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Register a new userA Tree Search Approach for Maximum-Likelihood Decoding of Reed-Muller Codes
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Added by
Seyyed Ali Hashemi
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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A low-complexity tree search approach is presented that achieves the maximum-likelihood (ML) decoding performance of Reed-Muller (RM) codes. The proposed approach generates a bit-flipping tree that is traversed to find the ML decoding result by performing successive-cancellation decoding after each node visit. A depth-first search (DFS) and a breadth-first search (BFS) scheme are developed and a log-likelihood-ratio-based bit-flipping metric is utilized to avoid redundant node visits in the tree. Several enhancements to the proposed algorithm are presented to further reduce the number of node visits. Simulation results confirm that the BFS scheme provides a lower average number of node visits than the existing tree search approach to decode RM codes.
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This paper presents a novel successive factor-graph permutation (SFP) scheme that significantly improves the error-correction performance of Reed-Muller (RM) codes under successive-cancellation list (SCL) decoding. In particular, we perform maximum-likelihood decoding on the symmetry group of RM codes to carefully select a good factor-graph permutation on the fly. We further propose an SFP-aided fast SCL decoding that significantly reduces the decoding latency while preserving the error-correction performance of the code. The simulation results show that for the third and fourth order RM codes of length $256$, the proposed decoder reduces up to $85%$ of the memory consumption, $78%$ of the decoding latency, and more than $99%$ of the computational complexity of the state-of-the-art recursive projection-aggregation decoder at the frame error rate of $10^{-3}$.
In this paper we propose efficient decoding techniques to significantly improve the error-correction performance of fast successive-cancellation (FSC) and FSC list (FSCL) decoding algorithms for short low-order Reed-Muller (RM) codes. In particular, we first integrate Fast Hadamard Transform (FHT) into FSC (FHT-FSC) and FSCL (FHT-FSCL) decoding algorithms to optimally decode the first-order RM subcodes. We then utilize the rich permutation group of RM codes by independently running the FHT-FSC and the FHT-FSCL decoders on a list of random bit-index permutations of the codes. The simulation results show that the error-correction performance of the FHT-FSC decoders on a list of $L$ random code permutations outperforms that of the FSCL decoder with list size $L$, while requiring lower memory requirement and computational complexity for various configurations of the RM codes. In addition, when compared with the state-of-the-art recursive projection-aggregation (RPA) decoding, the permuted FHT-FSCL decoder can obtain a similar error probability for the RM codes of lengths $128$, $256$, and $512$ at various code rates, while requiring several orders of magnitude lower computational complexity.
The well known Plotkin construction is, in the current paper, generalized and used to yield new families of Z2Z4-additive codes, whose length, dimension as well as minimum distance are studied. These new constructions enable us to obtain families of Z2Z4-additive codes such that, under the Gray map, the corresponding binary codes have the same parameters and properties as the usual binary linear Reed-Muller codes. Moreover, the first family is the usual binary linear Reed-Muller family.
A projective Reed-Muller (PRM) code, obtained by modifying a (classical) Reed-Muller code with respect to a projective space, is a doubly extended Reed-Solomon code when the dimension of the related projective space is equal to 1. The minimum distance and dual code of a PRM code are known, and some decoding examples have been represented for low-dimensional projective space. In this study, we construct a decoding algorithm for all PRM codes by dividing a projective space into a union of affine spaces. In addition, we determine the computational complexity and the number of errors correctable of our algorithm. Finally, we compare the codeword error rate of our algorithm with that of minimum distance decoding.
The famous Barnes-Wall lattices can be obtained by applying Construction D to a chain of Reed-Muller codes. By applying Construction ${{D}}^{{(cyc)}}$ to a chain of extended cyclic codes sandwiched between Reed-Muller codes, Hu and Nebe (J. London Math. Soc. (2) 101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes-Wall lattices. In this paper, we explicitly determine the minimum weight codewords of those codes for some special cases.
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