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arXiv:2106.07964 · IEEE International Symposium on Information Theory (ISIT)

Improving the List Decoding Version of the Cyclically Equivariant Neural Decoder

Xiangyu Chen, Min Ye

Published Jun 26, 2022• 3 citations
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Abstract

The cyclically equivariant neural decoder was recently proposed in [Chen-Ye, International Conference on Machine Learning, 2021] to decode cyclic codes. In the same paper, a list decoding procedure was also introduced for two widely used classes of cyclic codes—BCH codes and punctured Reed-Muller (RM) codes. While the list decoding procedure significantly improves the Frame Error Rate (FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of the list decoding procedure is even worse than the unique decoding algorithm when the list size is small. In this paper, we propose an improved version of the list decoding algorithm for BCH codes and punctured RM codes. Our new proposal significantly reduces the BER while maintaining the same (in some cases even smaller) FER. More specifically, our new decoder provides up to 2dB gain over the previous list decoder when measured by BER, and the running time of our new decoder is 15% smaller. Code available at github.com/improvedlistdecoder/code

DOI
10.1109/isit50566.2022.9834337

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The cyclically equivariant neural decoder was recently proposed in [Chen-Ye, International Conference on Machine Learning, 2021] to decode cyclic codes. In the same paper, a list decoding procedure was also introduced for two widely used classes of cyclic codes—BCH codes and punctured Reed-Muller (RM) codes. While the list decoding procedure significantly improves the Frame Error Rate (FER) of the cyclically equivariant neural decoder, the Bit Error Rate (BER) of the list decoding procedure is even worse than the unique decoding algorithm when the list size is small. In this paper, we propose an improved version of the list decoding algorithm for BCH codes and punctured RM codes. Our new proposal significantly reduces the BER while maintaining the same (in some cases even smaller) FER. More specifically, our new decoder provides up to 2dB gain over the previous list decoder when measured by BER, and the running time of our new decoder is 15% smaller. Code available at github.com/improvedlistdecoder/code
Neural decoders were introduced as a generalization of the classic Belief Propagation (BP) decoding algorithms, where the Trellis graph in the BP algorithm is viewed as a neural network, and the weights in the Trellis graph are optimized by training the neural network. In this work, we propose a novel neural decoder for cyclic codes by exploiting their cyclically invariant property. More precisely, we impose a shift invariant structure on the weights of our neural decoder so that any cyclic shift of inputs results in the same cyclic shift of outputs. Extensive simulations with BCH codes and punctured Reed-Muller (RM) codes show that our new decoder consistently outperforms previous neural decoders when decoding cyclic codes. Finally, we propose a list decoding procedure that can significantly reduce the decoding error probability for BCH codes and punctured RM codes. For certain high-rate codes, the gap between our list decoder and the Maximum Likelihood decoder is less than 0.10.10.1dB. Code available at https://github.com/cyclicallyneuraldecoder/CyclicallyEquivariantNeuralDecoders
In this work, we investigate the problem of neural-based error correction decoding, and more specifically, the new so-called syndrome-based decoding technique introduced to tackle scalability in the training phase for larger code sizes. We improve on previous works in terms of allowing full decoding of the message rather than codewords, allowing thus the application to non-systematic codes, and proving that the single-message training property is still viable. The suggested system is implemented and tested on polar codes of sizes (64,32) and (128,64), and a BCH of size (63,51), leading to a significant improvement in both Bit Error Rate (BER) and Frame Error Rate (FER), with gains between 0.3dB and 1dB for the implemented codes in the high Signal-to-Noise Ratio (SNR) regime.
Recursive list decoding of Reed-Muller (RM) codes, with moderate list size, is known to approach maximum-likelihood (ML) performance of short length(≤256)(\leq 256)(≤256)RM codes. Recursive decoding employs the Plotkin construction to split the original code into two shorter RM codes with different rates. In contrast to the standard approach which decodes the lower-rate code first, the method in this paper decodes the higher-rate code first. This modification enables an efficient permutation-based decoding technique, with permutations being selected on the fly from the automorphism group of the code using soft information from a channel. Simulation results show that the error-rate performance of the proposed algorithms, enhanced by a permutation selection technique, is close to that of the automorphism-based recursive decoding algorithm with similar complexity for short RM codes, while our decoders perform better for longer RM codes. In particular, it is demonstrated that the proposed algorithms achieve near-ML performance for short RM codes and for RM codes of length2m2^{m}2mand orderm−3m-3m−3with reasonable complexity.
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes.
Reed-Muller (RM) codes are classical codes that have enjoyed unabated interest since their introduction in 1954 due to their simple recursive structure and low-complexity decoding. The main goal of the thesis is to develop new coding techniques that can improve the performance of RM codes. In this work, we wish to keep the recursive low-complexity structure of RM codes while extending the sets of possible parameters and decoding algorithms. We address this task using two different methods. First, we consider new code constructions that extend code parameters using puncturing. Second, we design new decoding algorithms that repeatedly use the gradient properties of the classic recursive structure to improve code performance of RM codes. In the first part of the thesis, we propose a new class of codes that enjoy the recursive structure of RM codes, but reduce the exponential length of RM codes to a polynomial in m. These codes are obtained by puncturing classical RM codes, i.e. by restricting binary RM codes to small subsets of their positions. We study new recursive properties of these codes and analyze their parameters. In the second part of the thesis we develop new decoding algorithms for binary RM codes of second order that achieve near optimal performance for short and moderate code lengths. RM codes enjoy some simple decoding procedures; however, these algorithms fall short of optimal performance on moderate block lengths. We propose a suboptimal algorithm that reduces the complexity of maximum-likelihood decoding from super-polynomial order O(n(m+1)/2) to polynomial order O(mn2). At the same time, our numerical simulations show that the new algorithm performs very closely to its maximum-likelihood counterpart for blocks of up to several thousand bits.