Related papers: Machine Learning-Aided Efficient Decoding of Reed-…
Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes. They are used in many areas of coding theory in both electrical engineering and computer science. Yet, many of their important properties are…
Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple…
Short-length Reed--Muller codes under majority-logic decoding are of particular importance for efficient hardware implementations in real-time and embedded systems. This paper significantly improves Chen's two-step majority-logic decoding…
We explore the relationship between polar and RM codes and we describe a coding scheme which improves upon the performance of the standard polar code at practical block lengths. Our starting point is the experimental observation that RM…
We consider near maximum-likelihood (ML) decoding of short linear block codes based on neural belief propagation (BP) decoding recently introduced by Nachmani et al.. While this method significantly outperforms conventional BP decoding, the…
We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of…
This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family…
Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes such as cyclic codes, Reed-Solomon codes, and Reed-Muller codes have nice decoding algorithms.…
Reed-Muller (RM) codes have undergone significant analytical advancements over the past decade, particularly for binary memoryless symmetric (BMS) channels. We extend the scope of RM codes development and analysis to multiple-access…
The question of whether Reed-Muller (RM) codes achieve capacity on binary memoryless symmetric (BMS) channels has drawn attention since it was resolved positively for the binary erasure channel by Kudekar et al. in 2016. In 2021, Reeves and…
An efficient decoder for the generalized first-order Reed-Muller code RM_q(1,m) is essential for the decoding of various block-coding schemes for orthogonal frequency-division multiplexing with reduced peak-to-mean power ratio. We present…
In this paper, by treating Reed-Muller (RM) codes as a special class of low-density parity-check (LDPC) codes and assuming that sub-blocks of the parity-check matrix are randomly interleaved to each other as Gallager's codes, we present a…
We consider weighted Reed-Muller codes over point ensemble $S_1 \times...\times S_m$ where $S_i$ needs not be of the same size as $S_j$. For $m = 2$ we determine optimal weights and analyze in detail what is the impact of the ratio…
One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is based on using multiple trials of a simple RS decoding algorithm in combination with erasing or flipping a set of symbols or bits in each trial. This paper…
We show that Reed-Muller codes achieve capacity under maximum a posteriori bit decoding for transmission over the binary erasure channel for all rates $0 < R < 1$. The proof is generic and applies to other codes with sufficient amount of…
Assuming that we have a soft-decision list decoding algorithm of a linear code, a new hard-decision list decoding algorithm of its repeated code is proposed in this article. Although repeated codes are not used for encoding data, due to…
Define the codewords of the Tensor Reed-Muller code $\mathsf{TRM}(r_1,m_1;r_2,m_2;\dots;r_t,m_t)$ to be the evaluation vectors of all multivariate polynomials in the variables $\left\{x_{ij}\right\}_{i=1,\dots,t}^{j=1,\dots m_i}$ with…
In the search for highly efficient decoders for short LDPC codes approaching maximum likelihood performance, a relayed decoding strategy, specifically activating the ordered statistics decoding process upon failure of a neural min-sum…
Algebraic decoding algorithms are commonly applied for the decoding of Reed-Solomon codes. Their main advantages are low computational complexity and predictable decoding capabilities. Many algorithms can be extended for correction of both…
This article is focused on some variations of Reed-Muller codes that yield improvements to the rate for a prescribed decoding performance under the Berlekamp-Massey-Sakata algorithm with majority voting. Explicit formulas for the…