Binary Reed-Muller Codes Research Articles

Two classes of few-Lee weight [formula omitted]-linear codes using simplicial complexes and minimal codes via Gray map

Recently certain few-Lee weight additive codes over some mixed alphabet rings with optimal or minimal Gray images are constructed using suitable defining sets. Inspired by these works, we construct two special classes of linear codes over Z2[u] with u2=0 by using the mixed alphabet ring Z2Z2[u] and employing simplicial complexes having one or more maximal elements. We show that these codes have few-Lee weights by determining their Lee weight distributions. As applications, we are getting the 1st order binary Reed-Muller codes of length 23m−2 (where m>1) and some infinite families of binary self-orthogonal minimal codes via Gray map.

Several families of binary cyclic codes with good parameters

Binary odd-like duadic codes have parameters [n,(n+1)/2], where n is odd. It is well known that the minimum odd weight of odd-like duadic codes has the lower bound n. The binary quadratic-residue codes and the punctured binary Reed-Muller codes of certain order are two families of binary cyclic codes with parameters [n,(n+1)/2,d≥n]. It is very hard to construct an infinite family of binary cyclic codes with length n and dimension near (n+1)/2 whose minimum distances have a square-root bound. Recently, Tang and Ding constructed an infinite family of binary cyclic codes with parameters [2m−1,2m−1,d] whose minimum distances have lower bounds close to the square-root bound. The objective of this paper is to construct several infinite families of binary cyclic codes of length 2m−1 and dimension near 2m−1 whose minimum distance d much exceeds the square-root bound. For m≥9 being odd, an infinite family of binary cyclic codes with parameters [2m,2m−1,d≥3⋅2m−12−1] is presented. For m≥8 being even, an infinite family of binary cyclic codes with parameters [2m,2m−1−1+m2,d≥3⋅2m−22+1] is presented.

DECODING BINARY REED-MULLER CODES VIA GROEBNER BASES

The binary Reed-Muller codes can be characterized as the radical powers of a modular group algebra. In this paper, we deal with the Groebner bases to decode these codes.

Classification of the Codewords of Weights 16 and 18 of the Reed-Muller Code RM(n-3,n)

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.

Iterative Hard-Decision Decoding Algorithms for Binary Reed-Muller Codes

In this paper, novel hard-decision iterative decoding algorithms for binary Reed-Muller (RM) codes are presented. First, two algorithms are devised based on the majority-logic decoding algorithm with reliability measures of the received sequence. The bit-flipping (BF) and the normalized bit-flipping (NBF) decoding algorithms are hard-decision decoding algorithms. According to the updated hard reliability measures, the BF and NBF algorithms flip one bit of the received hard-decision sequence at a time in each iteration. The NBF decoding algorithm performs better than the BF decoding algorithm by normalizing the reliability measures of the information bits. Moreover, the BF and NBF algorithms are modified to flip multiple bits in one iteration to reduce the average number of iterations. The modified decoding algorithms are called the multiple-bits-flipping (MBF) algorithm and the normalized multiple-bits-flipping (NMBF) algorithm, respectively. The proposed algorithms have low computational complexities and can converge rapidly after a small number of iterations. The simulation results show that the proposed hard-decision decoding algorithms outperform the conventional decoding algorithm.

New classes of few-weight ternary codes from simplicial complexes

<abstract><p>In this article, we describe two classes of few-weight ternary codes, compute their minimum weight and weight distribution from mathematical objects called simplicial complexes. One class of codes described here has the same parameters with the binary first-order Reed-Muller codes. A class of (optimal) minimal linear codes is also obtained in this correspondence.</p></abstract>

On metric regularity of Reed–Muller codes

In this work we study metric properties of the well-known family of binary Reed–Muller codes. Let A be an arbitrary subset of the Boolean cube, and $${\widehat{A}}$$ be the metric complement of A—the set of all vectors of the Boolean cube at the maximal possible distance from A. If the metric complement of $${\widehat{A}}$$ coincides with A, then the set A is called a metrically regular set. The problem of investigating metrically regular sets appeared when studying bent functions, which have important applications in cryptography and coding theory and are also one of the earliest examples of a metrically regular set. In this work we describe metric complements and establish the metric regularity of the codes $${{\mathcal {R}}}{{\mathcal {M}}}(0,m)$$ and $${{\mathcal {R}}}{{\mathcal {M}}}(k,m)$$ for $$k \geqslant m-3$$ . Additionally, the metric regularity of the codes $${{\mathcal {R}}}{{\mathcal {M}}}(1,5)$$ and $${{\mathcal {R}}}{{\mathcal {M}}}(2,6)$$ is proved. Combined with previous results by Tokareva (Discret Math 312(3):666–670, 2012) concerning duality of affine and bent functions, this establishes the metric regularity of most Reed–Muller codes with known covering radius. It is conjectured that all Reed–Muller codes are metrically regular.

An Upper Bound on $\ell_q$ Norms of Noisy Functions

Let ${T}_{ \epsilon }$ , $0 \le \epsilon \le 1/2$ , be the noise operator acting on functions on the boolean cube $\{0,1\}^{n}$ . Let $f$ be a nonnegative function on $\{0,1\}^{n}$ and let ${q} \ge 1$ . We upper bound the $\ell _{{q}}$ norm of ${T}_{ \epsilon } {f}$ by the average $\ell _{{q}}$ norm of conditional expectations of f, given sets of roughly $(1-2 \epsilon)^{{r}({q})} \cdot {n}$ variables, where r is an explicitly defined function of q . We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of BEC-capacity achieving binary linear codes and their duals. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for all (constant) rates.

T-Count Optimization and Reed–Muller Codes

In this paper, we study the close relationship between Reed-Muller codes and single-qubit phase gates from the perspective of $T$-count optimization. We prove that minimizing the number of $T$ gates in an $n$-qubit quantum circuit over CNOT and $T$, together with the Clifford group powers of $T$, corresponds to finding a minimum distance decoding of a length $2^n-1$ binary vector in the order $n-4$ punctured Reed-Muller code. Moreover, we show that the problems are polynomially equivalent in the length of the code. As a consequence, we derive an algorithm for the optimization of $T$-count in quantum circuits based on Reed-Muller decoders, along with a new upper bound of $O(n^2)$ on the number of $T$ gates required to implement an $n$-qubit unitary over CNOT and $T$ gates. We further generalize this result to show that minimizing small angle rotations corresponds to decoding lower order binary Reed-Muller codes. In particular, we show that minimizing the number of $R_Z(2\pi/d)$ gates for any integer $d$ is equivalent to minimum distance decoding in $\mathcal{RM}(n - k - 1, n)^*$, where $k$ is the highest power of $2$ dividing $d$.

Batch codes from Hamming and Reed-Muller codes

Batch codes, introduced by Ishai et al., encode a string $x \in \Sigma^{k}$ 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-Muller codes. We look at locality and availability properties of first order Reed-Muller codes over any finite field. We then show that binary first order Reed-Muller codes are optimal batch codes when the number of users is 4 and generalize our study to the family of binary Reed-Muller codes which have order less than half their length.

Another generalisation of the binary Reed–Muller codes and its applications

The punctured binary Reed–Muller code is cyclic and was generalised into the punctured generalised Reed–Muller code over GF(q) in the literature. The first objective of this paper is to present another generalisation of the punctured binary Reed–Muller code and the binary Reed–Muller code, and analyse these codes. The second objective of this paper is to consider two applications of the new codes in constructing LCD codes and 2-designs. The major motivation of constructing and studying the new codes and their extended codes is the construction of 2-designs, which is an interesting topic in combinatorics. It is remarkable that the family of newly generalised cyclic codes contains a subclass of optimal ternary codes with parameters [3m−1,3m−1−2m,4] for all m≥2. Their extended codes have parameters [3m,3m−1−2m,5] for all m≥2, and are also optimal.

Modified majority logic decoding of Reed–Muller codes using factor graphs

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.

New bounds on the covering radius of the second order Reed-Muller code of length 128

In 1981, Schatz proved that the covering radius of the binary Reed-Muller code RM(2, 6) is 18. It was previously shown that the covering radius of RM(2, 7) is between 40 and 44. In this paper, we prove that the covering radius of RM(2, 7) is at most 42. As a corollary, we also find new upper bounds for RM(2, n), n = 8, 9, 10. Moreover, we give a sufficient and necessary condition for the covering radius of RM(2, 7) to be equal to 42. Using this condition, we prove that the covering radius of RM(2, 7) in RM(4, 7) is exactly 40, and as a by-product, we conclude that the covering radius of RM(2, 7) in the set of 2-resilient Boolean functions is at most 40, which improves the bound given by Borissov et al. (IEEE Trans. Inf. Theory 51(3):1182–1189, 2005).

Matrix embedding in steganography with binary Reed–Muller codes

This study presents a modified majority-logic decoding algorithm of Reed-Muller (RM) codes for matrix embedding (ME) in steganography. An ME algorithm uses linear block code to improve the embedding efficiency in steganography. The optimal embedding algorithm in steganography is equivalent to the maximum likelihood decoding (MLD) algorithm in error-correcting codes. The main disadvantage of ME is that the equivalent MLD algorithm of lengthy embedding codes requires highly complex embedding. This study used RM codes to embed data in binary host images. The authors propose a novel low-complexity embedding algorithm that uses a modified majority-logic algorithm to decode RM codes, in which a message-passing algorithm (i.e. sum-product, min-sum, or bias propagation) is performed on the highest order of information bits in the RM codes. The experimental results indicate that integrating bias propagation into the proposed scheme achieves superior embedding efficiency (relative to when the sum-product or min-sum algorithm is used) and can even achieve the embedding bound of RM codes.

Golay sequences coded coherent optical OFDM for long-haul transmission

We propose to use binary Golay sequences in coherent optical orthogonal frequency division multiplexing (CO-OFDM) to improve the long-haul transmission performance. The Golay sequences are generated by binary Reed-Muller codes, which have low peak-to-average power ratio and certain error correction capability. A low-complexity decoding algorithm for the Golay sequences is then proposed to recover the signal. Under same spectral efficiency, the QPSK modulated OFDM with binary Golay sequences coding with and without discrete Fourier transform (DFT) spreading (DFTS-QPSK-GOFDM and QPSK-GOFDM) are compared with the normal BPSK modulated OFDM with and without DFT spreading (DFTS-BPSK-OFDM and BPSK-OFDM) after long-haul transmission. At a 7% forward error correction code threshold (Q2 factor of 8.5dB), it is shown that DFTS-QPSK-GOFDM outperforms DFTS-BPSK-OFDM by extending the transmission distance by 29% and 18%, in non-dispersion managed and dispersion managed links, respectively.

Coding to reduce the interference to carrier ratio of OFDM signals

In this paper, we propose a simple approach to reduce the sensitivity of OFDM system to the carrier frequency offset (CFO). The main idea is to choose a well-known channel code and to rotate each coordinate by a fixed phase shift such that the maximum inter-carrier interference (ICI) taken over all sub-carriers is minimized. This approach is based on a geometric interpretation of the peak interference to carrier ratio (PICR) of OFDM signals. Simulation results show that a reduction in PICR of 7 dB can be easily achieved. Furthermore, we addressed the fundamental limit of the proposed technique by providing both an exact and an approximate lower bound of PICR of phase-shifted binary codes. The bounds are applied to some codes: non-redundant binary code, BCH codes, and Reed-Muller codes. Simulation results demonstrated that phase-shift designs approach lower bounds and are resilient to CFO change.

Spherically Punctured Reed-Muller Codes

Consider a binary Reed–Muller code RM $(r,m)$ defined on the $m$ -dimensional hypercube $\mathbb {F}_{2}^{m}$ . In this paper, we study punctured Reed–Muller codes $P_{r}(m,b)$ , whose positions are restricted to the $m$ -tuples of a given Hamming weight $b$ . In combinatorial terms, this paper concerns $m$ -variate Boolean polynomials of any degree $r$ , which are evaluated on a Hamming sphere of some radius $b$ in $\mathbb {F}_{2}^{m}$ . Codes $P_{r}(m,b)$ inherit some recursive properties of RM codes. In particular, they can be built from the shorter codes, by decomposing a spherical $b$ -layer into sub-layers of smaller dimensions. However, these sub-layers have different sizes and do not form the classical Plotkin construction. We analyze recursive properties of the spherically punctured codes $P_{r}(m,b)$ and find their distances for the arbitrary values of parameters $r,m$ , and $b$ . Finally, we describe recursive (successive cancellation) decoding of these codes.

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 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.

Effective attack on the McEliece cryptosystem based on Reed-Muller codes

AbstractMcEliece cryptosystem is an public-key cryptosystem; its security is based on the complexity of decoding problem for an arbitrary error-correcting code. V. M. Sidel’nikov in 1994 suggested to construct the cryptosystem on the base of binary Reed-Muller code. In 2007 L. Minder and A. Shokrollahi had designed a structural attack on theMcEliece cryptosystem based on the Reed-Muller codes.Herewe improve their attack and suggest a polynomial attack on the McEliece cryptosystem based on Reed-Muller codes RM(r, m) such that GCD(r,m − 1) = 1.

Spherically Punctured Biorthogonal Codes

Consider a binary Reed-Muller code RM(r,m) defined on the hypercube \BB F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> and let all code positions be restricted to the m-tuples of a given Hamming weight b. In this paper, we specify this single-layer construction obtained from the biorthogonal codes RM(1,m) and the Hadamard codes H(m). Both punctured codes inherit some recursive properties of the original RM codes; however, they cannot be formed by the recursive Plotkin construction. We first observe that any code vector in these codes has Hamming weight defined by the weight w of its information block. More specifically, this weight depends on the absolute values of the Krawtchouk polynomials K <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">b</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> (w). We then study the properties of the Krawtchouk polynomials and show that the minimum code weight of a single-layer code RM(1,m,b) is achieved at the minimum input weight w = 1 for any . We further refine our codes by limiting the possible weights w of the input information blocks. As a result, some of the designed code sequences meet or closely approach the Griesmer bound. Finally, we consider more general punctured codes whose positions form several spherical layers.