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… 

Figures and Tables from this paper

Automorphism Ensemble Decoding of Reed–Muller Codes
TLDR
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.
Iterative Reed–Muller Decoding
TLDR
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.
Reed-Muller Subcodes: Machine Learning-Aided Design of Efficient Soft Recursive Decoding
TLDR
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.
Decoding of Reed-Muller Codes Starting With a Higher-Rate Constituent Code
TLDR
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
TLDR
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.
Pruning Neural Belief Propagation Decoders
TLDR
A method to tailor an overcomplete parity-check matrix to (neural) BP decoding using machine learning and considers the weights in the Tanner graph as an indication of the importance of the connected check nodes (CNs) to decoding and uses them to prune unimportant CNs.
Recursive projection-aggregation decoding of Reed-Muller codes
  • Min Ye, E. Abbe
  • Computer Science
    2019 IEEE International Symposium on Information Theory (ISIT)
  • 2019
TLDR
A new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels based on projecting the code on its cosets, recursively decoding the projected codes, and aggregating the reconstructions.
Reed-Muller Codes Achieve Capacity on BMS Channels
This paper considers the performance of long Reed–Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori decoding. Its main result is that
Recursive Projection-Aggregation Decoding of Reed-Muller Codes
  • Min Ye, E. Abbe
  • Computer Science
    IEEE Transactions on Information Theory
  • 2020
TLDR
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.
Pruning and Quantizing Neural Belief Propagation Decoders
TLDR
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.
...
...

References

SHOWING 1-10 OF 20 REFERENCES
Reed–Muller Codes Achieve Capacity on Erasure Channels
TLDR
This work shows that symmetry alone implies near-optimal performance in any sequence of linear codes where the blocklengths are strictly increasing, the code rates converge, and the permutation group of each code is doubly transitive.
Decomposition Methods for Large Scale LP Decoding
TLDR
This paper draws on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding, and develops an efficient algorithm for Euclidean norm projection onto the parity polytope.
On Iterative Soft-Decision Decoding of Linear Binary Block Codes and Product Codes
TLDR
A decoding algorithm which only uses parity check vectors of minimum weight is proposed, which gives results close to soft decision maximum likelihood (SDML) decoding for many code classes like BCH codes.
Iterative Soft-Input Soft-Output Decoding of Reed–Solomon Codes by Adapting the Parity-Check Matrix
TLDR
An iterative algorithm is presented for soft-input soft-output (SISO) decoding of Reed-Solomon (RS) codes that uses the sum-product algorithm (SPA) in conjunction with a binary parity-check matrix of the RS code.
Performance comparison of short-length error-correcting codes
TLDR
A universal decoder that can decode any linear binary block code is used: Gaussian-elimination based Maximum-Likelihood decoder on the erasure channel and probabilistic Ordered Statistics Decoder on the Gaussian channel.
Using linear programming to Decode Binary linear codes
TLDR
The definition of a pseudocodeword unifies other such notions known for iterative algorithms, including "stopping sets," "irreducible closed walks," "trellis cycles," "deviation sets," and "graph covers," which is a lower bound on the classical distance.
Multiple-bases belief-propagation decoding of high-density cyclic codes
TLDR
It is shown that the inherent code property of a code has many structurally diverse parity-check matrices, capable of detecting different error patterns leads to decoding algorithms with significantly better performance when compared to standard belief-propagation decoding.
List decoding of polar codes
  • I. Tal, A. Vardy
  • Computer Science
    2011 IEEE International Symposium on Information Theory Proceedings
  • 2011
TLDR
It appears that the proposed list decoder bridges the gap between successive-cancellation and maximum-likelihood decoding of polar codes, and devise an efficient, numerically stable, implementation taking only O(L · n log n) time and O( L · n) space.
Multiple-Bases Belief-Propagation for Decoding of Short Block Codes
TLDR
It is shown that for selected codes the proposed scheme approaches near maximum likelihood (ML) performance for very small data processing delays.
Random Redundant Soft-In Soft-Out Decoding of Linear Block Codes
  • T. Halford, K. Chugg
  • Computer Science
    2006 IEEE International Symposium on Information Theory
  • 2006
TLDR
This work presents iterative soft-in soft-out (SISO) decoding algorithms in a common framework and presents a related algorithm - random redundant iterative decoding - that is both practically realizable and applicable to arbitrary linear block codes.
...
...