Gleason's 1970 theorem on weight enumerators of self-dual codes has played a crucial role for research in coding theory during the last four decades. Plenty of generalizations have been proved but, to our knowledge, they are all based on the symmetries given by MacWilliams' identities. This paper is intended to be a first step towards a more general investigation of symmetries of weight enumerators. We list the possible groups of symmetries, dealing both with the finite and infinite case, we develop a new algorithm to compute the group of symmetries of a given weight enumerator and apply these methods to the family of Reed-Muller codes, giving, in the binary case, an analogue of Gleason's theorem for all parameters.
It is well-known that Reed-Solomon codes and extended Reed-Solomon codes are two special classes of MDS codes with wide applications in practice. The complete weight enumerators of these codes are very important for determining the capability of both error-detection and error-correction. In this paper, for any positive integer $m$ and prime $p$, basing on the character sums, we determine the complete weight enumerators of the Reed-Solomon code and the extended Reed-Solomon code with dimension $k$ $(k=2,3)$ over $\mathbb{F}_{p^m}$, explictly, which are generalizations of the corresponding resul…
We give a description of the weighted Reed-Muller codes over a prime field in a modular algebra. A description of the homogeneous Reed-Muller codes in the same ambient space is presented for the binary case. A decoding procedure using the Landrock-Manz method is developed.
The punctured binary Reed-Muller code is cyclic and was generalized into the punctured generalized Reed-Muller code over $\gf(q)$ in the literature. The major objective of this paper is to present another generalization of the punctured binary Reed-Muller code. Another objective is to construct a family of reversible cyclic codes that are related to the newly generalized Reed-Muller codes.
Recently, Bhaintwal and Wasan studied the Generalized Reed-Muller codes over the prime power integer residue ring. In this paper, we give a generalization of these codes to Generalized Reed-Muller codes over Galois rings.
This paper considers '$δ$-almost Reed-Muller codes', i.e., linear codes spanned by evaluations of all but a $δ$ fraction of monomials of degree at most $d$. It is shown that for any $δ> 0$ and any $\varepsilon>0$, there exists a family of $δ$-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 approach is …
This paper is concerned with the affine-invariant ternary codes which are defined by Hermitian functions. We compute the incidence matrices of 2-designs that are supported by the minimum weight codewords of these ternary codes. The linear codes generated by the rows of these incidence matrix are subcodes of the extended codes of the 4-th order generalized Reed-Muller codes and they also hold 2-designs. Finally, we give the dimensions and lower bound of the minimum weights of these linear codes.
In distributed storage systems, both the repair bandwidth and locality are important repair cost metrics to evaluate the performance of a storage code. Recently, Guruswami and Wooters proposed an optimal linear repair scheme based on Reed-Solomon codes for a single failure, improved the bandwidth of the classical repair scheme. In this paper, we consider the repair bandwidth of Generalized Reed-Muller (GRM) codes, which have good locality property. We generalize Guruswami and Wooters' repairing scheme to GRM codes for single failure, which has nontrivial bandwidth closing to the lower bound wh…
First, a new proof of Berman and Charpin's characterization of the Reed-Muller codes over the binary field or over an arbitrary prime field is presented. These codes are considered as the powers of the radical of a modular algebra. Secondly, the same method is used for the study of the Generalized Reed-Muller codes over a non prime field.
The weight distribution of second order $q$-ary Reed-Muller codes have been determined by Sloane and Berlekamp (IEEE Trans. Inform. Theory, vol. IT-16, 1970) for $q=2$ and by McEliece (JPL Space Programs Summary, vol. 3, 1969) for general prime power $q$. Unfortunately, there were some mistakes in the computation of the latter one. This paper aims to provide a precise account for the weight distribution of second order $q$-ary Reed-Muller codes. In addition, the weight distributions of second order $q$-ary homogeneous Reed-Muller codes and second order $q$-ary projective Reed-Muller codes are …
In this paper, we study the third weight of generalized Reed-Muller codes. We prove under some restrictive condition that the third weight of generalized Reed-Muller codes depends on the third weight of generalized Reed-Muller codes of small order with two variables. In some cases, we are able to determine the third weight and the third weight codewords of generalized Reed-Muller codes.
In this note, we give a very simple description of the generalized Hamming weights of Reed--Muller codes. For this purpose, we generalize the well-known Macaulay representation of a nonnegative integer and state some of its basic properties.
We give a new proof of Delsarte, Goethals and Mac williams theorem on minimal weight codewords of generalized Reed-Muller codes published in 1970. To prove this theorem, we consider intersection of support of minimal weight codewords with affine hyperplanes and we proceed by recursion.
In 1981, Schatz proved that the covering radius of the binary Reed-Muller code $RM(2,6)$ is 18. For $RM(2,7)$, we only know that its covering radius is between 40 and 44. In this paper, we prove that the covering radius of the binary Reed-Muller code $RM(2,7)$ is at most 42. Moreover, we give a sufficient and necessary condition for Boolean functions of 7-variable to achieve the second-order nonlinearity 42.
We generalize to any q a theorem about covering radius of linear codes proved by Helleseth, Klove and Mykkelvit. Then we determine the covering radius of first order generalized Reed-Muller codes in second order generalized Reed-Muller codes. Using these results, we are able to give bounds for the covering radius of first order generalized Reed-Muller codes. Finaly, using Magma, we get some improvements for q=3.
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 $|S_1|/|S_2|$ on the minimum distance. In conclusion the weighted Reed-Muller code construction is much better than its reputation. For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms. With a small modification one of these algorithms is able to correct up to 31 errors of the [49, 11, 28] Joyn…
Batch codes, introduced by Ishai et al. encode a string $x \in Σ^{k}$ into an $m$-tuple of strings, called buckets. In this paper we consider multiset batch codes wherein a set of $t$-users wish to access one bit of information each from the original string. We introduce a concept of optimal batch codes. We first show that binary Hamming codes are optimal batch codes. The main body of this work provides batch properties of Reed-Müller codes. We look at locality and availability properties of first order Reed-Müller codes over any finite field. We then show that binary first order Reed-Müller c…
Recent work have shown that Reed-Muller (RM) codes achieve the erasure channel capacity. However, this performance is obtained with maximum-likelihood decoding which can be costly for practical applications. In this paper, we propose an encoding/decoding scheme for Reed-Muller codes on the packet erasure channel based on Plotkin construction. We present several improvements over the generic decoding. They allow, for a light cost, to compete with maximum-likelihood decoding performance, especially on high-rate codes, while significantly outperforming it in terms of speed.
We present an asymptotic limit between correctable and uncor-rectable errors on the Reed-Muller codes of any order. This limit is theoretical and does not depend of any decoding algorithm.
First we study some properties of the modular group algebra $\mathbb{F}_{p^r}[G]$ where $G$ is the additive group of a Galois ring of characteristic $p^r$ and $\mathbb{F}_{p^r}$ is the field of $p^r$ elements. Secondly a description of the Generalized Reed-Muller codes over $\mathbb{F}_{p^r}$ in $\mathbb{F}_{p^r}[G]$ is presented.
A novel permutation decoding method for Reed-Muller codes is presented. The complexity and the error correction performance of the suggested permutation decoding approach are similar to that of the recursive lists decoder. It is demonstrated that the proposed decoding technique can take advantage of several early termination methods leading to a significant reduction of the operations number required for the decoding, with the error correction performance being the same.
Recursive decoding techniques are considered for Reed-Muller (RM) codes of growing length $n$ and fixed order $r.$ An algorithm is designed that has complexity of order $n\log n$ and corrects most error patterns of weight up to $n(1/2-\varepsilon)$ given that $\varepsilon$ exceeds $n^{-1/2^{r}}.$ This improves the asymptotic bounds known for decoding RM codes with nonexponential complexity.
Projective Reed-Muller codes correspond to subcodes of the Reed-Muller code in which the polynomials being evaluated to yield codewords, are restricted to be homogeneous. The Generalized Hamming Weights (GHW) of a code ${\cal C}$, identify for each dimension $ν$, the smallest size of the support of a subcode of ${\cal C}$ of dimension $ν$. The GHW of a code are of interest in assessing the vulnerability of a code in a wiretap channel setting. It is also of use in bounding the state complexity of the trellis representation of the code. In prior work by the same authors, a code-shortening algori…
In this paper we describe a variation of the classical permutation decoding algorithm that can be applied to any affine-invariant code with respect to certain type of information sets. In particular, we can apply it to the family of first-order Reed-Muller codes with respect to the information sets introduced in [2]. Using this algortihm we improve considerably the number of errors we can correct in comparison with the known results in this topic.
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 o…
We study the quadratic residue weight enumerators of the dual projective Reed-Solomon codes of dimensions $5$ and $q-4$ over the finite field $\mathbb{F}_q$. Our main results are formulas for the coefficients of the the quadratic residue weight enumerators for such codes. If $q=p^v$ and we fix $v$ and vary $p$ then our formulas for the coefficients of the dimension $q-4$ code involve only polynomials in $p$ and the trace of the $q$th and $(q/p^2)$th Hecke operators acting on spaces of cusp forms for the congruence groups $\operatorname{SL}_2 (\mathbb{Z}), Γ_0(2)$, and $Γ_0(4)$. The main tool w…
The second weight of the Generalized Reed-Muller code of order $d$ over the finite field with $q$ elements is now known for $d <q$ and $d>(n-1)(q-1)$. In this paper, we determine the second weight for the other values of $d$ which are not multiple of $q-1$ plus 1. For the special case $d=a(q-1)+1$ we give an estimate.
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 symmetry as well. The main idea is to combine the following observations: (i) monotone functions experience a sharp threshold behavior, (ii) the extrinsic information transfer (EXIT) functions are monotone, (iii) Reed--Muller codes are 2-transitive and thus the EXIT functions associated with their codeword bits are all equal, and (iv) therefore the Area Theore…
Over a finite field $\F_q$ the $(n,d,q)$-Reed-Muller code is the code given by evaluations of $n$-variate polynomials of total degree at most $d$ on all points (of $\F_q^n$). The task of testing if a function $f:\F_q^n \to \F_q$ is close to a codeword of an $(n,d,q)$-Reed-Muller code has been of central interest in complexity theory and property testing. The query complexity of this task is the minimal number of queries that a tester can make (minimum over all testers of the maximum number of queries over all random choices) while accepting all Reed-Muller codewords and rejecting words that ar…
The paper proposes to decode Reed-Muller (RM) codes by projecting onto only a few subspaces such that the number of projections is significantly reduced. It reveals that the probability that error pairs are canceled simultaneously in two different projections is determined by their intersection size. Then, correlation coefficient which indicates the intersection size of two subspaces in a collection is introduced for collecting subspaces. Simulation results show that our proposed approach with a small number of projections onto collected subspaces performs close to the original approach.
Motivated by applications in combinatorial design theory and constructing LCD codes, C. Ding et al \cite{DLX} introduced cyclic codes $\mho(q,m,h)$ and $\bar\mho(q,m,h)$ over $\mathbb{F}_q$ as new generalization and version of the punctured binary Reed-Muller codes. In this paper, we show several new results on minimum distance of $\mho(q,m,h)$ and $\bar\mho(q,m,h)$ which are generalization or improvement of previous results given in \cite{DLX}.
We consider recursive decoding for Reed-Muller (RM) codes and their subcodes. Two new recursive techniques are described. We analyze asymptotic properties of these algorithms and show that they substantially outperform other decoding algorithms with nonexponential complexity known for RM codes. Decoding performance is further enhanced by using intermediate code lists and permutation procedures. For moderate lengths up to 512, near-optimum decoding with feasible complexity is obtained.
This paper introduces the Reed Muller Sieve, a deterministic measurement matrix for compressed sensing. The columns of this matrix are obtained by exponentiating codewords in the quaternary second order Reed Muller code of length $N$. For $k=O(N)$, the Reed Muller Sieve improves upon prior methods for identifying the support of a $k$-sparse vector by removing the requirement that the signal entries be independent. The Sieve also enables local detection; an algorithm is presented with complexity $N^2 \log N$ that detects the presence or absence of a signal at any given position in the data doma…
We propose new results on low weight codewords of affine and projective generalized Reed-Muller codes. In the affine case we prove that if the size of the working finite field is large compared to the degree of the code, the low weight codewords are products of affine functions. Then in the general case we study some types of codewords and prove that they cannot be second, thirds or fourth weight depending on the hypothesis. In the projective case the second distance of generalized Reed-Muller codes is estimated, namely a lower bound and an upper bound of this weight are given.
We study generalized covering radii, a fundamental property of linear codes that characterizes the trade-off between storage, latency, and access in linear data-query protocols such as PIR. We prove lower and upper bounds on the generalized covering radii of Reed-Muller codes, as well as finding their exact value in certain extreme cases. With the application to linear data-query protocols in mind, we also construct a covering algorithm that gets as input a set of points in space, and find a corresponding set of codewords from the Reed-Muller code that are jointly not farther away from the inp…
It was proved by J. Schatz that the covering radius of the second order Reed--Muller code $RM(2, 6)$ is 18 (IEEE Trans Inf Theory 27: 529--530, 1985). However, the covering radius of $RM(2,7)$ has been an open problem for many years. In this paper, we prove that the covering radius of $RM(2,7)$ is 40, which is the same as the covering radius of $RM(2,7)$ in $RM(3,7)$. As a corollary, we also find new upper bounds for $RM(2,n)$, $n=8,9,10$.
