Decoding Reed-Muller Codes Using Minimum- Weight Parity Checks

@article{Santi2018DecodingRC,
  title={Decoding Reed-Muller Codes Using Minimum- Weight Parity Checks},
  author={Elia Santi and Christian H{\"a}ger and Henry D. Pfister},
  journal={2018 IEEE International Symposium on Information Theory (ISIT)},
  year={2018},
  pages={1296-1300}
}
Reed-Muller (RM) codes exhibit good performance under maximum-likelihood (ML) decoding due to their highly-symmetric structure. In this paper, we explore the question of whether the code symmetry of RM codes can also be exploited to achieve near-ML performance in practice. The main idea is to apply iterative decoding to a highly-redundant parity-check (PC) matrix that contains only the minimum-weight dual codewords as rows. As examples, we consider the peeling decoder for the binary erasure… 
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A versatile decoding architecture for RM codes based on their rich automorphism group is presented and the theoretical limitations of this method with respect to polar codes are proved.
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This work presents a belief propagation (BP) decoding architecture for RM codes based on their rich automorphism group and achieves the best performance of all iterative RM decoders presented thus far.
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The soft-decision based version of the RPA algorithm, called soft-subRPA, is derived that is shown to improve upon the performance of subRPA and enables training a machine learning (ML) model to search for good sets of projections in the sense of minimizing the decoding error rate.
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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 length RM codes, while the decoders perform better for longer RM codes.
Recursive Decoding of Reed-Muller Codes Starting With the Higher-Rate Constituent Code
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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 the decoders perform better for longer RM codes.
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    2019 IEEE International Symposium on Information Theory (ISIT)
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A new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels is proposed, based on projecting the code on its cosets, recursively decoding the projected codes, and aggregating the reconstructions.
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This work proposes a novel decoding approach based on neural belief propagation (NBP) decoding recently introduced by Nachmani et al. in which a different parity-check matrix is allowed in each iteration of the algorithm.
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