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International Symposium on Turbo Codes and Iterative Information Processing

Neural Network Decoding of Polar Codes with Large Kernels

Valerio Bioglio, Gastón De Boni Rovella, Meryem Benammar

Published Aug 18, 2025• 1 citation
Publisher

Abstract

Polar codes with large kernels offer improved error-correction performance but suffer from high decoding complexity due to costly marginalization operations. In this work, we propose neural network (NN)-based approximations of these marginalization functions, enabling efficient decoding for arbitrary linear kernels. We show that training on all-zero partial codewords is sufficient for linear kernels, significantly reducing the dataset size and allowing for smaller, faster NN models. Our method is evaluated within a Successive Cancellation (SC) decoder, being compatible with any blocklength, code rate, or SNR. Experimental results for kernel size 16 demonstrate strong performance and substantial complexity savings over exact marginalization.

DOI
10.1109/istc65386.2025.11154538

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Polar codes with large kernels offer improved error-correction performance but suffer from high decoding complexity due to costly marginalization operations. In this work, we propose neural network (NN)-based approximations of these marginalization functions, enabling efficient decoding for arbitrary linear kernels. We show that training on all-zero partial codewords is sufficient for linear kernels, significantly reducing the dataset size and allowing for smaller, faster NN models. Our method is evaluated within a Successive Cancellation (SC) decoder, being compatible with any blocklength, code rate, or SNR. Experimental results for kernel size 16 demonstrate strong performance and substantial complexity savings over exact marginalization.
Polar codes using successive-cancellation decoding always suffer from high latency for its serial nature. Fast simplified successive-cancellation decoding algorithm improves the situation in theoretically but not performs well as expected in practical for the workload of nodes identification and the existence of many short blocks. Meanwhile, Neural network (NN) based decoders have appeared as potential candidates to replace conventional decoders for polar codes. But the exponentially increasing training complexity with information bits is unacceptable which means it is only suitable for short codes. In this paper, we present an improvement that increases decoding efficiency without degrading the error-correction performance. The long polar codes are divided into several sub-blocks, some of which can be decoded adopting fast maximum likelihood decoding method and the remained parts are replaced by several short codes NN decoders. The result shows that time steps the proposed algorithm need only equal to 79.8% of fast simplified successive-cancellation decoders require. Moreover, it has up to 21.2 times faster than successive-cancellation decoding algorithm. More importantly, the proposed algorithm decreases the hardness when applying in some degree.
Theoretically, conventional decoders for polar codes can be entirely replaced by neural network (NN) with enough size and enough training, which called NN decoder. But the exponentially increasing training complexity becomes unacceptable when information length increases, which means only decoders for short codes can be trained practically. However, a successive cancellation (SC) decoder for long polar codes can be divided into several SC decoders for short codes, which can be replaced by several short codes NN decoders, then the whole decoder becomes our NN aided SC (NNSC) decoder. Besides, we defined Universal Set of NN, which can be combined into NNSC decoders for any long polar codes. In this paper, the main purpose of constructing NNSC decoder is increasing decoding efficiency of polar codes by taking advantage of NN, and in the meantime ensuring an acceptable performance compared to conventional decoding algorithms.
The traditional fast successive-cancellation (SC) decoding algorithm can effectively reduce the decoding steps, but the decoding adopts a sub-optimal algorithm, so it cannot improve the bit error performance. In order to improve the bit error performance while maintaining low decoding steps, we introduce a neural network subcode that can achieve optimal decoding performance and combine it with the traditional fast SC decoding algorithm. While exploring how to combine neural network node (NNN) with R1, R0, single-parity checks (SPC), and Rep, we find that the decoding failed sometimes when the NNN was not the last subcode. To solve the problem, we propose two neural network-assisted decoding schemes: a key-bit-based subcode NN-assisted decoding (KSNNAD) scheme and a last subcode NN-assisted decoding (LSNNAD) scheme. The LSNNAD scheme recognizes the last subcode as an NNN, and the NNN with nearly optimal decoding performance gives rise to some performance improvements. To further improve performance, the KSNNAD scheme recognizes the subcode with a key bit as an NNN and changes the training data and label accordingly. Computer simulation results confirm that the two schemes can effectively reduce the decoding steps, and their bit error rates (BERs) are lower than those of the successive-cancellation decoder (SCD).
Deep learning recently shows strong competitiveness to improve polar code decoding. However, suffering from prohibitive training and computation complexity, the conventional deep neural network (DNN) is only possible for very short code length. In this paper, the main problems of deep learning in decoding are well solved. We first present the multiple scaled belief propagation (BP) algorithm, aiming at obtaining faster convergence and better performance. Based on this, deep neural network decoder (NND) with low complexity and latency, is proposed for any code length. The training only requires a small set of zero codewords. Besides, its computation complexity is close to the original BP. Experiment results show that the proposed (64,32) NND with 5 iterations achieves even lower bit error rate (BER) than the 30-iteration conventional BP and (512, 256) NND also outperforms conventional BP decoder with same iterations. The hardware architecture of basic computation block is given and folding technique is also considered, saving about 50% hardware cost.
Neural Network Decoders (NNDs) have been recently considered and investigated as an alternative to the classical algorithms for decoding Polar Codes. In particular, a class of partitioned decoders has been proposed where modified Successive Cancellation (SC) or Belief Propagation (BP) algorithms exploit the neural network decoding of short sublocks. Although NNDs work as one-shot decoders with small decoding latency, the performance of these decoders, in comparison with classic approaches, still suffers from some losses; this fact limits their application to codes with short/medium length. In this article we introduce the novel Neural Successive Cancellation List (NSCL) decoder, as a partitioned version of the SC decoder operating on a list of L multiple decoding candidates that are built with the decoded sub-blocks obtained from a set of NNDs working in parallel at each partition. This decoder significantly improves the BER performance of the non-list version under the same code length. Hence, it reduces the intrinsic loss of performance given by the use of NNDs and allows the application of NN-based decoding to longer codewords.
A method for efficiently successive cancellation (SC) decoding of polar codes with high-dimensional linear binary kernels (HDLBK) is presented and analyzed. We devise a lll-expressions method which can obtain simplified recursive formulas of SC decoder in likelihood ratio form for arbitrary linear binary kernels to reduce the complexity of corresponding SC decoder. By considering the bit-channel transition probabilities WG(⋅)(⋅∣0)W_{G}^{(\cdot)}(\cdot|0)WG(⋅)​(⋅∣0) and WG(⋅)(⋅∣1)W_{G}^{(\cdot)}(\cdot|1)WG(⋅)​(⋅∣1) separately, a WWW-expressions method is proposed to further reduce the complexity of HDLBK based SC decoder. For a m×mm\times mm×m binary kernel, the complexity of straightforward SC decoder is O(2mNlog⁡N)O(2^{m}N\log N)O(2mNlogN). With WWW-expressions, we reduce the complexity of straightforward SC decoder to O(m2Nlog⁡N)O(m^{2}N\log N)O(m2NlogN) when m≤16m\leq 16m≤16. Simulation results show that 16×1616\times1616×16 kernel polar codes offer significant advantages in terms of error performances compared with 2×22\times22×2 kernel polar codes under SC and list SC decoders.