Layered Normalized Min-Sum Decoding

• Layered Normalized Min-Sum is a decoding method that partitions check nodes into non-conflicting groups and updates variable-node beliefs immediately, achieving faster convergence than flooding schedules.
• It utilizes conflict graph coloring to form layers, enabling parallel updates that lower computational complexity while approximating the sum-product algorithm for LDPC and FDPC codes.
• When combined with syndrome-guided bit flipping, LNMS delivers significant coding gains and reduced latency, making it a practical choice for modern error correction in communication systems.

Searching arXiv for recent and foundational papers on Layered Normalized Min-Sum and normalized min-sum decoding. Layered Normalized Min-Sum (LNMS) is a layered version of the standard normalized min-sum decoder in which the normalized min-sum message update rule is retained, but the schedule is changed so that check nodes are partitioned into non-conflicting groups and variable-node beliefs are updated immediately after each group is processed. In the FDPC setting, this layered schedule is constructed by conflict graph coloring and is presented as a low-complexity message-passing decoder that converges faster than flooding-style normalized min-sum while preserving the sign-product and minimum-magnitude approximation characteristic of min-sum decoding (Hosseinzadeh et al., 1 Oct 2025). More broadly, normalized min-sum itself has been described both as a good approximation to the sum-product algorithm and, in a distinct theoretical line, as an algorithm derivable directly from cooperative optimization, a perspective offered as an alternative mathematical basis for its power, limitation, and possible extensions [0609088].

1. Position within the min-sum and belief-propagation family

Normalized min-sum (NMS) is a lower-complexity approximation of belief propagation (BP). In the formulation used for FDPC decoding, the check-node rule replaces the exact BP update by an outgoing sign equal to the product of incoming signs, an outgoing magnitude equal to the minimum magnitude among incoming messages excluding the target edge, and a constant normalization factor $\alpha<1$ to compensate for the optimistic nature of the min approximation; in the reported simulations, $\alpha=0.75$ (Hosseinzadeh et al., 1 Oct 2025).

LNMS does not change that message rule. The defining modification is the schedule. Standard BP with a flooding schedule updates all check nodes using previous variable-to-check messages and only afterward updates all variable nodes, whereas LNMS processes groups of non-conflicting check nodes and immediately injects the resulting check-node information into the connected variable-node log-likelihood ratios (LLRs). This makes new information propagate within the same iteration rather than being delayed by a full half-iteration (Hosseinzadeh et al., 1 Oct 2025).

The theoretical context of NMS is not limited to approximation arguments. A separate account states that the normalized min-sum algorithm can achieve near-optimal performance at decoding LDPC codes, that it was traditionally regarded as a good approximation to the sum-product algorithm used for decoding LDPC codes and Turbo codes, and that it can alternatively be derived directly from cooperative optimization. That account presents cooperative optimization as a newly discovered general method for global/combinatorial optimization and argues that it offers another theoretical basis for the algorithm and a general framework for designing new decoding algorithms [0609088].

2. Layer formation by conflict graph coloring

In the FDPC formulation, LNMS relies on a conflict graph to create the layered schedule. Let $\mathbf{H}$ be the $M \times N$ parity-check matrix. The conflict graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$ has one vertex per check node, $\mathcal{V}=\{1,\dots,M\}$, and an edge between two check nodes if they share at least one variable node. Its adjacency matrix is defined by

$$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$

Thus, $A_{m,k}>0$ indicates that check nodes $m$ and $k$ conflict because they touch at least one common variable (Hosseinzadeh et al., 1 Oct 2025).

Layers are obtained by greedy graph coloring. Check nodes are processed in order $\alpha=0.75$0; for each check node $\alpha=0.75$1, the set of colors already used by its neighbors is

$\alpha=0.75$2

and the assigned color is

$\alpha=0.75$3

All check nodes with the same color form a layer,

$\alpha=0.75$4

where $\alpha=0.75$5 is the number of colors used. Because adjacent vertices never share a color, check nodes inside a layer do not share any variable node and therefore can be updated in parallel safely. In the reported simulations, the number of layers is set to $\alpha=0.75$6 (Hosseinzadeh et al., 1 Oct 2025).

This construction is central to the meaning of “layered” in LNMS. The layering is not an incidental implementation detail but the mechanism that permits sequential layer processing combined with immediate variable-node updates. A plausible implication is that the effectiveness of LNMS depends not only on the min-sum approximation but also on the structure of the parity-check matrix and the resulting conflict graph.

3. Channel model, message updates, and stopping rule

The reported LNMS decoder assumes BPSK transmission over the binary-input AWGN (BI-AWGN) channel:

$\alpha=0.75$7

with $\alpha=0.75$8. The initial channel LLRs are

$\alpha=0.75$9

and all check-to-variable messages are initialized to zero:

$\mathbf{H}$0

For each check node $\mathbf{H}$1, the neighboring variable-node set is

$\mathbf{H}$2

These definitions provide the iterative state on which the layered update operates (Hosseinzadeh et al., 1 Oct 2025).

For each edge $\mathbf{H}$3 with $\mathbf{H}$4, the variable-to-check update is

$\mathbf{H}$5

The check-to-variable update uses normalized min-sum:

$\mathbf{H}$6

where $\mathbf{H}$7 is the normalization factor, the sign is the product of all incoming signs except from $\mathbf{H}$8, and $\mathbf{H}$9 denotes the minimum absolute value among incoming messages excluding the message on edge $M \times N$0. The updated check-to-variable messages for check node $M \times N$1 are collected as

$M \times N$2

and the delta update is

$M \times N$3

After each check node in the current layer is processed, the variable-node beliefs are updated immediately:

$M \times N$4

This immediate a posteriori LLR update is the central layered feature (Hosseinzadeh et al., 1 Oct 2025).

After all layers have been processed in an iteration, the decoder forms hard decisions

$M \times N$5

then computes the syndrome and stops early if the syndrome is zero. The same broad stopping principle appears in related NMS work on BCH codes, where the hard decision is tested by the parity-check condition $M \times N$6 (Hosseinzadeh et al., 1 Oct 2025, Li et al., 2024).

4. Convergence behavior and complexity profile

The reported reason for faster convergence is information reuse within the same iteration. Under flooding schedules, every check node computes messages from stale variable beliefs, and all variable nodes are updated only after every check node has finished. In LNMS, variable beliefs are updated as soon as each layer is done, so later layers in the same iteration use fresher information that already reflects the influence of earlier layers. The stated practical consequence is that LNMS usually converges in fewer iterations than standard flooding BP or flooding NMS and often lowers the average decoding latency (Hosseinzadeh et al., 1 Oct 2025).

The same source states that the proposed LNMS decoder has complexity comparable to NMS decoding, with per-iteration complexity proportional to the number of edges in the Tanner graph. LNMS helps reduce average iterations through early termination, while the SGBF stage adds extra cost only when decoding fails. Since failures become rare at moderate-to-high SNR, the average additional complexity of SGBF is described as small (Hosseinzadeh et al., 1 Oct 2025).

A useful boundary condition appears in related BCH work. That decoder is explicitly characterized as a revised NMS decoder rather than a layered decoder in the classic LDPC sense; it keeps the standard NMS flooding updates and improves performance by an optimized redundant parity-check matrix and multiple automorphisms processed in parallel. This distinction is technically important because it separates schedule changes of the LNMS type from other ways of strengthening NMS, such as automorphism-based input dilation or parity-check-matrix redesign (Li et al., 2024).

5. LNMS for FDPC codes and syndrome-guided bit flipping

In the FDPC setting, LNMS is introduced as the message-passing core of a hybrid decoder that combines layered normalized min-sum with syndrome-guided bit flipping (SGBF). FDPC codes are described as fair-density parity-check codes that have been recently introduced and that demonstrate improved performance compared to low-density parity-check codes standardized in 5G systems particularly in high-rate regimes. The LNMS decoder leverages conflict graph coloring for efficient layered scheduling, while SGBF is activated only in the event of decoding failure after the prescribed maximum number of iterations (Hosseinzadeh et al., 1 Oct 2025).

The SGBF stage uses a reliability metric that combines the magnitude of the final a posteriori LLR and the number of unsatisfied parity checks incident on a bit. For each bit $M \times N$7, the syndrome-derived failure count is

$M \times N$8

and the reliability score is

$M \times N$9

Small LLR magnitude indicates low confidence, while large $\mathcal{G}=(\mathcal{V},\mathcal{E})$0 means that the bit participates in many unsatisfied parity checks; thus a smaller $\mathcal{G}=(\mathcal{V},\mathcal{E})$1 indicates a more suspicious bit. The algorithm selects the $\mathcal{G}=(\mathcal{V},\mathcal{E})$2 bits with the smallest reliabilities. In the simulations, $\mathcal{G}=(\mathcal{V},\mathcal{E})$3 is the main choice, and the paper also studies $\mathcal{G}=(\mathcal{V},\mathcal{E})$4 (Hosseinzadeh et al., 1 Oct 2025).

For each selected position $\mathcal{G}=(\mathcal{V},\mathcal{E})$5, a candidate LLR vector is generated by flipping the sign of that single channel LLR, and each candidate is re-decoded using the same LNMS procedure. Candidate selection is based on the syndrome weight

$\mathcal{G}=(\mathcal{V},\mathcal{E})$6

with

$\mathcal{G}=(\mathcal{V},\mathcal{E})$7

If $\mathcal{G}=(\mathcal{V},\mathcal{E})$8, the corresponding valid codeword is used; otherwise the original LNMS result is retained. The source characterizes SGBF as a selective, syndrome-informed list of single-bit perturbations rather than an exhaustive search (Hosseinzadeh et al., 1 Oct 2025).

6. Experimental conditions, reported gains, and scope limits

The reported simulations are performed over the BI-AWGN channel with BPSK modulation, and performance is measured primarily by frame error rate (FER) versus $\mathcal{G}=(\mathcal{V},\mathcal{E})$9. The evaluated FDPC codes include FDPC$\mathcal{V}=\{1,\dots,M\}$0, FDPC$\mathcal{V}=\{1,\dots,M\}$1, FDPC$\mathcal{V}=\{1,\dots,M\}$2, and FDPC$\mathcal{V}=\{1,\dots,M\}$3. The parity-check matrices are generated using the FDPC construction from the cited work, specifically Algorithm 2 of Moradi et al. The decoder uses a maximum of $\mathcal{V}=\{1,\dots,M\}$4 iterations, $\mathcal{V}=\{1,\dots,M\}$5 layers in LNMS, normalization factor $\mathcal{V}=\{1,\dots,M\}$6, and mainly SGBF list size $\mathcal{V}=\{1,\dots,M\}$7 (Hosseinzadeh et al., 1 Oct 2025).

For comparison, the study includes NMS decoding of 5G-LDPC codes using the MATLAB 5G Toolbox, normalized BP decoding of polar codes, and, for length $\mathcal{V}=\{1,\dots,M\}$8, polar list decoding with list size $\mathcal{V}=\{1,\dots,M\}$9. It also specifies a polar-code scaling factor $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$0 and a 5G-LDPC scaling factor $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$1, and notes that for fair comparison under similar average complexity, the 5G-LDPC iteration budget is adjusted to match the proposed decoder’s average complexity (Hosseinzadeh et al., 1 Oct 2025).

The principal reported performance claims are numerical. For FDPC$$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$2, adding SGBF to LNMS yields about a $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$3 coding gain at FER $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$4 relative to standalone LNMS, using $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$5 and $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$6 iterations. For FDPC$$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$7 with a bit-flipping set size of $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$8 and a maximum of $$\mathbf{A} = \mathbf{H}\mathbf{H}^T - \operatorname{diag}(\mathbf{H}\mathbf{H}^T).$$9 iterations, the proposed decoder achieves approximately a $A_{m,k}>0$0 coding gain over standalone LNMS decoding at a frame error rate of $A_{m,k}>0$1, while providing coding gains of $A_{m,k}>0$2 over other state-of-the-art codes including polar codes and 5G-LDPC codes at the same length and rate and also under belief propagation decoding. The same qualitative behavior is reported for FDPC$A_{m,k}>0$3 and FDPC$A_{m,k}>0$4, with larger $A_{m,k}>0$5 improving FER and diminishing returns from $A_{m,k}>0$6 to $A_{m,k}>0$7 in the explicit FDPC$A_{m,k}>0$8 study (Hosseinzadeh et al., 1 Oct 2025).

A common source of confusion is the tendency to group all strong NMS variants under the label “layered.” The short-BCH work provides a counterexample: despite substantial gains, that decoder is described as a parallelizable flooding-style NMS with multiple permuted inputs per iteration, not as a classical layered decoder. It also warns that for very short BCH codes, undetected errors are non-negligible, especially when NMS is combined with order-1 OSD and the decoder may produce a wrong codeword that still satisfies parity checks. This suggests that the layered schedule of LNMS should be distinguished from post-processing or automorphism-based enhancements, even when all of them are built around normalized min-sum updates (Li et al., 2024).