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 $0.1$dB. Code available at https://github.com/cyclicallyneuraldecoder/CyclicallyEquivariantNeuralDecoders

本研究提出了基于循环编码的一种新型神经解码器，通过利用循环不变性，将我们的神经解码器的权重施加了移位不变性结构，从而实现任何输入的循环移位都能得到相同的循环移位输出。与BCH代码和RM代码考虑戳孔情况的早期神经解码器相比，我们的系统在解码循环码时性能更优，而新的列表解码过程可以显著降低BCH代码和戳孔RM代码的解码错误概率，特定高速代码的列表解码器与最大似然译码器之间的差距小于0.1dB。

循环等变神经解码器用于循环码