Subcode Ensemble Decoding (SCED)

• SCED is a decoding strategy that enhances error correction by operating on diverse subcodes of a parent code.
• It employs parallel decoding paths—such as BP or list decoders—with an ML-in-the-list rule to select the optimal codeword.
• Hierarchical and affine extensions of SCED enable scalability, balancing decoding latency, complexity, and performance gains.

Subcode Ensemble Decoding (SCED) is an ensemble-decoding paradigm in which multiple constituent decoders operate on different subcodes of an original code and the final estimate is selected from the resulting candidate list. In the recent literature, the term is introduced for short block length linear block codes in “Subcode Ensemble Decoding of Linear Block Codes” (Mandelbaum et al., 21 Jan 2025), and it is subsequently specialized to polar codes in “Subcode Ensemble Decoding of Polar Codes” (Lulei et al., 24 Apr 2025). Across these formulations, the central objective is to improve short-block-length decoding by creating decoder diversity through subcodes rather than through only automorphisms, alternate schedules, or equivalent parity-check representations.

1. Definition and conceptual scope

SCED belongs to the broader class of ensemble decoding methods in which a set of independent constituent decoders works in parallel on the received sequence, each proposing a codeword candidate, and the maximum likelihood decision is designated as the decoder output (Krieg et al., 2024). Its distinctive feature is that the diversity source is the code itself: different paths decode different subcodes of the parent code rather than only different representations of the same code.

For binary linear block codes, the parent code is written as

$$\mathcal{C}(n,k)=\{\mathbf{x}\in\mathbb{F}_2^n:\mathbf{H}\mathbf{x}=\mathbf{0}\}=\mathrm{Null}(\mathbf{H}),$$

and a typical SCED path uses a parity-check matrix obtained by appending additional rows,

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$

thereby inducing a proper subcode when the appended row is linearly independent of the rows of $\mathbf{H}$ (Mandelbaum et al., 21 Jan 2025). The final output is selected by an ML-in-the-list rule,

$$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$

where the list $\mathcal L$ contains the valid outputs in the parent code if such outputs exist, and otherwise all path outputs (Mandelbaum et al., 21 Jan 2025).

This construction differs structurally from automorphism ensemble decoding (AED), which changes the received-word representation through automorphisms, and from multiple-bases belief propagation (MBBP), which changes the parity-check representation of the same code. SCED instead changes the decoded constraint set itself. In the short block length regime, this is intended to reduce the gap between stand-alone belief propagation and maximum-likelihood decoding without requiring knowledge of the automorphism group of the code or the NP-complete search for low-weight dual codewords (Mandelbaum et al., 21 Jan 2025).

2. Algebraic basis: subcodes, linear coverings, and hierarchy

A central theoretical notion in SCED is the linear covering property. A set of subcodes $\{\mathcal C_i\}$ is a linear covering of $\mathcal C$ if

$\bigcup \mathcal C_i=\mathcal C.$

This guarantees that every codeword of the parent code is contained in at least one constituent subcode (Mandelbaum et al., 21 Jan 2025, Jo et al., 9 Feb 2026). In the original linear-block-code formulation, the smallest number of proper linear subcodes needed to cover a binary linear code is $3$, and a constructive covering is obtained by choosing two appended rows $\mathbf h_1,\mathbf h_2$ that are linearly independent of the rows of $$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$0, then setting

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$1

The corresponding three subcodes collectively satisfy the covering property (Mandelbaum et al., 21 Jan 2025).

This three-way relation became the starting point for hierarchical generalizations. In Hierarchical Subcode Ensemble Decoding (HSCED), if

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$2

with

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$3

then $$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$4. Recursively applying the same construction at every node of a depth-$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$5 tree produces $$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$6 leaf subcodes while preserving

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$7

This gives a systematic way to scale the ensemble size while preserving linear covering (Jo et al., 9 Feb 2026).

A later development, affine SCED (aSCED), relaxes the restriction to linear subcodes by allowing strictly affine subcodes

$$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$8

In that formulation, the minimum covering number drops from $$\mathbf{H}_\ell= \begin{pmatrix} \mathbf{H}\ \mathbf{h}_\ell \end{pmatrix},$$9 to $\mathbf{H}$0, but only if the covering consists of one linear subcode and one affine subcode (Mandelbaum et al., 8 Apr 2026). This suggests that the original linear-only SCED occupies one point in a larger family of covering-based ensemble constructions.

3. Polar-code specialization through pre-transformations

The polar-code version of SCED is motivated by the short block length regime in which pre-transformed polar codes together with successive cancellation list (SCL) decoding possess excellent error correction capabilities, but practical list size is limited due to the suboptimal scaling of the required area in hardware implementations (Lulei et al., 24 Apr 2025). AED can improve performance for a fixed list size by running multiple parallel SCL decodings on permuted received words, yet AED is limited to appropriately designed polar codes. The stated purpose of polar SCED is to avoid that design constraint by using multiple decodings in different subcodes whose selected family jointly covers the original code (Lulei et al., 24 Apr 2025).

The abstract of the polar paper states that polar SCED is realized by expressing polar subcodes through suitable pre-transformations (PTs), and that it introduces a framework classifying pre-transformations for pre-transformed polar codes based on their role in encoding and decoding. Within this framework, it proposes a new type of PT enabling SCED for polar codes, analyzes its properties, and discusses how to construct an efficient ensemble (Lulei et al., 24 Apr 2025). The graphical notation reported for that work associates an input vector $\mathbf{H}$1, pre-transformations $\mathbf{H}$2 and $\mathbf{H}$3, a mother code $\mathbf{H}$4, a transformed vector $\mathbf{H}$5, a codeword $\mathbf{H}$6, and a highlighted region $\mathbf{H}$7, which is consistent with a pre-transformed polar-subcode construction (Lulei et al., 24 Apr 2025).

A related polar-subcode viewpoint had already been used in CRC-aided polar decoding. For an invertible binary matrix $\mathbf{H}$8, one may write

$\mathbf{H}$9

and

$$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$0

thereby reinterpreting transformed representations as polar subcodes (Geiselhart et al., 2022). This suggests a concrete algebraic route by which PT-induced polar subcodes can be used as SCED constituents, especially for pre-transformed constructions such as CRC-aided polar codes.

4. Decoding architectures and neighboring ensemble methods

In practice, SCED has been instantiated primarily with BP-type constituent decoders for linear block codes and with transformed polar subdecoders in iterative/list polar decoding. In the original linear-block-code setting, each path is a BP decoder on either the parent code or a subcode PCM, and all path outputs are combined through ML-in-the-list selection (Mandelbaum et al., 21 Jan 2025). In HSCED for polar codes, the ensemble consists of one BP decoder for the base graph plus one BP decoder per leaf subcode, and at depth $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$1 a total of $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$2 parallel BP decoders operate on distinct subcodes (Jo et al., 9 Feb 2026).

SCED is adjacent to several other ensemble-decoding realizations. MBBP uses multiple parity-check matrices of the same code; AED applies automorphisms; scheduling ensemble decoding changes the update order; noise-aided ensemble decoding perturbs the channel observations; saturated BP saturates a small set of unreliable positions (Krieg et al., 2024). SCED is closest in spirit to MBBP and AED, but its path diversity comes from subcode restriction rather than from equivalent Tanner graphs or code symmetries.

The distinction is especially sharp in polar coding. The polar SCED abstract positions the method against AED by noting that AED runs multiple parallel SCL decodings on permuted received words, whereas SCED uses multiple decodings in different subcodes and does not impose the same design constraints (Lulei et al., 24 Apr 2025). Closely related automorphism work on shortened polar codes shows that the surviving automorphism group is “limited but non-empty,” that valid permutations must preserve the shortening set through

$$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$3

and that ensemble gains can coexist with lower decoding latency (Pillet et al., 2024). SCED departs from that symmetry-based route by using subcodes as the primary source of diversity.

5. Performance, complexity, and empirical trade-offs

The first SCED simulations on LDPC codes report improved decoding performance compared to stand-alone decoding and automorphism ensemble decoding for three LDPC codes (Mandelbaum et al., 21 Jan 2025). For the $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$4 code, SCED-11 gives about $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$5 dB gain over stand-alone MSA and about $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$6 dB gain over AED-11 at FER $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$7; SCED-43 reduces the gap to OSD-4 to about $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$8 dB (Mandelbaum et al., 21 Jan 2025). For the irregular PEG-LDPC code $$\hat{\mathbf x}:=\arg\max_{\mathbf x\in\mathcal L}L(\mathbf y\mid \mathbf x),$$9, SCED-11 yields approximately $\mathcal L$0 dB gain over both MSA and SPA (Mandelbaum et al., 21 Jan 2025).

The same paper emphasizes a latency advantage: if $\mathcal L$1 is the number of iterations used by path $\mathcal L$2, then worst-case latency scales as

$\mathcal L$3

whereas total complexity scales as

$\mathcal L$4

Because the paths run in parallel, SCED-11 with $\mathcal L$5 per path can outperform equal-complexity stand-alone MSA with $\mathcal L$6 while having significantly reduced latency (Mandelbaum et al., 21 Jan 2025).

Empirically, the role of linear covering is more nuanced than its algebraic importance might suggest. In the 5G-code construction study, a Bernoulli-sampled maximum-coverage ensemble that did not form a linear covering left $\mathcal L$7 of 10,000 random codewords outside all three auxiliary subcodes, yet all three tested design strategies delivered roughly the same FER gain, about $\mathcal L$8 dB at FER $\mathcal L$9 over stand-alone SPA (Mandelbaum et al., 21 Jan 2025). This suggests that graph diversity and complementarity on dominant BP failures can be as important as exact codeword coverage.

Hierarchical and affine extensions strengthen the performance picture. HSCED delivers significant block-error-rate improvements over standard BP and conventional SCED under the same decoding-latency constraint, and at $\{\mathcal C_i\}$0 it closely approaches SCL32 reliability while keeping worst-case latency fixed at $\{\mathcal C_i\}$1 cycles (Jo et al., 9 Feb 2026). Affine SCED improves error correction performance over competing existing ensemble schemes on two LDPC codes and two BCH codes, and for one BCH code it achieves near-maximum likelihood performance using only $\{\mathcal C_i\}$2 BP decoding paths (Mandelbaum et al., 8 Apr 2026).

6. Developments, misconceptions, and open directions

A recurrent misconception is to treat SCED as merely another name for MBBP or AED. The literature does not support that equivalence. MBBP keeps the code fixed and varies the parity-check basis; AED keeps the code fixed and varies the received-word representation; SCED changes the decoded code constraints through linear or affine subcodes (Krieg et al., 2024, Mandelbaum et al., 21 Jan 2025, Mandelbaum et al., 8 Apr 2026). A second misconception is that linear covering alone determines practical performance. The early LDPC experiments show that LC is theoretically central but not always the empirically dominant design criterion for finite-iteration BP (Mandelbaum et al., 21 Jan 2025).

For polar codes, a further caution is that not every valid transformed representation is equally decoder-friendly. In a related CRC-aided polar-subcode BP-list setting, LTA permutations perform best, stage-shuffle permutations lose about $\{\mathcal C_i\}$3 dB at BLER $\{\mathcal C_i\}$4, and full $\{\mathcal C_i\}$5 random permutations lose about another $\{\mathcal C_i\}$6 dB (Geiselhart et al., 2022). This suggests that polar SCED must balance ensemble richness against the quality of each constituent subcode.

The current research trajectory moves in three directions. One is scalability: HSCED replaces a flat three-subcode template by recursive $\{\mathcal C_i\}$7-leaf constructions with preserved covering (Jo et al., 9 Feb 2026). A second is generalization from linear to affine subcodes, which simplifies covering, equalizes protection across codewords, and reuses the same graph inside a coset batch through syndrome-dependent sign changes (Mandelbaum et al., 8 Apr 2026). The third is polar specialization: the polar SCED abstract points toward a PT-based classification framework, a new type of PT enabling SCED for polar codes, and efficient-ensemble construction for pre-transformed polar codes (Lulei et al., 24 Apr 2025).

Taken together, these developments position SCED as a general short-block-length ensemble-decoding methodology in which coverage, constituent-graph diversity, and hardware-parallel complexity must be designed jointly. The polar-code branch of that program is explicitly framed around pre-transformations and polar subcodes, while the broader SCED family continues to expand through hierarchical and affine generalizations (Lulei et al., 24 Apr 2025, Jo et al., 9 Feb 2026, Mandelbaum et al., 8 Apr 2026).