Affine Subcode Ensemble Decoding of Linear Block Codes

Abstract: In the short block length regime, ensemble decoding schemes with their inherently parallel structure can improve error correction performance and reduce latency compared to stand-alone suboptimal decoders such as belief propagation (BP). In this work, we introduce affine subcode ensemble decoding (aSCED), which uses an ensemble of decoders operating on linear block codes and both linear and strictly affine subcodes. This generalizes the recently proposed subcode ensemble decoding (SCED), which is restricted to linear subcodes. We derive BP update rules for affine subcodes and show that aSCED simplifies ensemble design compared to SCED, multiple bases BP, and automorphism ensemble decoding. Monte-Carlo simulations of two low-density parity-check codes and two Bose-Chaudhuri-Hocquenghem (BCH) codes demonstrate improved error correction performance of aSCED over competing existing ensemble schemes. Notably, for one BCH code, when combining ensemble design with algorithms for constructing high-performance parity-check matrices, aSCED achieves near-maximum likelihood performance using only 64 BP decoding paths.

• The paper presents aSCED, which generalizes traditional SCED by incorporating affine subcodes to achieve uniform codeword protection with fewer subcodes.
• It employs modified belief propagation with adjusted check-node updates to account for affine offsets, enabling hardware reuse and streamlined decoder design.
• Empirical results show that aSCED outperforms BP, AED, and SCED, approximating maximum likelihood performance with significantly fewer decoding paths.

Affine Subcode Ensemble Decoding of Linear Block Codes: A Technical Analysis

Introduction and Motivation

The persistent challenge of achieving reliable short-block-length error correction with low-complexity iterative decoders such as belief propagation (BP) has motivated significant research in both code and decoder design. While LDPC codes under BP excel at long block lengths, applications demanding ultra-reliable, low-latency communications, e.g., IoT or 6G URLLC, require improved BP performance at shorter codes. Prior efforts have explored optimizing parity-check matrices (PCMs) and enhancing decoding via ensemble approaches—combining multiple suboptimal decoders in parallel.

This paper introduces Affine Subcode Ensemble Decoding (aSCED), which generalizes the previously established Subcode Ensemble Decoding (SCED) by incorporating affine subcodes in addition to linear subcodes for ensemble design. This innovation further simplifies ensemble construction, guarantees uniform codeword coverage, and delivers improved error correction at fixed or reduced decoding complexity compared to existing strategies such as Multiple-Basis BP (MBBP) and Automorphism Ensemble Decoding (AED).

Theoretical Contributions and New Ensemble Construction

Subcode Ensembles: SCED improves BP performance by decoding over proper linear subcodes formed by appending independent rows to the code's PCM. Ensuring a linear covering—every codeword lies in at least one subcode—is essential for full code protection. Typically, at least three proper linear subcodes are needed for such a covering.

Affine Subcodes: aSCED extends this by including cosets (affine subcodes) of linear subcodes. The key theoretical result shows that, unlike with only linear subcodes, a covering with just two proper subcodes is achievable: one linear and one affine (Theorem 1). For a $(k-1)$-dimensional linear subcode, all $2^{k-k'}$ cosets (one linear, $2^{k-k'}-1$ affine) partition the original code.

This reduction in required ensemble size—made possible by affine subcodes—yields significant design and implementation advantages. Notably, aSCED achieves uniform protection: every codeword is included in the same number of decoding paths, a property not generally assured by SCED.

ML Decoding Limit with Affine Paths: The authors further prove that a sufficiently large aSCED ensemble ($2^k$ paths, where $k$ is the code dimension) achieves exact maximum likelihood (ML) performance (Proposition 2), though practical ensembles attain near-ML with far fewer paths.

BP Decoding of Affine Subcodes: Algorithmic Adaptation

Standard BP decoding can be directly applied to linear subcodes, but affine subcodes require a modification: the check-node (CN) update must account for the affine offset (coset syndrome). The update equation at a CN is multiplied by $(-1)^{s_j}$, where $s_j$ is the corresponding entry in the affine syndrome. The variable node (VN) update and initialization remain unchanged. This minimal change allows hardware reuse across all decoders in an aSCED batch, leading to streamlined implementations.

Design Principle for aSCED Batches

A fundamental design guideline is to select a parent subcode with PCM $\bm{H}_s$ and rank deficiency $\Delta$, then include the linear subcode and all $2^\Delta-1$ associated affine cosets as an aSCED batch. Batches can be combined for broader diversity by varying the appended rows, trading off latency/complexity and hardware reuse.

Structural Advantages: Exploiting Subcode Duals for PCM Optimization

Optimized BP performance for short, dense codes hinges on constructing sparse, cycle-free PCMs. The paper leverages the fact that dual codes of subcodes $2^{k-k'}$0 satisfy $2^{k-k'}$1; new, lower-weight dual codewords become available for constructing structured sparse PCMs (ssPCMs). This enrichment of the search space enables building superior PCMs for the ensemble, often with fewer short cycles and reduced edge count.

Numerical Results: Empirical Validation and Performance Leadership

LDPC Codes: For both the 5G $2^{k-k'}$2 and CCSDS $2^{k-k'}$3 LDPC codes, aSCED with $2^{k-k'}$4 (adding a single linearly independent row per batch) provides 0.2–0.4 dB FER improvement over BP, AED, and SCED at target FERs of $2^{k-k'}$5, while requiring fewer designed PCMs and offering hardware reuse within a batch.

Figure 2: Performance of BP-based decoders for the 5G LDPC code $2^{k-k'}$6. aSCED achieves superior FER compared to both AED and SCED at equal complexity (see text for detailed comparison).

BCH Codes: For classical BCH codes $2^{k-k'}$7 and $2^{k-k'}$8, aSCED leverages subcode ssPCMs to fundamentally improve performance:

• Appending single or multiple independent rows of moderate weight to the PCM results in subcodes whose duals contain many new, low-weight codewords. These enable constructing much sparser ssPCMs.
• Monte Carlo results show that even modest aSCED ensembles (e.g., 64 decoding paths) achieve near-ML performance, substantially outperforming MBBP ensembles of equal size in both FER and decoding complexity (TEC). The gap over equal-complexity MBBP is typically 0.1–0.3 dB at practical FERs.

Figure 3: List error rate (LER) of aSCED batches for the BCH $2^{k-k'}$9 code as a function of check node degree. Appending low-weight rows yields strong gains for the ensemble.

• Increasing the rank deficiency $2^{k-k'}-1$0 (i.e., appending more independent rows per batch) reduces FER and approaches ML very rapidly, but the best complexity-performance tradeoff is often found by using multiple batches with $2^{k-k'}-1$1, due to enhanced diversity.

Practical and Theoretical Implications

aSCED provides a formally justified, practically effective method for constructing ensemble decoders with:

• Reduced design and implementation complexity: Fewer, more regular PCMs (hardware reuse within batches).
• Uniform codeword protection: All codewords are equally likely to be decodable in the ensemble.
• State-of-the-art performance: Outperforming other BP-based ensemble decoding (including MBBP and AED), especially at moderate path counts, and attaining ML performance with orders of magnitude fewer paths.
• Fundamental insight: The ability to exploit subcode duals for more effective PCM construction ties together recent advances in short-block-code design and decoding.

The results suggest that aSCED, especially when combined with structured PCM design techniques, is well-suited for advanced communication systems seeking low latency and near-ML error rates at short code lengths. Furthermore, the theoretical results imply opportunities for further exploration—e.g., systematic design of affine subcode ensembles for other code families, optimization of hardware architectures for batch decoding, and generalization to non-binary codes.

Conclusion

aSCED generalizes ensemble decoding by incorporating affine subcodes, providing provable and practical benefits in codeword protection, decoder design, and error-correction capability under low-complexity BP. In empirical tests, aSCED regularly surpasses alternative approaches and, for some codes, attains ML performance using realistic resources. These results highlight new directions for both coding theory and practical decoder engineering—suggesting that affine subcode ensembles offer an essential tool for future high-reliability short-block coding applications.