Reed–Muller Codes for Random Erasures and Errors

This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the BEC and BSC, and in particular when can they achieve capacity for these two classical channels. Necessarily, the paper also studies properties of evaluations of multi-variate GF(2) polynomials on random sets of inputs. For erasures, we prove that RM codes achieve capacity both for very high rate and very low rate regimes. For errors, we prove that RM codes achieve capacity for very low rate regimes, and for very high rates, we show that they can uniquely decode at about square root of the number of errors at capacity.The proofs of these four results are based on different techniques, which we find interesting in their own right. In particular, we study the following questions about E(m,r), the matrix whose rows are truth tables of all monomials of degree ≤ r in m variables. What is the most (resp. least) number of random columns in E(m,r) that define a submatrix having full column rank (resp. full row rank) with high probability? We obtain tight bounds for very small (resp. very large) degrees r, which we use to show that RM codes achieve capacity for erasures in these regimes.Our decoding from random errors follows from the following novel reduction. For every linear code C of sufficiently high rate we construct a new code C' obtained by tensoring C, such that for every subset S of coordinates, if $C$ can recover from erasures in $S$, then C' can recover from errors in S. Specializing this to RM codes and using our results for erasures imply our result on unique decoding of RM codes at high rate.Finally, two of our capacity achieving results require tight bounds on the weight distribution of RM codes. We obtain such bounds extending the recent [27] bounds from constant degree to linear degree polynomials.

This paper considers "delta-almost Reed-Muller codes", i.e., linear codes spanned by evaluations of all but a delta fraction of monomials of degree at most d. It is shown that for any delta > 0 and any epsilon > 0, there exists a family of delta-almost Reed-Muller codes of constant rate that correct 1/2-epsilon fraction of random errors with high probability. For exact Reed-Muller codes, the analogous result is not known and represents a weaker version of the longstanding conjecture that Reed-Muller codes achieve capacity for random errors (Abbe-Shpilka-Wigderson STOC '15). Our proof is based on the recent polarization result for Reed-Muller codes, combined with a combinatorial approach to establishing inequalities between the Reed-Muller code entropies.

Reed-Muller (RM) codes were introduced in 1954 and have long been conjectured to achieve Shannon’s capacity on symmetric channels. The activity on this conjecture has recently been revived with the emergence of polar codes. RM codes and polar codes are generated by the same matrix $\begin{aligned} G_{m}= \left[{\begin{smallmatrix}1 & 0 1 & 1 \end{smallmatrix}}\right]^{\otimes m} \end{aligned}$ but using different subset of rows. RM codes select simply rows having largest weights. Polar codes select instead rows having the largest conditional mutual information proceeding top to down in $G_{m}$ ; while this is a more elaborate and channel-dependent rule, the top-to-down ordering allows Arikan to show that the conditional mutual information polarizes, and this gives directly a capacity-achieving code on any symmetric channel. RM codes are yet to be proved to have such a property, despite the recent success for the erasure channel. In this article, we connect RM codes to polarization theory. We show that proceeding in the RM code ordering, i.e., not top-to-down but from the lightest to the heaviest rows in $G_{m}$ , the conditional mutual information again polarizes. Here “polarization” means that almost all the conditional mutual information becomes either very close to 0 or very close to 1. Polarization itself is a necessary condition for RM codes to achieve capacity on symmetric channels while polarization together with a strong order on the conditional mutual information gives a sufficient condition, where strong order means that rows with larger weight always correspond to larger conditional mutual information. Although we are not able to prove the strong order, we establish a partial order on the conditional mutual information, which is a subset of the strong order. While the main results of this article–polarization together with the partial order–provide some advances on the capacity-achieving conjecture of RM codes, we emphasize that our results do not allow us to prove the conjecture.

Reed–Muller (RM) codes encode an $m$ -variate polynomial of degree at most $r$ by evaluating it on all points in $\{0,1\}^{m}$ . We denote this code by $RM(r,m)$ . The minimum distance of $RM(r,m)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length $n=2^{m}$ ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree $r=o(\sqrt {m})$ ), we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt {m/\log m})$ ), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , for which the same erasure pattern can be corrected in $RM(m-(r+1),m)$ . The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kotter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes.

Reed-Muller codes are some of the oldest and most widely studied error-correcting codes, of interest for both their algebraic structure as well as their many algorithmic properties. A recent beautiful result of Saptharishi, Shpilka and Volk [SSV17] showed that for binary Reed-Muller codes of length n and distance d = O(1), one can correct polylog(n) random errors in poly(n) time (which is well beyond the worst-case error tolerance of O(1)). In this paper, we consider the problem of syndrome decoding Reed-Muller codes from random errors. More specifically, given the polylog(n)-bit long syndrome vector of a codeword corrupted in polylog(n) random coordinates, we would like to compute the locations of the codeword corruptions. This problem turns out to be equivalent to a basic question about computing tensor decomposition of random low-rank tensors over finite fields. Our main result is that syndrome decoding of Reed-Muller codes (and the equivalent tensor decomposition problem) can be solved efficiently, i.e., in polylog(n) time. We give two algorithms for this problem: 1. The first algorithm is a finite field variant of a classical algorithm for tensor decomposition over real numbers due to Jennrich. This also gives an alternate proof for the main result of [SSV17]. 2. The second algorithm is obtained by implementing the steps of [SSV17]'s Berlekamp-Welch-style decoding algorithm in sublinear-time. The main new ingredient is an algorithm for solving certain kinds of systems of polynomial equations.

This work proves new results on the ability of binary Reed-Muller codes to decode from random errors and erasures. Specifically, we prove that RM codes with m variables and degree γm, for some explicit constant γ achieve capacity for random erasures (i.e. for the binary erasure channel) and for random errors (for the binary symmetric channel). Earlier, it was known that RM codes achieve capacity for the binary symmetric channel for degrees r = o(m). For the binary erasure channel it was known that RM codes achieve capacity for degree . Thus, our results provide a new range of parameters for which RM achieve capacity for these two well studied channels. In addition, our results imply that for every ϵ > 0 (in fact we can get up to RM codes of degree r < (1/2 – ϵ)m can correct a fraction of 1 – o(1) random erasures with high probability. We also show that, information theoretically, such codes can handle a fraction of random errors with high probability. For example, given noisy evaluations of a degree 0.499m polynomial, it is possible to interpolate it even if a random 0. 499 fraction of the evaluations were corrupted, with high probability. While the o(1) terms are not the correct ones to ensure capacity, these results show that RM codes of rates up to 1/poly(log n) (where n = 2m is the block length) are is some sense as good as capacity achieving codes. We obtain these results by proving improved bounds on the weight distribution of Reed-Muller codes of high degrees. Namely, given weight β ϵ (0, 1) we prove an upper bound on the number of codewords of relative weight at most β. We obtain new results in two different settings: for weights β < 1/2 and for weights that are close to 1/2. Our results for weights close to 1/2 also answer an open problem posed by Beame et al. [10].

Recently, it has been verified that the Reed-Muller (RM) code, which has been predicted for a long time in the field of code theory, achieves capacity in a binary erasure channel (BEC) with a MAP (maximum a posteriori) decoder. Through weight distribution interpretation, it was also discovered that the RM code achieves a binary memoryless symmetric (BMS) channel capacity. A successive cancellation list (SCL) technique has been studied as a practical decoder instead of a MAP decoder. In this paper, the performance improvement potential that can be improved when a generalized concatenated code (GCC) with an outer code is designed to improve the practical SCL decoder performance is analyzed. When information bits are used as frozen bits in these code-word designs, we analyze an inter-code-word distance and the decoding outage probability of the designed code. We expect that our results can be used as a conceptual base for improving communication methods in the future development of channel encoding technology.

Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result.

We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as ${\mathcal{C}_n}(r,m)$, and their duals are denoted as ${{\mathcal{B}}_n}(r,m)$. The length of these codes is n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , where n ≥ 2, and r is their ‘order’. When n = 2, ${\mathcal{C}_n}(r,m)$ is the RM code of order r and length 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> . The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to ${{\mathcal{B}}_n}(r,m)$ as the Berman code and ${\mathcal{C}_n}(r,m)$ as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al. (2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding.

A new class of binary codes is introduced. The codes have length n = 1.5 · 2m (m ≥ 3), and have a similar ‘double inductive’ structure to that possessed by the class of Reed-Muller (RM) codes. The new codes are structured such that their length-to-distance ratio n/d = 2m, m ≥ 2. For rates less than 1/2, the new codes have a higher rate than the next Reed-Muller code of the same order and length n = 2 · 2m.

This letter proposes a modified PTS technique using binary Reed-Muller (RM) codes for error correction and PAPR control in BPSK OFDM systems. A RM code is divided into the direct sum of a correcting subcode for encoding information bits and a scrambling subcode for encoding PAPR bits. The transmitted signal of the resulting OFDM sequence is selected with minimum PAPR from a number of candidates which are codewords of a coset of the scrambling subcode. We consider the RM codes in natural and cyclic orderings. Numerical results show that RM codes in cyclic ordering achieve better performance in PAPR reduction than RM codes in natural ordering.

In this paper we construct new subcodes of the second-order binary Reed-Muller code by using the permutation group and by projecting the code onto codes with smaller parameters. The permutation group of Reed-Muller codes is the general affine group and can be decomposed into the semi-direct product of the translation group and the general linear group. The action of the translation group projects the second order Reed-Muller code onto copies of the first order Reed-Muller code. The general linear group projects the code onto codes for which we can control the useful length and the dimension. These parameters depend on the dimension of the eigenspace of the chosen element of the general linear group for the eigenvalue 1.

The approach to error correction coding taken by modern digital communication systems started in the late 1940&amp;apos;s with the ground breaking work of Shannon, Hamming and Golay. Reed-Muller (RM) codes were an important step beyond the Hamming and Golay codes because they allowed more flexibility in the size of the code word and the number of correctable errors per code word. Whereas the Hamming and Golay codes were specific codes with particular values for q; n; k; and t, the RM codes were a class of binary codes with a wide range of allowable design parameters. Binary Reed-Muller codes are among the most prominent families of codes in coding theory. They have been extensively studied and employed for practical applications. In this research, the performance simulation of Reed-Muller Codec was presented. An introduction on Reed-Muller codes, were introduced that consists of defining the key terms and operation used with the binary numbers. Reed-Muller codes were defined and encoding matrices were discussed. The decoding process was given and some examples were demonstrated to clarify the method. The results and the performance of Reed-Muller encoding were presented and the messages been encoded using the defined matrices were shown. The simulation of the decoding part also been shown. The performance of Reed-Muller codes were then analyzed in terms of its code rate, code length and minimum Hamming distance. The analysis that performed also successfully examines the relationship between the parameters of Reed-Muller coding. The decoding part of the Reed-Muller codes can detect one error and correct it as shown in the examples.

Reed-Muller codes are error-correcting codes used in many areas related to coding theory, such as electrical engineering and computer science. The binary rth-order Reed-Muller code RM(r, n) can be viewed as the set of all n-variable Boolean functions of algebraic degree at most r. Despite the intense work on these codes, many problems are known to be hard (notably, determining their covering radius) and remain open to this day. Fourteen years ago, Carlet and Mesnager improved in [IEEE Transactions on Information Theory, “Improving the Upper Bounds on the Covering Radii of Binary Reed-Muller Codes”, 53(1), 2007] the upper bound on the covering radius of the Reed-Muller code of order 2, and they deduced improved upper bounds on the covering radii of the Reed-Muller codes of higher orders. Until 2021, these upper bounds remain the best ones in the literature. The Reed-Muller code RM(n -3, n), which corresponds to the dual of the Reed-Muller code RM(2, n), has attracted much attention. One of the main reasons is that it is precisely the code that has been considered to get the upper bounds derived by Carlet and Mesnager. Those upper bounds have been obtained thanks to the characterization of the codewords of the Reed-Muller code, whose Hamming weights are strictly less than 2.5 times the minimum distance 2n-r due to Kasami, Tokura, and Azumi. Despite their impressive work in the seventieth, a more refined study and profound description of those codewords of RM(n -3, n) whose Hamming weight equals 16, and especially 18, seem necessary, as it could help us significantly in improving the covering radius of Reed-Muller codes. In this paper, we push further the known results on the Reed-Muller codes by focusing on the Reed-Muller code RM(n -3, n). We provide a classification of the codewords of weight 16 and 18 of the Reed-Muller code RM(n -3, n). Our algebraic descriptions allow us to count the number of such codewords and to enumerate all of them explicitly.

This work establishes the connection between the finite and infinite algebraic structure of a full row or column rank polynomial matrix and its Moore Penrose inverse. For non-singular matrices, it is known that the inverse has no finite zeros and its finite poles are the finite zeros of the original matrix, while the infinite poles and zeros of the original matrix are the infinite zeros and poles of its inverse respectively. The fact that the Moore Penrose inverse is unique for every matrix along with the fact that for non-singular matrices it coincides with the normal inverse leads to the assumption that a similar connection holds for the structure, both finite and infinite, of the Moore Penrose inverse of a polynomial matrix. We prove that the Moore Penrose inverse of a full row or column rank polynomial matrix has no finite zeros, and that extra poles appear, compared to the non-singular case. Furthermore, we prove that the result regarding the infinite structure of nonsingular matrices is extended to the case of the Moore Penrose inverse of a full rank polynomial matrix. Finally we determine a connection between the minimal indices of the polynomial matrix and its Moore Penrose inverse. The results are also presented through an illustrative example.