Reed-Muller Research Articles

Automorphism Ensemble Decoding of Reed–Muller Codes

Reed–Muller (RM) codes are known for their good maximum likelihood (ML) performance in the short block-length regime. Despite being one of the oldest classes of channel codes, finding a low complexity soft-input decoding scheme is still an open problem. In this work, we present a versatile decoding architecture for RM codes based on their rich automorphism group. The decoding algorithm can be seen as a generalization of multiple-bases belief propagation (MBBP) and may use any polar or RM decoder as constituent decoders. We provide extensive error-rate performance simulations for successive cancellation (SC)-, SC-list (SCL)- and belief propagation (BP)-based constituent decoders. We furthermore compare our results to existing decoding schemes and report a near-ML performance for the RM(3,7)-code (e.g., 0.04 dB away from the ML bound at BLER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">−3</sup> ) at a competitive computational cost. Moreover, we provide some insights into the automorphism subgroups of RM codes and SC decoding and, thereby, prove the theoretical limitations of this method with respect to polar codes.

A Novel Application of Boolean Functions With High Algebraic Immunity in Minimal Codes

Boolean functions with high algebraic immunity are important cryptographic primitives in some stream ciphers. In this paper, two methodologies for constructing minimal binary codes from sets, Boolean functions and vectorial Boolean functions with high algebraic immunity, are proposed. More precisely, a general construction of new minimal codes using minimal codes contained in Reed-Muller codes and sets without nonzero low degree annihilators is presented. The other construction allows us to yield minimal codes from certain subcodes of Reed-Muller codes and vectorial Boolean functions with high algebraic immunity. Via these general constructions, infinite families of minimal binary linear codes of dimension m and length less than or equal to m(m+1)/2 are obtained. Besides, a lower bound on the minimum distance of the proposed minimal linear codes is established. Conjectures and open problems are also presented. The results of this paper show that Boolean functions with high algebraic immunity have nice applications in several fields additionally to symmetric cryptography, such as coding theory and secret sharing schemes.

Coded Computing via Binary Linear Codes: Designs and Performance Limits

We consider the problem of coded distributed computing where a large linear computational job, such as a matrix multiplication, is divided into <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> smaller tasks, encoded using an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(n,k)$ </tex-math></inline-formula> linear code, and performed over <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> distributed nodes. The goal is to reduce the average execution time of the computational job. We provide a connection between the problem of characterizing the average execution time of a coded distributed computing system and the problem of analyzing the error probability of codes of length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n$ </tex-math></inline-formula> used over erasure channels. Accordingly, we present closed-form expressions for the execution time using binary random linear codes and the best execution time any linear-coded distributed computing system can achieve. It is also shown that there exist <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">good</i> binary linear codes that not only attain (asymptotically) the best performance that any linear code (not necessarily binary) can achieve but also are numerically stable against the inevitable rounding errors in practice. We then develop a low-complexity algorithm for decoding Reed-Muller (RM) codes over erasure channels. Our decoder only involves additions, subtractions, and inversion of relatively small matrices of dimensions at most <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\log n+1$ </tex-math></inline-formula> , and enables coded computation over real-valued data. Extensive numerical analysis of the fundamental results as well as RM- and polar-coded computing schemes demonstrate the excellence of the RM-coded computation in achieving close-to-optimal performance while having a low-complexity decoding and explicit construction. The proposed framework in this paper enables efficient designs of distributed computing systems given the rich literature in the channel coding theory.

Shortened Linear Codes Over Finite Fields

The puncturing and shortening techniques are two important approaches to constructing new linear codes from old ones. In the past 70 years, a lot of progress on the puncturing technique has been made, and many works on punctured linear codes have been done. Many families of linear codes with interesting parameters have been obtained with the puncturing technique. However, little research on the shortening technique has been done and there are only a handful references on shortened linear codes. The first objective of this paper is to prove some general theory for shortened linear codes. The second objective is to study some shortened codes of the Hamming codes, Simplex codes, some Reed-Muller codes, and ovoid codes. Eleven families of optimal shortened codes over finite fields are presented in this paper. As a byproduct, five infinite families of 2-designs are also constructed from some of the shortened codes presented in this paper.

Projective toric codes

Any integral convex polytope [Formula: see text] in [Formula: see text] provides an [Formula: see text]-dimensional toric variety [Formula: see text] and an ample divisor [Formula: see text] on this variety. This paper gives an explicit construction of the algebraic geometric error-correcting code on [Formula: see text], obtained by evaluating global section of the line bundle corresponding to [Formula: see text] on every rational point of [Formula: see text]. This work presents an extension of toric codes analogous to the one of Reed–Muller codes into projective ones, by evaluating on the whole variety instead of considering only points with nonzero coordinates. The dimension of the code is given in terms of the number of integral points in the polytope [Formula: see text] and an algorithmic technique to get a lower bound on the minimum distance is described.

List Decoding of Arıkan's PAC Codes.

Polar coding gives rise to the first explicit family of codes that provably achieve capacity with efficient encoding and decoding for a wide range of channels. However, its performance at short blocklengths under standard successive cancellation decoding is far from optimal. A well-known way to improve the performance of polar codes at short blocklengths is CRC precoding followed by successive-cancellation list decoding. This approach, along with various refinements thereof, has largely remained the state of the art in polar coding since it was introduced in 2011. Recently, Arıkan presented a new polar coding scheme, which he called polarization-adjusted convolutional (PAC) codes. At short blocklengths, such codes offer a dramatic improvement in performance as compared to CRC-aided list decoding of conventional polar codes. PAC codes are based primarily upon the following main ideas: replacing CRC codes with convolutional precoding (under appropriate rate profiling) and replacing list decoding by sequential decoding. One of our primary goals in this paper is to answer the following question: is sequential decoding essential for the superior performance of PAC codes? We show that similar performance can be achieved using list decoding when the list size L is moderately large (say, ). List decoding has distinct advantages over sequential decoding in certain scenarios, such as low-SNR regimes or situations where the worst-case complexity/latency is the primary constraint. Another objective is to provide some insights into the remarkable performance of PAC codes. We first observe that both sequential decoding and list decoding of PAC codes closely match ML decoding thereof. We then estimate the number of low weight codewords in PAC codes, and use these estimates to approximate the union bound on their performance. These results indicate that PAC codes are superior to both polar codes and Reed–Muller codes. We also consider random time-varying convolutional precoding for PAC codes, and observe that this scheme achieves the same superior performance with constraint length as low as .

A Modified Symmetric Key Fully Homomorphic Encryption Scheme Based on Read-Muller Code

Homomorphic encryption became popular and powerful cryptographic primitive for various cloud computing applications. In the recent decades several developments has been made. Few schemes based on coding theory have been proposed but none of them support unlimited operations with security. We propose a modified Reed-Muller Code based symmetric key fully homomorphic encryption to improve its security by using message expansion technique. Message expansion with prepended random fixed length string provides one-to-many mapping between message and codeword, thus one-to many mapping between plaintext and ciphertext. The proposed scheme supports both (MOD 2) additive and multiplication operations unlimitedly. We make an effort to prove the security of the scheme under indistinguishability under chosen-plaintext attack (IND-CPA) through a game-based security proof. The security proof gives a mathematical analysis and its complexity of hardness. Also, it presents security analysis against all the known attacks with respect to the message expansion and homomorphic operations.

Counting plane cubic curves over finite fields with a prescribed number of rational intersection points

For each integer $k \in [0,9]$, we count the number of plane cubic curves defined over a finite field $\mathbb{F}_q$ that do not share a common component and intersect in exactly $k\ \mathbb{F}_q$-rational points. We set this up as a problem about a weight enumerator of a certain projective Reed-Muller code. The main inputs to the proof include counting pairs of cubic curves that do share a common component, counting configurations of points that fail to impose independent conditions on cubics, and a variation of the MacWilliams theorem from coding theory.

Long-Time Asymptotics for the Focusing Hirota Equation with Non-Zero Boundary Conditions at Infinity Via the Deift-Zhou Approach

Support constrained generator matrix for a linear code has been an active topic in recent years. The necessary and sufficient condition for the existence of MDS codes over small fields with support constrained generator matrices were conjectured by Dau, Song, Yuen in 2014. This GM-MDS conjecture was proved independently by Lovett and Yildiz-Hassibi in 2018. In this paper we propose the necessary and sufficient conditions for support constrained generator matrices of general linear codes based on the generalized Hamming weights. It is proved that the direct generalization of the GM-MDS conjecture for $2$-MDS codes and algebraic geometry codes is not true over arbitrary fields. We propose and prove a GHW-based sufficient condition for support constrained matrices of general linear codes. This is the first sufficient condition for the existence of support constrained generator matrices for general linear codes over arbitrary finite fields. Moreover a weaker GHW-based sufficient condition for support constrained generator matrices are given for the binary simplex code and the first order $q$-ary Reed-Muller codes.

Performance Evaluation of MRMED Algorithm by Monitoring BER

In this research work, we have developed a communication system (transmitter / receiver) to control peak to average power (PAPR) with small bit error rate (BER) for a 4G system called multicode code division multiple access (MCCDMA). Proposed communication system works on modified Reed Muller encoded data (MRMED) string. In MRMED data is first encoded with Reed Muller (RM) code. Thereafter, encoded RM message is XORed with optimal binary string, which results lower Peak to Average Power ratio (PAPR). A well-known fact is that, bit error rate (BER) is the best performance measurement tool for a communication system. To check the integrity of our communication system, we have run the simulation for monitoring BER using MRMED sequence. Simulation work conducted, with multipath Rayleigh fading, Minimum Shift Keying (MSK) modulation and several orders of RM codes. Our results show that implementing MRMED sequences of the suggested MCCDMA communication structure, returns noticeable lower BER. For instance, in case of RM(1,4), that has error improvement proficiency of 3 (three) errors , returns BER = 8.2x10−5 adopting MSK, at SNR = 12dB. Similarly, for RM(2,3), which has error improvement efficiency of 0 error and shows distinct BER of 4.9x10−4 at 12dB (SNR).In addition to using simulation for checking BER performance of our communication system, we have also shown in our results that, as the error improvement capability of different RM codes surges, correspondingly we get a lower BER.

A mΘ spectrum of Reed-Muller codes

In this paper, we construct mΘ spectrum of Reed Muller codes, a family of mΘ linear codes [5] with large mΘ minimum distance. First, we cover the necessary material on the mΘ Boolean algebra. We then define the mΘ spectrum of Reed Muller codes RM Θ (r, m) as certain subspaces of the mΘ Boolean algebra . We determinate the parameters of RM Θ (r, m).

Linear codes of 2-designs as subcodes of the generalized Reed-Muller codes

This paper is devoted to the affine-invariant ternary codes defined by Hermitian functions. We first compute the incidence matrices of the 2-designs supported by the minimum weight codewords of these ternary codes. Then we show that the linear codes spanned by the rows of these incidence matrices are subcodes of the 4-th order generalized Reed-Muller codes and also hold 2-designs. Finally, we determine the dimension and develop a lower bound on the minimum distance of the ternary linear codes.

Fast En/Decoding of Reed-Solomon Codes for Failure Recovery

Reed-Solomon (RS) codes are used in many storage systems for failure recovery. In popular software implementations, RS codes are defined by using a parity check matrix that is either a Cauchy matrix padded with an identity or a Vandermonde matrix. The encoding complexity can be reduced by searching for a Cauchy matrix that has a smaller number of ‘1's in its bit matrices or exploiting Reed-Muller (RM) transform in the Vandermonde matrix multiplication. This article proposes two new approaches that improve upon the previous schemes. In our first approach, different constructions of finite fields are explored to further reduce the number of ‘1's in the bit matrices of the Cauchy matrix and a new searching method is developed to find the matrices with minimum number of ‘1's. Our second approach defines RS codes using a parity check matrix in the format of a Vandermonde matrix concatenated with an identity matrix so that the multiplication with the inverse erasure columns in the encoding is eliminated and the decoding can be carried out using simplified formulas. The Vandermonde matrix in such an unconventional RS code definition needs to be constructed using finite field elements in non-consecutive order. A modification is also developed in this article to enable the application of the RM transform in this case to reduce the matrix multiplication complexity. For 4-erasure-correcting RS codes over <inline-formula><tex-math notation="LaTeX">$GF(2^8)$</tex-math></inline-formula> , the two proposed approaches increase the encoding throughput by 40 and 15 percent on average over the prior works based on Cauchy matrix and Vandermonde matrix with RM transform, respectively, for a range of codeword length. Moreover, the decoding throughput is also significantly improved.

Extended Length and Error-Correcting-Capacity of First-Order Reed–Muller Code in Extended Length Range

In this paper all the parameters in the conditions of rather comparatively larger probability of the value of n, have been discussed. In this extended range $$\upmu$$ < n $$\le$$ $$4\upmu$$ , the relation between existence of vector v(S) having the lowest weight in the coset RMm + v(S), and different values of r and d, have been analyzed. The relation between the number ( $$\tau$$ ) of vectors having lowest weight in RMm + v(S) and various values of n, r, and d have been analyzed; and the corresponding values of $$\tau$$ have been found. Also, the number of vectors (t) with lowest weight d for various values of d depending upon different quantities constituting the values of n and µ, has been found, n lying in extended range. All this is aimed at analyzing and improving the error-correcting-capacity of Reed–Muller code (first-order). Efforts are made to discuss the validity of the extended length of code and error-correcting-capacity of such a code of increased length.

The linear codes of t-designs held in the Reed-Muller and Simplex codes

A fascinating topic of combinatorics is the study of t-designs, which has a very long history. The incidence matrix of a t-design generates a linear code over GF(q) for any prime power q, which is called the linear code of the t-design over GF(q). On the other hand, some linear codes hold t-designs with t ≥ 1. The purpose of this paper is to study the linear codes of t-designs held in the Reed-Muller and Simplex codes. Some general theory for the linear codes of t-designs held in linear codes is presented. Several open problems are also presented.

Threshold-Secure Coding With Shared Key

Cryptographic protocols are often implemented at upper layers of communication networks, while error-correcting codes are employed at the physical layer. In this paper, we consider utilizing readily-available physical layer functions, such as encoders and decoders, together with shared keys to provide a threshold-type security scheme. To this end, we first consider a scenario where the effect of the physical layer is omitted and all the channels between the involved parties are assumed to be noiseless. We introduce a model for threshold-secure coding, where the legitimate parties communicate using a shared key such that an eavesdropper does not get any information, in an information-theoretic sense, about the key as well as about any subset of the input symbols of size up to a certain threshold. Then, a framework is provided for constructing threshold-secure codes from linear block codes while characterizing the requirements to satisfy the reliability and security conditions. Moreover, we propose a threshold-secure coding scheme, based on Reed-Muller (RM) codes, that meets security and reliability conditions. It is shown that the encoder and the decoder of the scheme can be implemented efficiently with quasi-linear time complexity. In particular, a successive cancellation decoder is shown for the RM-based coding scheme. Then we extend the setup to the scenario where the channel between the legitimate parties is no longer noiseless. The reliability condition for noisy channels is then modified accordingly, and a method is described to construct codes attaining threshold security as well as desired reliability. Also, we propose a coding scheme based on RM codes for threshold security and robustness designed for binary erasure channels along with a unified successive cancellation decoder. The proposed threshold-secure coding schemes are flexible and can be adapted for different key lengths.

Incremental Massive Random Access Exploiting the Nested Reed-Muller Sequences

Massive machine-type communication (mMTC) is expected to provide reliable and low-latency connectivity for an enormous number of devices, which turn active sporadically or frequently. In this highly dynamic situation, it is crucial to design efficient random access (RA) procedures to cope both with the flood of simultaneous access requests and with the potential access failures. In this article, by exploiting the large sequence space, the excellent correlation property and especially the elegant nested structure of Reed-Muller (RM) sequences, we propose a new RA scheme, which facilitates both instantaneous access for newly active users and incremental access for the existing users who suffer from detection failures. In particular, when a failure occurs, the user continues accessing the channel employing an expanded RM sequence, which is combined with the previously received ones at the access point (AP) to form a longer sequence so as to attain potentially better detection probability. Furthermore, a recursive detection algorithm is designed for jointly detecting the resultant RM sequences and the channel coefficients of both the newly active users and the existing ones. The performance of the proposed algorithm is analyzed in detail. Our simulation results validate the analysis and show the scheme’s superior access probability, access latency and computational complexity.

Secure Codes With Accessibility for Distributed Storage

A distributed storage system must support efficient access to stored data while ensuring recovery of temporally unavailable nodes. Another important aspect of a distributed storage system is security. In this paper, we bring these features together and investigate the problem of efficient access to stored data in presence of a passive eavesdropper with access to limited number of nodes. The access efficiency is measured in two different terms, namely, the number of accessed nodes and the volume of generated network traffic. These quantities possess a natural connection to locality and repair bandwidth in distributed storage system. For each of them we derive bounds on parameters and provide explicit constructions based on maximum distance separable codes. Motivated by practical perspectives we propose the techniques to ensure the same workload on each node as well as constructions over small fields based on subfield subcodes, Euclidean geometry codes and Reed-Muller codes. Finally, we derive an asymptotic random coding bound on parameters of a secure distributed storage system and propose further research directions.

Binary Golay Spreading Sequences and Reed-Muller Codes for Uplink Grant-Free NOMA

Non-orthogonal multiple access (NOMA) is an emerging technology for massive connectivity in machine-type communications (MTC). In code-domain NOMA, non-orthogonal spreading sequences are uniquely assigned to all devices, where active ones attempt a grant-free access to a system. In this paper, we study a set of user-specific, non-orthogonal, binary spreading sequences for uplink grant-free NOMA. Based on binary Golay complementary sequences, each spreading sequence provides the peak-to-average power ratio (PAPR) of at most 3 dB for multicarrier transmission. Exploiting the theoretical connection to Reed-Muller codes, we conduct a probabilistic analysis to search for a permutation set for Golay sequences, which presents theoretically bounded low coherence for the spreading matrix. Simulation results confirm that the maximum PAPR of transmitted multicarrier signals via the spreading sequences is significantly lower than those for random bipolar, Gaussian, and pseudo-random sequences. Thanks to the low coherence, the performance of compressed sensing (CS) based joint channel estimation and multiuser detection using the spreading sequences turns out to be superior or similar to those for random bipolar, Gaussian, pseudo-random, and Zadoff-Chu sequences. The binary Golay spreading sequences have only two phases regardless of the sequence length, which can make them suitable for low cost MTC devices.

Pruning and Quantizing Neural Belief Propagation Decoders

We consider near maximum-likelihood (ML) decoding of short linear block codes. In particular, we propose a novel decoding approach based on neural belief propagation (NBP) decoding recently introduced by Nachmani et al. in which we allow a different parity-check matrix in each iteration of the algorithm. The key idea is to consider NBP decoding over an overcomplete parity-check matrix and use the weights of NBP as a measure of the importance of the check nodes (CNs) to decoding. The unimportant CNs are then pruned. In contrast to NBP, which performs decoding on a given fixed parity-check matrix, the proposed pruning-based neural belief propagation (PB-NBP) typically results in a different parity-check matrix in each iteration. For a given complexity in terms of CN evaluations, we show that PB-NBP yields significant performance improvements with respect to NBP. We apply the proposed decoder to the decoding of a Reed-Muller code, a short low-density parity-check (LDPC) code, and a polar code. PB-NBP outperforms NBP decoding over an overcomplete parity-check matrix by 0.27-0.31 dB while reducing the number of required CN evaluations by up to 97%. For the LDPC code, PB-NBP outperforms conventional belief propagation with the same number of CN evaluations by 0.52 dB. We further extend the pruning concept to offset min-sum decoding and introduce a pruning-based neural offset min-sum (PB-NOMS) decoder, for which we jointly optimize the offsets and the quantization of the messages and offsets. We demonstrate performance 0.5 dB from ML decoding with 5-bit quantization for the Reed-Muller code.