Reed-Muller (RM) codes achieve the capacity of general binary-input memoryless symmetric channels and are conjectured to have a comparable performance to that of random codes in terms of scaling laws. However, such results are established assuming maximum-likelihood decoders for general code parameters. Also, RM codes only admit limited sets of rates. Efficient decoders such as successive cancellation list (SCL) decoder and recently-introduced recursive projection-aggregation (RPA) decoders are available for RM codes at finite lengths. In this paper, we focus on subcodes of RM codes with flexible rates. We first extend the RPA decoding algorithm to RM subcodes. To lower the complexity of our decoding algorithm, referred to as subRPA, we investigate different approaches to prune the projections. Next, we derive the soft-decision based version of our algorithm, called soft-subRPA, that not only improves upon the performance of subRPA but also enables a differentiable decoding algorithm. Building upon the soft-subRPA algorithm, we then provide a framework for training a machine learning (ML) model to search for \textit{good} sets of projections that minimize the decoding error rate. Training our ML model enables achieving very close to the performance of full-projection decoding with a significantly smaller number of projections. We also show that the choice of the projections in decoding RM subcodes matters significantly, and our ML-aided projection pruning scheme is able to find a \textit{good} selection, i.e., with negligible performance degradation compared to the full-projection case, given a reasonable number of projections.

翻译：Reed-Muller (RM) 码能够达到一般二进制输入无记忆对称信道的容量，并且根据标度律被推测具有与随机码相当的性能。然而，这些结果是在假设通用码参数的极大似然解码器下建立的。此外，RM码仅支持有限的码率集合。针对有限长度的RM码，已有高效解码器，例如逐次消除列表（SCL）解码器和近期提出的递归投影聚合（RPA）解码器。本文聚焦于具有灵活码率的RM子码。我们首先将RPA解码算法扩展到RM子码。为降低所提出的子RPA（subRPA）算法的复杂度，我们研究了不同的投影剪枝方法。随后，我们推导了算法的软判决版本（soft-subRPA），该版本不仅提升了subRPA的性能，还实现了可微分的解码算法。基于soft-subRPA算法，我们进一步构建了一个训练机器学习（ML）模型的框架，以搜索能够最小化解码错误率的"最优"投影集合。通过训练ML模型，我们能够以显著更少的投影数量达到近乎全投影解码的性能。我们还证明，在解码RM子码时，投影的选择至关重要，而我们所提出的ML辅助投影剪枝方案能够在合理数量的投影下，找出与全投影情况相比性能损失可忽略的"最优"投影选择。