On the weight distribution of the coset leaders of the first-order Reed - Muller code (Corresp.)

The approach to error correction coding taken by modern digital communication systems started in the late 1940'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.

[Formula: see text]A hyperbolic code is an evaluation code that improves a Reed–Muller code because the dimension increases while the minimum distance is not penalized. We give necessary and sufficient conditions, based on the basic parameters of the Reed–Muller code, to determine whether a Reed–Muller code coincides with a hyperbolic code. Given a hyperbolic code [Formula: see text], we find the largest Reed–Muller code contained in [Formula: see text] and the smallest Reed–Muller code containing [Formula: see text]. We then prove that similar to Reed–Muller and affine Cartesian codes, the [Formula: see text]th generalized Hamming weight and the [Formula: see text]th footprint of the hyperbolic code coincide. Unlike for Reed–Muller and affine Cartesian codes, determining the [Formula: see text]th footprint of a hyperbolic code is still an open problem. We give upper and lower bounds for the [Formula: see text]th footprint of a hyperbolic code that, sometimes, are sharp.

We prove that the families of quaternary Reed – Muller codes obtained by the BQ-Plotkin construction 2009 have bases of minimum weight codewords. In 2020 we found that the quaternary Reed – Muller codes constructed by the quaternary Plotkin approach have the minimum weight bases. Combining these two constructions we prove that all known quaternary linear Reed – Muller codes have bases of minimum weight codewords. The bases are obtained iteratively.

Reed Muller (RM) codes are known for their good minimum distance. One can use their structure to construct polar-like codes with good distance properties by choosing the information set as the rows of the polarization matrix with the highest Hamming weight, instead of the most reliable synthetic channels. However, the information length options of RM codes are quite limited due to their specific structure. In this work, we present sufficient conditions to increase the information length by at least one bit for some underlying RM codes and in order to obtain pre-transformed polar-like codes with the same minimum distance than lower rate codes. Moreover, our findings are combined with the method presented in [1] to further reduce the number of minimum weight codewords. Numerical results show that the designed codes perform close to the meta-converse bound at short blocklengths and better than the polarized adjusted convolutional polar codes with the same parameters.

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.

This paper studies the parameters for which binary Reed-Muller (RM) codes can be decoded successfully on the binary erasure channel and binary symmetry channel, and, in particular, when can they achieve capacity for these two classical channels. Necessarily, this paper also studies the properties of evaluations of multivariate GF(2) polynomials on the 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 the 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 the truth tables of all the 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 tensorizing 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 the RM codes and using our results for erasures imply our result on the unique decoding of the 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 bounds from constant degree to linear degree polynomials.

In this paper, we propose efficient maximum-likelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed- Muller codes are well known examples of such KPB codes. The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the “tree folding” technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κ-folding operation transforms the binary tree with N levels into a non-binary tree with N=2κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find (n over κ) different folding which lead to decoders with different complexity, for a given code. Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.

The notion of a Private Information Retrieval (PIR) code was recently introduced by Fazeli, Vardy and Yaakobi [1] who showed that this class of codes permit PIR at reduced levels of storage overhead in comparison with rephcated-server PIR. In the present paper, the construction of an (n, k) τ-server binary linear PIR code having parameters n = lΣ i=0 (m i ), k = (m i ) and τ = 2l for any integer m ≥ l ≥ 0 is presented. These codes are obtained through homogeneous-polynomial evaluation and correspond to the binary. Projective Reed Muller (PRM) code. The construction can be extended to yield PIR codes for any τ = ∊ {2l, 2l − 1 | l ∊ Z, l ≥ 0} and any value of k, through a combination of single-symbol puncturing and shortening of the PRM code. Each of these code constructions above, have smaller storage overhead in comparison with known short block length codes in [1]. For the particular case of τ = 3,4, we show that the codes constructed here are optimal, systematic PIR codes by providing an improved lower bound on the block length n{k, τ) of a systematic PIR code. It follows from a result by Vardy and Yaakobi [2], that these codes also yield optimal, systematic primitive multi-set {n, k, τ) B batch codes for τ = 3,4. The PIR code constructions presented here also yield upper bounds on the generahzed Hamming weights of binary PRM codes.

Reed—Muller (RM) codes are a class of low-rate codes that have been largely replaced by Reed—Solomon codes. However, they have certain attributes which may well see their re-emergence in modern applications. In particular, coding and decoding is very fast. With Hamming codes (and the CRC used for correction), the message size was n bits where n = 2 r − 1 and r was the number of redundant bits. Reed—Muller codes are based on maximal length codes which are, in some ways, the opposite of Hamming codes. Taking a typical Hamming code, say (15, 11), with four redundancy bits, the corresponding maximal length code would be a (15, 4) code so now, n = 2 k − 1 and k is the number of information bits. This gives a very low coding rate of about 0.27.

This study investigates the decoding algorithms of binary Reed–Muller (RM) codes. The main goal of the study is to modify majority‐logic decoding using sum‐product or min‐sum decoding algorithms at the highest‐order information bits in RM codes. Numerical results show that compared with the traditional majority‐logic decoding algorithm, the modified majority‐logic decoding method proposed in this study can reduce the bit error rate by a large margin. This is achieved without incurring a significant increase in decoding complexity because soft‐decision decoding is only used in the highest information bits in RM codes whereas majority‐logic decoding is still applied at the other orders of information bits in RM codes.

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 $\varepsilon >0$ , there exists a family of $\delta $ -almost Reed–Muller codes of constant rate that correct $1/2- \varepsilon $ 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.

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.

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.

The paper proposes a new multi-step soft-decision decoding for the generic Reed–Muller (RM) code family, which is based on a simple multi-step soft-input soft-output (SISO) module and inherent merits of the RM codes. In an AWGN channel, the simulation results show that for RM codes of different code rates and lengths the introduced approach can offer pronounced performance gains over other soft-decision sub-optimal approaches and a conventional hard majority logic decoder. The results also indicate that the new algorithm can achieve a performance very close to that provided by a maximum likelihood decoder under the same conditions.

In this study, a concatenated coding scheme based on Reed–Muller (RM) codes and bit‐extension codes is proposed for equivocation of a wiretap channel. RM codes and their cosets are adopted for message encoding, and bit‐extension codes are used to enhance the equivocation capability for a wiretapper's channel. The average equivocation is discussed when only RM codes are used in the system, and the probability causing imperfect secrecy is also determined. Analytical results show that the proposed code can be used for the equivocation capability of wiretap channels and suggest a proper management over a wiretap channel.