This extension guarantees that MBBP-LD outputs the $t=1$2 highest likelihood candidates discovered in the ensemble (0905.0079).

Beyond the standard ML-in-the-list metric, alternative post-processing—such as frequency-weighted scoring (FWS), which rewards recurrent outputs from different bases and penalizes higher-weight errors—can further improve performance, as demonstrated for quantum LDPC codes (Rabeti et al., 4 Nov 2025).

5. Complexity, Latency, and Hardware Efficiency

The computational complexity of MBBP-LD scales linearly in the number of bases:

$t=1$3

for block length $t=1$4, average variable node degree $t=1$5, $t=1$6 iterations, and $t=1$7 bases, for standard LDPC codes (Krieg et al., 2024). Memory consumption is $t=1$8. The main advantage is that all $t=1$9 decoder branches are independent and can be fully parallelized either at the algorithm or hardware level.

Hardware implementations for polar codes demonstrate that MBBP-LD achieves high throughput with negligible area overhead for additional list size, since the permutation network can be shared among candidate graphs (Ren et al., 2022). Early stopping mechanisms further reduce average latency, as most candidate BP decoders converge before $N_{\text{it}}$0 in moderate-to-high SNR regimes (Geiselhart et al., 2022).

6. Performance Evaluation and Empirical Results

In classical codes, MBBP-LD achieves block error rates (BLER) close to maximum-likelihood decoding. For example, for the [24,12,8] Golay code, MBBP-LD with $N_{\text{it}}$1 achieves performance within 0.05 dB of the ML bound, and with $N_{\text{it}}$2 is indistinguishable from ML (0905.0079). For polar codes, $N_{\text{it}}$3 in the range of 8–32 suffices to close most of the gap to SCL and approach the ML bound, e.g., for $N_{\text{it}}$4, MBBP-LD with $N_{\text{it}}$5 achieves similar performance to SCL-4 at frame error rate $N_{\text{it}}$6 (Ren et al., 2022, Elkelesh et al., 2018).

Quantum LDPC codes also benefit: on the $N_{\text{it}}$7 bicycle code, MBBP-LD reduces logical error rate by up to 40% versus BP-OSD, while retaining linear time complexity (Rabeti et al., 4 Nov 2025).

7. Extensions, Limitations, and Practical Considerations

The gain from MBBP-LD typically saturates for $N_{\text{it}}$8 in short-to-moderate blocklength regimes. The construction of effective bases requires a sufficiently rich supply of low-weight dual codewords or automorphisms; performance diminishes for codes lacking this structure. Memory and compute costs are proportional to $N_{\text{it}}$9, but full parallelization yields low latency (Krieg et al., 2024).

In hybrid MBBP-LD/SCL approaches (notably in automorphism-ensemble decoders for RM codes), constituent decoders can themselves be list decoders, further widening the candidate pool (Geiselhart et al., 2020). Early-stopping, soft-output fusion, and structural frozen-bit optimization are auxiliary enhancements that further improve error rates or integration with outer/concatenated codes (Elkelesh et al., 2018, Geiselhart et al., 2022).

• (0905.0079) Multiple-Bases Belief-Propagation Decoding of High-Density Cyclic Codes
• (Elkelesh et al., 2018) Belief Propagation List Decoding of Polar Codes
• (Geiselhart et al., 2022) A Polar Subcode Approach to Belief Propagation List Decoding
• (Ren et al., 2022) High-Throughput Flexible Belief Propagation List Decoder for Polar Codes
• (Krieg et al., 2024) A Comparative Study of Ensemble Decoding Methods for Short Length LDPC Codes
• (Rabeti et al., 4 Nov 2025) List Decoding and New Bicycle Code Constructions for Quantum LDPC Codes
• (Geiselhart et al., 2020) Automorphism Ensemble Decoding of Reed-Muller Codes