Reed-Muller Research Articles

New advances in permutation decoding of first-order Reed-Muller codes

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 algorithm we improve considerably the number of errors we can correct in comparison with the known results in this topic.

Dynamic Frozen-Function Design for Reed-Muller Codes With Automorphism-Based Decoding

In this letter, we propose to add dynamic frozen bits to underlying polar codes with a Reed-Muller information set with the aim of maintaining the same sub-decoding structure in Automorphism Ensemble (AE) and lowering the Maximum Likelihood (ML) bound by reducing the number of minimum weight codewords. We provide the dynamic freezing constraint matrix that remains identical after applying a permutation linear transformation. This feature also permits to drastically reduce the memory requirements of an AE decoder with polar-like codes having dynamic frozen bits. We show that, under AE decoding, the proposed dynamic freezing constraints lead to a gain of up to 0.25dB compared to the ML bound of the R(3,7) Reed-Muller code, at the cost of small increase in memory requirements.

The Structure of Reed–Muller Codes Over a Nonprime Field

It is well known that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. The case of a nonprime field is less studied in terms of equalities and inclusions between Reed–Muller codes and radical powers. In this paper, we prove that Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers and provide necessary and sufficient conditions for inclusions between these codes and the powers of the radical.

Extended Cyclic Codes Sandwiched Between Reed–Muller Codes

The famous Barnes–Wall lattices can be obtained by applying Construction D to a chain of Reed–Muller codes. By applying Construction <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\text {D}}^{\text {(cyc)}}$ </tex-math></inline-formula> to a chain of extended cyclic codes sandwiched between Reed–Muller codes, Hu and Nebe (J. London Math. Soc. <xref ref-type="disp-formula" rid="deqn2" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">(2)</xref> 101 (2020) 1068-1089) constructed new series of universally strongly perfect lattices sandwiched between Barnes–Wall lattices. In this paper, we first extend their construction to generalized Reed–Muller codes, and then explicitly determine the minimum vectors of those new sandwiched Reed–Muller codes for some special cases.

Identification Codes: A Topical Review With Design Guidelines for Practical Systems

A wide range of information technology applications require the identification of a particular message or label that represents the identity of an object at a distance, e.g., over a wireless channel. Conventionally, the underlying information that represents the identity is transmitted over the channel, following the information-theoretic concept of message transmission. If the purpose of the interaction over the channel is only to verify (match) an identity, then the concept of identification over channels—utilizing the identification codes that have been developed by the information theory community—can provide an exponential efficiency gain over message transmission. This topical review article conducts for the first time a comprehensive detailed evaluation of the existing identification codes for the practically relevant regime of finite parameters. We examine essentially all published identification codes, including codes based on inner constant weight codes that are concatenated with outer linear block codes, such as Reed-Solomon and Reed-Muller codes. Specifically, we conduct a holistic identification code comparison based on the logarithm of the number of representable identities (in shannon), the size (in bit) of the transmitted cue that represents an identity, and the corresponding type II error probability bound for essentially all existing identification codes. Based on the resulting insights, we formulate guidelines for the design of practical (finite-parameter) identification codes. For instance, we find that a linear block code (without concatenation with a sophisticated inner constant-weight code) is sufficient for most practical identification code usages.

A Weighted Sum Based Construction of PAC Codes

The polarization-adjusted convolutional (PAC) codes concatenate an outer convolutional transform with an inner polar transform to improve the error-correction performance of polar codes. In the short-to-medium codeword length regime, it can approach the normal approximation (NA) bound. However, its optimal rate-profile remains unknown. This letter proposes a new construction for PAC codes based on a weighted sum (WS) metric, which considers both the cutoff rates and the utilizations of the polarized subchannels. In comparison with the existing Reed-Muller (RM) rate-profiles, it can yield a flexible rate. By adjusting the design signal-to-noise ratio (SNR), an appropriate tradeoff between the decoding performance and complexity can be achieved. Simulation results show the designed codes can outperform the ones that are designed by the existing techniques.

A Modified Selective Mapping Technique for Peak-to-Average Power Ratio Reduction in Orthogonal Frequency Division Multiplexing with Index Modulation Systems

In this study, we propose a modified selective mapping (SLM) technique to reduce the Peak-to-Average Power Ratio (PAPR) of Orthogonal Frequency Division Multiplexing with Index Modulation (OFDM-IM) systems. The main reason for using Reed–Muller and Polar codes is to protect the information bits to be transmitted while providing error correction capability and improved PAPR performance to the transmitted signal. Although OFDM-IM technology is a popular communication transmission technology recently, one of its disadvantages is that the transmitted signal has a high PAPR. The proposed method uses the Hamming weight distribution of the specific order Reed–Muller code to create a corresponding lookup table to select the subcarriers that are active in the OFDM-IM system. Moreover, it can correct errors in the information used. The proposed method also uses Polar codes as a channel encoder of the OFDM-IM system to analyze its performance in improving the PAPR. The simulation results show that method can not only improve the PAPR performance of OFDM-IM systems but also provide OFDM-IM transmission-signal error correction capacity.

Efficient multivariate low-degree tests via interactive oracle proofs of proximity for polynomial codes

We consider the proximity testing problem for error-correcting codes which consist in evaluations of multivariate polynomials either of bounded individual degree or bounded total degree. Namely, given an oracle function $$f :L^m \rightarrow {\mathbb {F}}_q$$ , where $$L\subset {\mathbb {F}}_q$$ , a verifier distinguishes whether f is the evaluation of a low-degree polynomial or is far (in relative Hamming distance) from being one, by making only a few queries to f. This topic has been studied in the context of locally testable codes, interactive proofs, probalistically checkable proofs, and interactive oracle proofs. We present the first interactive oracle proofs of proximity (IOPP) for tensor products of Reed–Solomon codes (evaluation of polynomials with bounds on individual degrees) and for Reed–Muller codes (evaluation of polynomials with a bound on the total degree) that simultaneously achieve logarithmic query complexity, logarithmic verification time, linear oracle proof length and linear prover running time. Such low-degree polynomials play a central role in constructions of probabilistic proof systems and succinct non-interactive arguments of knowledge with zero-knowledge. For these applications, highly-efficient multivariate low-degree tests are desired, but prior probabilistic proofs of proximity required super-linear proving time. In contrast, for multivariate codes of length N, our constructions admit a prover running in time linear in N and a verifier which is logarithmic in N. Our constructions are directly inspired by the IOPP for Reed–Solomon codes of [Ben-Sasson et al., ICALP 2018] named “FRI protocol”. Compared to the FRI protocol, our IOPP for tensor products of Reed–Solomon codes achieves the same efficiency parameters. As for Reed–Muller codes, for fixed constant number of variables m, the concrete efficiency of our IOPP for Reed–Muller codes compares well, all things equal.

On Hard and Soft Decision Decoding of BCH Codes

The binary primitive BCH codes are cyclic and are constructed by choosing a subset of the cyclotomic cosets. Which subset is chosen determines the dimension, the minimum distance and the weight distribution of the BCH code. We construct possible BCH codes and determine their coderate, true minimum distance and the non-equivalent codes. A particular choice of cyclotomic cosets gives BCH codes which are, extended by one bit, equivalent to Reed-Muller codes, which is a known result from the sixties. We show that BCH codes have possibly better parameters than Reed-Muller codes, which are related in recent publications to polar codes. We study the decoding performance of these different BCH codes using information set decoding based on minimal weight codewords of the dual code. We show that information set decoding is possible even in case of a channel without reliability information since the decoding algorithm inherently calculates reliability information. Different BCH codes of the same rate are compared and different decoding performances and complexity are observed. Some examples of hard decision decoding of BCH codes have the same decoding performance as maximum likelihood decoding. All presented decoding methods can possibly be extended to include reliability information of a Gaussian channel for soft decision decoding. We show simulation results for soft decision list information set decoding and compare the performance to other methods.

Self-Orthogonality Matrix and Reed-Muller Codes

Kim et al. (2021) gave a method to embed a given binary <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> code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {C}\,\,(k = 3, 4)$ </tex-math></inline-formula> into a self-orthogonal code of the shortest length which has the same dimension <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> and minimum distance <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d' \ge d(\mathcal {C})$ </tex-math></inline-formula> . We extend this result by proposing a new method related to a special matrix, called the self-orthogonality matrix <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$SO_{k}$ </tex-math></inline-formula> , obtained by shortening a Reed-Muller code <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${\mathcal R}(2,k)$ </tex-math></inline-formula> . Using this approach, we can extend binary linear codes to many optimal self-orthogonal codes of dimensions 5 and 6. Furthermore, we partially disprove the conjecture (Kim et al. (2021)) by showing that if <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$31 \le n \le 256$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n\equiv 14,22,29 \pmod {31}$ </tex-math></inline-formula> , then there exist optimal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n], [5]$ </tex-math></inline-formula> codes which are self-orthogonal. We also construct optimal self-orthogonal <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[n], [6]$ </tex-math></inline-formula> codes when <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$41 \le n \le 256$ </tex-math></inline-formula> satisfies <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \ne 46, 54, 61$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n \equiv \!\!\!\!\!/~7, 14, 22, 29, 38, 45, 53, 60 \pmod {63}$ </tex-math></inline-formula> .

Decoding Reed-Muller Codes With Successive Codeword Permutations

A novel recursive list decoding (RLD) algorithm for Reed-Muller (RM) codes based on successive permutations (SP) of the codeword is presented. A low-complexity SP scheme applied to a subset of the symmetry group of RM codes is first proposed to carefully select a good codeword permutation on the fly. Then, the proposed SP technique is integrated into an improved RLD algorithm that initializes different decoding paths with random codeword permutations, which are sampled from the full symmetry group of RM codes. Finally, efficient latency and complexity reduction schemes are introduced that virtually preserve the error-correction performance of the proposed decoder. Simulation results demonstrate that at the target frame error rate of 10−3 for the RM code of length 256 with 163 information bits, the proposed decoder reduces 6% of the computational complexity and 22% of the decoding latency of the state-of-the-art semi-parallel simplified successive-cancellation decoder with fast Hadamard transform (SSC-FHT) that uses 96 permutations from the full symmetry group of RM codes, while relatively maintaining the error-correction performance and memory consumption of the semi-parallel permuted SSC-FHT decoder.

Classification of Hadamard products of one-codimensional subcodes of Reed–Muller codes

Abstract For Reed–Muller codes we consider subcodes of codimension 1. A classification of Hadamard products of such subcodes is obtained. With the use of this classification it has been shown that in most cases the problem of recovery of the secret key of a code-based cryptosystem employing such subcodes is equivalent to the problem of recovery of the secret key of the same cryptosystem based on Reed–Muller codes, which is known to be tractable.

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.

High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder

In this paper, we show that applying a machine learning technique called auto-decoder (AD) to high-rate and short length Reed–Muller (RM) decoding enables it to achieve maximum likelihood decoding (MLD) performance and faster decoding speed than when fast Hadamard transform (FHT) is applied in additive white Gaussian noise (AWGN) channels. The decoding speed is approximately 1.8 times and 125 times faster than the FHT decoding for R(1,4) and R(2,4), respectively. The number of nodes in the hidden layer of AD is larger than that of the input layer, unlike the conventional auto-encoder (AE). Two ADs are combined in parallel and merged together, and then cascaded to one fully connected layer to improve the bit error rate (BER) performance of the code.

Designing the Quantum Channels Induced by Diagonal Gates

The challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. This paper introduces a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). It focuses on CSS codes, and describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends very strongly on the signs of Z-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. The paper derives necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provides an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate (introduced by Rengaswamy et al.), the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find application in magic state distillation and elsewhere. When all the signs are positive, the paper characterizes all possible CSS codes, invariant under transversal Z-rotation through &amp;#x03C0;/2l, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on l. The generator coefficient framework extends to arbitrary stabilizer codes but there is nothing to be gained by considering the more general class of non-degenerate stabilizer codes.

Quantum Pin Codes

We introduce quantum pin codes: a class of quantum CSS codes. Quantum pin codes are a generalization of quantum color codes and Reed-Muller codes and share a lot of their structure and properties. Pin codes have gauge operators, an unfolding procedure and their stabilizers form so-called <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -orthogonal spaces meaning that the joint overlap between any <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> stabilizer elements is always even. This last feature makes them interesting for devising magic-state distillation protocols, for instance by using puncturing techniques. We study examples of these codes and their properties.

An Information-Theoretic Perspective on Successive Cancellation List Decoding and Polar Code Design

This work identifies information-theoretic quantities that are closely related to the required list size on average for successive cancellation list (SCL) decoding to implement maximum-likelihood decoding over general binary memoryless symmetric (BMS) channels. It also provides upper and lower bounds for these quantities that can be computed efficiently for very long codes. For the binary erasure channel (BEC), we provide a simple method to estimate the mean accurately via density evolution. The analysis shows how to modify, e.g., Reed-Muller codes, to improve the performance when practical list sizes, e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$L\in {[{8, 1024}]}$ </tex-math></inline-formula> , are adopted. Exemplary constructions with block lengths <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N\in \{128,512\}$ </tex-math></inline-formula> outperform polar codes of 5G over the binary-input additive white Gaussian noise channel. It is further shown that there is a concentration around the mean of the logarithm of the required list size for sufficiently large block lengths, over discrete-output BMS channels. We provide the probability mass functions (p.m.f.s) of this logarithm, over the BEC, for a sequence of the modified RM codes with an increasing block length via simulations, which illustrate that the p.m.f.s concentrate around the estimated mean.

Morphing Quantum Codes

We introduce a morphing procedure that can be used to generate new quantum codes from existing quantum codes. In particular, we morph the 15-qubit Reed-Muller code to obtain a $[\![10,1,2]\!]$ code that is the smallest known stabilizer code with a fault-tolerant logical $T$ gate. In addition, we construct a family of hybrid color-toric codes by morphing the color code. Our code family inherits the fault-tolerant gates of the original color code, implemented via constant-depth local unitaries. As a special case of this construction, we obtain toric codes with fault-tolerant multi-qubit control-$Z$ gates. We also provide an efficient decoding algorithm for hybrid color-toric codes in two dimensions, and numerically benchmark its performance for phase-flip noise. We expect that morphing may also be a useful technique for modifying other code families such as triorthogonal codes.

On the Generalized Covering Radii of Reed-Muller Codes

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 input than the upper bound on the generalized covering radius of the code. We prove that the algorithm runs in time that is polynomial in the code parameters.

On Weight Distributions for a Class of Codes with Parameters of Reed-Muller Codes

On Weight Distributions for a Class of Codes with Parameters of Reed-Muller Codes