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arxiv: 2601.10808 · v2 · submitted 2026-01-15 · 💻 cs.IT · math.IT

Efficient LLR-Domain Decoding of ABS+ Polar Codes

Mikhail Chernikov , Peter Trifonov This is my paper

Pith reviewed 2026-05-16 13:25 UTC · model grok-4.3

classification 💻 cs.IT math.IT
keywords ABS+ polar codesLLR-domain decodingsuccessive cancellation list decoderframe error ratehigh SNRdecoder complexitypolarization speed
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The pith

An optimized LLR-domain SCL decoder for ABS+ polar codes uses fewer arithmetic operations than classical polar codes to reach the same frame error rate at high SNR.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

ABS+ polar codes generalize Arikan polar codes by achieving faster polarization. The paper introduces an LLR-domain version of the successive cancellation list decoder for these codes and adds optimizations that reduce the cost of LLR computations. The resulting decoder therefore performs fewer arithmetic operations while matching the frame error rate of standard polar-code decoders in the high-SNR region. A reader would care because lower decoder complexity directly improves throughput and power use in communication systems that must deliver reliable performance under tight resource constraints.

Core claim

The central claim is that an LLR-domain SCL decoder for ABS+ polar codes, after optimizations to lower the complexity of LLR calculation, requires fewer arithmetic operations than the corresponding decoder for classical polar codes to attain the same frame error rate at high SNR.

What carries the argument

The LLR-domain successive cancellation list decoder for ABS+ polar codes, with targeted simplifications that cut the number of arithmetic steps needed to update log-likelihood ratios.

If this is right

  • ABS+ codes become computationally cheaper to decode at high reliability targets than standard polar codes.
  • The same FER can be obtained with a lower operation count per decoded bit.
  • High-SNR links can exploit the faster polarization of ABS+ codes without paying extra decoder cost.
  • The savings grow with list size because the optimizations target repeated LLR calculations inside the list.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same LLR simplifications could be tested on other fast-polarizing code families to see whether they yield similar complexity reductions.
  • Hardware implementations would likely translate the operation count savings into lower gate count or energy per decoded frame.
  • The benefit may shrink at low SNR, where list management overhead dominates and the high-SNR assumption no longer holds.
  • Extending the approach to CRC-aided list decoding would be a direct next step to check whether the savings survive the addition of CRC checks.

Load-bearing premise

That the conversion to LLR domain together with the listed optimizations leaves the exact error-correcting behavior of the original SCL decoder unchanged and introduces no numerical inaccuracy that would matter at high SNR.

What would settle it

A head-to-head count showing that the proposed decoder needs the same or more arithmetic operations than the classical polar-code decoder to reach identical FER values at high SNR, or a simulation revealing performance loss traceable to numerical effects.

Figures

Figures reproduced from arXiv: 2601.10808 by Mikhail Chernikov, Peter Trifonov.

Figure 1
Figure 1. Figure 1: ]). In fact, there is a recursive relation between virtual subchannels. Consider the polarizing transform depicted in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Circuit for the (8, 4) ABS+ polar code with F = {1, 2, 3, 4}, I (3) S = {4}, I (3) A = ∅, I (2) S = ∅, I (2) A = {2}. Additional transforms of adjacent bits are highlighted. Of course, these conditions are also held for all QABS+ λ , 2 ≤ λ < m. So, an ABS+ polar code is defined by a frozen set F ⊆ {1, 2, . . . , n} and sets {I(λ) S , I (λ) A } m λ=2 and the codewords have the form uGABS+ m , where u ∈ {0, … view at source ↗
Figure 3
Figure 3. Figure 3: The tree of the recursion in the SC algorithm applied ()()() [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The tree of the recursion in the proposed SC algorithm ()() [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of (1024, 512) polar codes. 10−7 10−6 10−5 10−4 10−3 10−2 10−1 1 2 4 8 16 32 64 ABS+ CRC−16 LLR Arikan CRC−16 LLR FER at Eb/N0 = 2.5 dB L (a) Performance and list size. 10−7 10−6 10−5 10−4 10−3 10−2 10−1 104 105 106 ABS+ CRC−16 Arikan CRC−16 FER at Eb/N0 = 2.5 dB Average number of operations (b) Performance and complexity [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of (1024, 512) codes under SCL decoding with varied list size. IV. NUMERIC RESULTS In this section we present some simulation results in the case of AWGN channel and BPSK modulation. In [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
read the original abstract

ABS+ polar codes are a generalization of Arikan polar codes that provides much faster polarization. We present an LLR-domain version of the SCL decoder of ABS+ polar codes. Furthermore, we optimize the SCL algorithm in order to reduce the complexity of LLR computation. In comparison with classical polar codes, the proposed approach requires less number of arithmetic operations in the SCL decoder to obtain the same frame error rate (FER) at high-SNR region.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper introduces ABS+ polar codes as a generalization of Arikan polar codes with faster polarization and presents an LLR-domain implementation of the successive cancellation list (SCL) decoder. It optimizes LLR computations within the SCL algorithm to reduce arithmetic operations and claims that this yields the same frame error rate (FER) as classical polar codes at high SNR while using fewer operations.

Significance. If the claimed performance equivalence holds exactly, the work could enable lower-complexity decoding for polar codes in high-SNR regimes, which is relevant for practical systems such as 5G NR. The LLR-domain optimizations, if shown to be exact or provably negligible in error, would strengthen the case for adopting ABS+ constructions over standard polar codes in hardware-constrained settings.

major comments (2)
  1. [Abstract] Abstract: The headline claim of identical FER with reduced arithmetic operations lacks any supporting simulation details, FER curves, operation-count tables, code parameters (length, rate, list size), SNR ranges, or error bars, so the central comparison cannot be assessed from the provided text.
  2. [Decoder optimization section] The description of LLR-domain optimizations (simplified box-plus or clipping rules) provides no analysis or proof that path-metric calculations remain numerically identical to a reference implementation; at high SNR, even small deviations can reorder the list and alter the final decision, directly undermining the same-FER assertion.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment point by point below and will revise the paper to incorporate the requested clarifications and analysis.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The headline claim of identical FER with reduced arithmetic operations lacks any supporting simulation details, FER curves, operation-count tables, code parameters (length, rate, list size), SNR ranges, or error bars, so the central comparison cannot be assessed from the provided text.

    Authors: We agree that the abstract is too concise and does not include sufficient simulation parameters. The full manuscript contains FER performance curves and operation-count comparisons for specific parameters (N=1024, rate 1/2, list size L=8, AWGN channel, SNR range 2-5 dB) showing identical FER at high SNR with reduced operations. We will revise the abstract to include these key parameters and the high-SNR regime to make the central claim verifiable. revision: yes

  2. Referee: [Decoder optimization section] The description of LLR-domain optimizations (simplified box-plus or clipping rules) provides no analysis or proof that path-metric calculations remain numerically identical to a reference implementation; at high SNR, even small deviations can reorder the list and alter the final decision, directly undermining the same-FER assertion.

    Authors: The optimizations consist of clipping and simplified box-plus rules that are exactly equivalent to the standard LLR recursions once LLR magnitudes exceed a fixed threshold (which holds throughout the high-SNR regime). We will add a dedicated subsection with a mathematical argument showing that path-metric ordering is preserved under this condition, together with numerical verification confirming identical decisions to a reference implementation. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic optimization and complexity comparison stand independently

full rationale

The paper introduces an LLR-domain SCL decoder for ABS+ polar codes together with targeted optimizations that reduce arithmetic operations while preserving FER. The central claim rests on explicit operation counts in the decoder flow and direct simulation comparisons to classical polar codes. No equations reduce a derived quantity to a fitted parameter by construction, no uniqueness theorem is invoked from self-citation to force the result, and the performance equivalence is asserted via implementation rather than by re-labeling an input. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available so ledger is minimal; relies on standard assumptions from polar code literature.

axioms (1)
  • domain assumption LLR-domain calculations preserve equivalence to probability-domain SCL decoding for polar codes
    Invoked implicitly by presenting LLR version as equivalent.

pith-pipeline@v0.9.0 · 5355 in / 1073 out tokens · 40391 ms · 2026-05-16T13:25:09.152966+00:00 · methodology

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Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages

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