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arXiv:2507.16699 · International Symposium on Turbo Codes and Iterative Information Processing

Error Detection Based on Generalized Successive Cancellation List Decoding for Polar Codes

Alexander Sauter, Mustafa Cemil Coşkun, Gianluigi Liva

Published Aug 18, 2025• 0 citations
arXivHTMLPDF

Abstract

Successive cancellation list (SCL) decoding has been widely adopted for polar codes, which allows near maximum likelihood performance with sufficiently large list size. In this work, we show that, if the list size is 2γ, where γ is the fundamental quantity called mixing factor, then a modification to SCL decoding can implement Forney’s generalized decoding rule. Hence, it provides an efficient means to discard unreliable decisions. The performance achieved by short polar codes under the proposed generalized SCL decoding is analyzed via Monte Carlo simulations.

DOI
10.1109/istc65386.2025.11154550

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IEEE Access•Jan 1, 2020•4 citations
Successive cancellation list (SCL) decoding has been widely adopted for polar codes, which allows near maximum likelihood performance with sufficiently large list size. In this work, we show that, if the list size is 2γ, where γ is the fundamental quantity called mixing factor, then a modification to SCL decoding can implement Forney’s generalized decoding rule. Hence, it provides an efficient means to discard unreliable decisions. The performance achieved by short polar codes under the proposed generalized SCL decoding is analyzed via Monte Carlo simulations.
Successive-cancellation list (SCL) decoding is a widely used and studied decoding algorithm for polar codes. For short blocklengths, empirical evidence shows that SCL decoding with moderate list sizes (say,L≤32L \leq 32L≤32) closely matches the performance of maximum-likelihood (ML) decoding. Hashemi et al. proved that on the binary erasure channel (BEC), SCL decoding actually coincides with ML decoding for list sizesL≥2γL \geq 2^{\gamma}L≥2γ, whereγ\gammaγis a new parameter we call the mixing factor. Loosely speaking, the mixing factor counts the number of information bits mixed-in among the frozen bits; more preciselyγ=∣{i∈Fc:i≤max⁡{F}}∣\gamma=\vert\{i\in \mathcal{F}^{c}:i\leq \max\{\mathcal{F}\}\}\vertγ=∣{i∈Fc:i≤max{F}}∣, whereF⊂[n]\mathcal{F}\subset [n]F⊂[n]denotes the set of frozen indices. Herein, we extend the aforementioned result of Hashemi et al. from the BEC to arbitrary binary-input memoryless symmetric channels. Our proof is based on capturing all2γ2^{\gamma}2γdecoding paths that correspond to theγ\gammaγinformation bits appearing before the last frozen bit, and then finding the most-likely extension for each of these paths efficiently using a nearest coset decoding algorithm introduced herein. Furthermore, we present a hybrid successive-cancellation list (H -SCL) decoding algorithm, which is a hybrid between conventional SCL decoding and nearest coset decoding. We believe that the hybrid algorithm can outperform the conventional SCL decoder with lower decoding complexity.
The successive cancellation list decoder (SCL) is an efficient decoder for classical polar codes with low decoding error, approximating the maximum likelihood decoder (MLD) for small list sizes. Here we adapt the SCL to the task of decoding quantum polar codes and show that it inherits the high performance and low complexity of the classical case, and can approximate the quantum MLD for certain channels. We apply SCL decoding to a novel version of quantum polar codes based on the polarization weight (PW) method, which entirely avoids the need for small amounts of entanglement assistance apparent in previous quantum polar code constructions. When used to find the precise error pattern, the quantum SCL decoder (SCL-E) shows competitive performance with surface codes of similar size and low-density parity check codes of similar size and rate. The SCL decoder may instead be used to approximate the probability of each equivalence class of errors, and then choose the most likely class. We benchmark this class-oriented decoder (SCL-C) against the SCL-E decoder and find a noticeable improvement in the logical error rate. This improvement stems from the fact that the contributions from just the low-weight errors give a reasonable approximation to the error class probabilities. Both SCL-E and SCL-C maintain the complexity O(LN log N) of SCL for code sizeNNNand list size L.
For polar codes, successive cancellation list (SCL) decoding algorithm significantly improves finite-length performance compared to SC decoding. SCL-flip decoding can further enhance the performance but the gain diminishes as code length increases, due to the difficulty in locating the first error bit position. In this work, we introduce an SCL-perturbation decoding algorithm to address this issue. A basic version of the algorithm introduces small random perturbations to the received symbols before each SCL decoding attempt, and exhibits non-diminishing gain at large block lengths. Its enhanced version adaptively performs random perturbations or directional perturbation on each received symbol according to previous decoding results, and managed to correct more errors with fewer decoding attempts. Extensive simulation results demonstrate stable gains across various code rates, lengths and list sizes. To the best of our knowledge, this is the first SCL enhancement with non-diminishing gains as code length increases, and achieves unprecedented efficiency. With only one additional SCL-LLL decoding attempt (in total two), the proposed algorithm achieves SCL-2L2L2L-equivalent performance. Since the gain is obtained without increasing list size, the algorithm is best suited for hardware implementation.
Polar codes are the first provably capacity-achieving novel channel codes. Successive cancellation list (SCL) decoding algorithm has a performance close to maximum-likelihood decoding when code length is finite, though the complexity is high. To reduce the complexity of SCL, a partial path expansion with segmented check and pruning SCL (PPE-SCP-SCL) algorithm is proposed in this paper. Based on the successive decoding structure, if the decoding result of a bit is judged to be reliable enough, then a hard decision is made directly and no path expansion is necessary. Moreover, several parity-check points are introduced to perform segmented parity-check during the decoding process, and if the check fails, the current path is directly abandoned. The rationality of the operations is theoretically verified, and numerical simulations also verify that the proposed algorithm can significantly reduce the complexity of decoding with trivial loss of error performance.
In this paper, two improvements for successive cancellation list flip (SCL-Flip) decoding are presented for polar codes. First, a novel bit-selection metric based on path metric (PM) is introduced. With the proposed metric, the improved scheme can have lower complexity and saturate faster than other state-of-the-art decoders. Second, we generalize the SCL-Flip decoding, denoted SCL-Flip-ω, to flip the decision in path competition at most ω times in decoding process. We analyze the frequency of channel-induced error under various circumstances, measure the trade-off between error-correction performance and complexity, and then propose SCL-Flip-2 decoding which is the first SCL-based decoding that allows for multiple bits for flipping to the best of our knowledge. Simulation results reveal that the proposed decoding with a small list size can achieve a better performance than CA-SCL decoding with a large list size while keeping the complexity low.
The successive cancellation list (SCL) decoding is used to achieve good error-correcting performance for practical finite-length polar codes. However, the metric sorting that is repeatedly performed in SCL decoding increases the overall decoding latency as the list size increases, making it difficult to apply the SCL decoding to the applications with limited processing time. To reduce the latency of the metric sorting, this paper proposes a new sorting method derived by analyzing path extension cases encountered in SCL decoding. The proposed method can avoid unnecessary sorting operations by adaptively determining the sorting size, and can be combined with other metric sorting methods. Simulation results show that the proposed method significantly reduces the decoding latency for various list sizes and code rates without degrading the error-correcting performance. The proposed method also becomes more effective as the code rate or the list size increases. Combined with the state-of-the-art sorting method, the proposed method reduces the average SCL decoding latency for a (1024, 512) polar code by 33.1% when the list size is 8.