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arXiv:2503.18758 · IEEE Access

On the Optimality of Single-Label and Multi-Label Neural Network Decoders

Yunus Can Gültekin, Péter Scheepers, Yuncheng Yuan, Federico Corradi, Alex Alvarado

Published Jan 1, 2026• 0 citations
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Abstract

We investigate the design of two neural network (NN) architectures recently proposed as decoders for forward error correction: the so-called single-label NN (SLNN) and multi-label NN (MLNN) decoders. These decoders have been reported to achieve near-optimal codeword- and bit-wise performance, respectively. Results in the literature show near-optimality for a variety of short codes. In this paper, we analytically prove that certain SLNN and MLNN architectures can, in fact, always realize optimal decoding, regardless of the code. These optimal architectures and their binary weights are shown to be defined by the codebook, i.e., no training or network optimization is required. Our proposed architectures are in fact not NNs, but implement the maximum likelihood decoding rule differently. Optimal performance is numerically demonstrated for Hamming (7; 4), Polar (16; 8), and BCH (31; 21) codes. The results show that our optimal architectures (which are not NNs) are less complex than the SLNN and MLNN architectures proposed in the literature, which in fact only achieve near-optimal performance. Extension to longer codes is still hindered by the curse of dimensionality. Therefore, even though SLNN and MLNN can perform maximum likelihood decoding, such architectures cannot be used for medium and long codes.

DOI
10.1109/access.2026.3693469

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We investigate the design of two neural network (NN) architectures recently proposed as decoders for forward error correction: the so-called single-label NN (SLNN) and multi-label NN (MLNN) decoders. These decoders have been reported to achieve near-optimal codeword- and bit-wise performance, respectively. Results in the literature show near-optimality for a variety of short codes. In this paper, we analytically prove that certain SLNN and MLNN architectures can, in fact, always realize optimal decoding, regardless of the code. These optimal architectures and their binary weights are shown to be defined by the codebook, i.e., no training or network optimization is required. Our proposed architectures are in fact not NNs, but implement the maximum likelihood decoding rule differently. Optimal performance is numerically demonstrated for Hamming (7; 4), Polar (16; 8), and BCH (31; 21) codes. The results show that our optimal architectures (which are not NNs) are less complex than the SLNN and MLNN architectures proposed in the literature, which in fact only achieve near-optimal performance. Extension to longer codes is still hindered by the curse of dimensionality. Therefore, even though SLNN and MLNN can perform maximum likelihood decoding, such architectures cannot be used for medium and long codes.
This paper analyzes the design and competitiveness of four neural network (NN) architectures recently proposed as decoders for forward error correction (FEC) codes. We first consider the so-called single-label neural network (SLNN) and the multi-label neural network (MLNN) decoders which have been reported to achieve near maximum likelihood (ML) performance. Here, we show analytically that SLNN and MLNN decoders can always achieve ML performance, regardless of the code dimensions-although at the cost of computational complexityand no training is in fact required. We then turn our attention to two transformer-based decoders: the error correction code transformer (ECCT) and the cross-attention message passing transformer (CrossMPT). We compare their performance against traditional decoders, and show that ordered statistics decoding outperforms these transformer-based decoders. The results in this paper cast serious doubts on the application of NN-based FEC decoders in the short and medium blocklength regime.
The problem of decoding an (n, k, d) error-correcting block code, where a k-bit message word is mapped to an n-bit codeword, can be cast as a single-label classification problem. While it has been observed that the performance of such single-label neural decoders closely approaches that of the corresponding maximum likelihood decoder (MLD), the number of output nodes increases exponentially with k, making it prohibitive to implement for large k. To address this issue, we explore classification based multi-label neural decoders, in which the number of output nodes increases linearly with k. We consider well-known linear and non-linear block codes, as well as concatenated block codes, which have applications in emerging wireless networks. Our study finds that (i) although the number of output nodes linearly increases with k in a multi-label decoder, it requires more hidden layers and nodes in each hidden layer than the corresponding single-label decoder to achieve its best performance, and (ii) although one can design a multi-label decoder with bit error rate matching that of the MLD, it leaves more blocks in error leading to a reduced performance in terms of block error rate. We also note that the performance of the proposed decoder for concatenated codes is at least as good as that of a natural decoding algorithm in which the inner code is first decoded using the MLD and then the outer code is decoded with a polynomial-time decoding algorithm.
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).
In this work, we propose a fully differentiable graph neural network (GNN)-based architecture for channel decoding and showcase a competitive decoding performance for various coding schemes, such as low-density parity-check (LDPC) and BCH codes. The idea is to let a neural network (NN) learn a generalized message passing algorithm over a given graph that represents the forward error correction (FEC) code structure by replacing node and edge message updates with trainable functions. Contrary to many other deep learning-based decoding approaches, the proposed solution enjoys scalability to arbitrary block lengths and the training is not limited by the curse of dimensionality. We benchmark our proposed decoder against state-of-the-art in conventional channel decoding as well as against recent deep learning-based results. For the (63,45) BCH code, our solution outperforms weighted belief propagation (BP) decoding by approximately 0.4 dB with significantly less decoding iterations and even for 5G NR LDPC codes, we observe a competitive performance when compared to conventional BP decoding. For the BCH codes, the resulting GNN decoder can be fully parametrized with only 9640 weights.
Error correcting codes are a fundamental component in modern day communication systems, demanding extremely high throughput, ultra-reliability and low latency. Recent approaches using machine learning (ML) models as decoders offer both improved performance and great adaptability to unknown environments, where traditional decoders struggle. We introduce a general framework to further boost the performance and applicability of ML models. We propose to combine ML decoders with a competing discriminator network that tries to distinguish between codewords and noisy words, and, hence, guides the decoding models to recover transmitted codewords. Our framework is game-theoretic, motivated by generative adversarial networks (GANs), with the decoder and discriminator competing in a zero-sum game. The decoder learns to simultaneously decode and generate codewords while the discriminator learns to tell the difference between decoded outputs and codewords. Thus, the decoder is able to decode noisy received signals into codewords, increasing the probability of successful decoding. We show a strong connection of our framework with the optimal maximum likelihood decoder by proving that this decoder defines a Nash equilibrium point of our game. Hence, training to equilibrium has a good possibility of achieving the optimal maximum likelihood performance. Moreover, our framework does not require training labels, which are typically unavailable during communications, and, thus, seemingly can be trained online and adapt to channel dynamics. To demonstrate the performance of our framework, we combine it with recent neural decoders and show improved performance compared to the original models and traditional decoding algorithms on various codes.
Designing a practical, low complexity, close to optimal, channel decoder for powerful algebraic codes with short to moderate block length is an open research problem. Recently it has been shown that a feed-forward neural network architecture can improve on standard belief propagation decoding, despite the large example space. In this paper we introduce a recurrent neural network architecture for decoding linear block codes. Our method shows comparable bit error rate results compared to the feed-forward neural network with significantly less parameters. We also demonstrate improved performance over belief propagation on sparser Tanner graph representations of the codes. Furthermore, we demonstrate that the RNN decoder can be used to improve the performance or alternatively reduce the computational complexity of the mRRD algorithm for low complexity, close to optimal, decoding of short BCH codes.