Reed-Muller Research Articles

Algebraic hierarchical locally recoverable codes with nested affine subspace recovery

AbstractCodes with locality, also known as locally recoverable codes, allow for recovery of erasures using proper subsets of other coordinates. These subsets are typically of small cardinality to promote recovery using limited network traffic and other resources. Hierarchical locally recoverable codes allow for recovery of erasures using sets of other symbols whose sizes increase as needed to allow for recovery of more symbols. In this paper, we describe a hierarchical recovery structure arising from geometry in Reed–Muller codes and codes with availability from fiber products of curves. We demonstrate how the fiber product hierarchical codes can be viewed as punctured subcodes of Reed–Muller codes, uniting the two constructions. This point of view provides natural structures for local recovery with availability at each level in the hierarchy.

On a Group Under which Symmetric Reed-Muller Codes are Invariant

On a Group Under which Symmetric Reed-Muller Codes are Invariant

Entanglement-assisted Quantum Reed-Muller Tensor Product Codes

We present the construction of standard entanglement-assisted (EA) qubit Reed-Muller (RM) codes and their tensor product variants from classical RM codes. We show that the EA RM codes obtained using the CSS construction have zero coding rate and negative catalytic rate. We further show that EA codes constructed from these same classical RM codes using the tensor product code (TPC) construction have positive coding rate and provide a subclass of EA RM TPCs that have positive catalytic rate, thus establishing the coding analog of superadditivity for this family of codes, useful towards quantum communications. We also generalize this analysis to obtain conditions for EA TPCs from classical codes to have positive catalytic rate when their corresponding EA CSS codes have zero rate.

A Deterministic Algorithm for Computing the Weight Distribution of Polar Code

In this work, we present a deterministic algorithm for computing the entire weight distribution of polar codes. As the first step, we derive an efficient recursive procedure to compute the weight distribution that arises in successive cancellation decoding of polar codes along any decoding path. This solves the open problem recently posed by Polyanskaya, Davletshin, and Polyanskii. Using this recursive procedure, at code length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , we can compute the weight distribution of any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">polar cosets</i> in time <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). We show that any polar code can be represented as a disjoint union of such polar cosets; moreover, this representation extends to polar codes with dynamically frozen bits. However, the number of polar cosets in such representation scales exponentially with a parameter introduced herein, which we call the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">mixing factor</i> . To upper bound the complexity of our algorithm for polar codes being decreasing monomial codes, we study the range of their mixing factors. We prove that among all decreasing monomial codes with rates at most 1/2, self-dual Reed-Muller codes have the largest mixing factors. To further reduce the complexity of our algorithm, we make use of the fact that, as decreasing monomial codes, polar codes have a large automorphism group. That automorphism group includes the block lower-triangular affine group (BLTA), which in turn contains the lower-triangular affine group (LTA). We prove that a subgroup of LTA acts transitively on certain subsets of decreasing monomial codes, thereby drastically reducing the number of polar cosets that we need to evaluate. This complexity reduction makes it possible to compute the weight distribution of polar codes at length <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> = 128.

Secure ECDSA SRAM-PUF Based on Universal Single/Double Scalar Multiplication Architecture.

Physically unclonable functions (PUFs) are crucial for enhancing cybersecurity by providing unique, intrinsic identifiers for electronic devices, thus ensuring their authenticity and preventing unauthorized cloning. The SRAM-PUF, characterized by its simple structure and ease of implementation in various scenarios, has gained widespread usage. The soft-decision Reed-Muller (RM) code, an error correction code, is commonly employed in these designs. This paper introduces the design of an RM code soft-decision attack algorithm to reveal its potential security risks. To address this problem, we propose a soft-decision SRAM-PUF structure based on the elliptic curve digital signature algorithm (ECDSA). To improve the processing speed of the proposed secure SRAM-PUF, we propose a custom ECDSA scheme. Further, we also propose a universal architecture for the critical operations in ECDSA, elliptic curve scalar multiplication (ECSM), and elliptic curve double scalar multiplication (ECDSM) based on the differential addition chain (DAC). For ECSMs, iterations can be performed directly; for ECDSMs, a two-dimensional DAC is constructed through precomputation, followed by iterations. Moreover, due to the high similarity of ECSM and ECDSM data paths, this universal architecture saves hardware resources. Our design is implemented on a field-programmable gate array (FPGA) and an application-specific integrated circuit (ASIC) using a Xilinx Virtex-7 and an TSMC 40 nm process. Compared to existing research, our design exhibits a lower bit error rate (2.7×10-10) and better area-time performance (3902 slices, 6.615 μs ECDSM latency).

Jacobi polynomials for the first-order generalized Reed–Muller codes

In this paper, we give the Jacobi polynomials for first-order generalized Reed–Muller codes. We show as a corollary the nonexistence of combinatorial 3-designs in these codes.

Some algebraic questions about the Reed-Muller code

Let Rq(r,n) denote the rth order Reed-Muller code of length qn over Fq. We consider two algebraic questions about the Reed-Muller code. Let Hq(r,n)=Rq(r,n)/Rq(r−1,n). (1) When q=2, it is known that there is a “duality” between the actions of GL(n,F2) on H2(r,n) and on H2(r′,n), where r+r′=n. The result is false for a general q. However, we find that a slightly modified duality statement still holds when q is a prime or r<charFq. (2) Let F(Fqn,Fq) denote the Fq-algebra of all functions from Fqn to Fq. It is known that when q is a prime, the Reed-Muller codes {0}=Rq(−1,n)⊂Rq(0,n)⊂⋯⊂Rq(n(q−1),n)=F(Fqn,Fq) are the only AGL(n,Fq)-submodules of F(Fqn,Fq). In particular, Hq(r,n) is an irreducible GL(n,Fq)-module when q is a prime. For a general q, Hq(r,n) is not necessarily irreducible. We determine all its submodules and the factors in its composition series. The factors of the composition series of Hq(r,n) provide an explicit family of irreducible representations of GL(n,Fq) over Fq.

A lower bound on the third-order nonlinearity of the simplest [formula omitted] bent functions

Boolean functions used in symmetric-key encryption should have high higher-order nonlinearity to resist several known cryptographic attacks, such as algebraic attacks and low-degree approximation attacks. The higher-order nonlinearity also plays an important role in coding theory and theoretical computer science, since it relates to the covering radius of Reed–Muller codes and the Gowers norm, respectively. It is well-known that bent functions have the highest nonlinearity in an even number of variables and thus they possess the best ability to withstand fast correlation attacks and best affine approximation attacks. However, there is currently limited knowledge regarding the higher-order nonlinearity of bent functions because computing the higher-order nonlinearity, or even providing tight lower bounds, is an extremely hard task. In 1974, Dillon proposed two well-known classes of bent functions based on partial spread (in brief, PS), called PS− and PS+, respectively. He also exhibited a subclass of bent functions in PS−, known as partial spread affine plane(PSap for short). In this paper, we provide a lower bound on the third-order nonlinearity of the simplest PSap bent functions in n variables, where n≥6 is even, by calculating the nonlinearities of all second-order derivatives of this kind of bent functions. Compared to the two known lower bounds on the third-order nonlinearity given by Carlet and Tang et al. respectively, our lower bound is much better than these two ones.

Reed–Muller Codes on BMS Channels Achieve Vanishing Bit-Error Probability for all Rates Below Capacity

This paper considers the performance of Reed-Muller (RM) codes transmitted over binary memoryless symmetric (BMS) channels under bitwise maximum-a-posteriori (bit-MAP) decoding. Its main result is that, for a fixed BMS channel, the family of binary RM codes can achieve a vanishing bit-error probability at rates approaching the channel capacity. This partially resolves a long-standing open problem that connects information theory and error-correcting codes. In contrast with the earlier result for the binary erasure channel, the new proof does not rely on hypercontractivity. Instead, it combines a nesting property of RM codes with new information inequalities relating the generalized extrinsic information transfer function and the extrinsic minimum mean-squared error.

Identifying codewords in general Reed-Muller codes and determining their weights

&lt;abstract&gt;&lt;p&gt;Determining the weight distribution of all Reed-Muller codes is a huge and exciting problem that has been around since the sixties. Some progress has been made very recently, but we are still far from a solution. In this paper, we addressed the subproblem of determining as many codeword weights as possible in Reed-Muller codes of any lengths and any orders, which is decisive for determining their weight spectra (i.e., the lists of all possible weights in these codes). New approaches seem necessary for both the main problem and the subproblem. We first studied the difficulties and the limits of the approach, which consisted of using the usual primary and secondary constructions of Boolean functions for the purpose of determining as many weights as possible in Reed-Muller codes. We then introduced a way, different from the usual constructions, to generate Boolean functions in $ n $ variables having an algebraic degree bounded from above, without any restriction on $ n $, and whose Hamming weights can be determined. This provided weights in Reed-Muller codes of any lengths $ 2^n $ and any orders, allowing us to reach potentially new values in the weight spectra of Reed-Muller codes (as we illustrate with all Reed-Muller codes of lengths up to $ 2^{21} $), with the related codewords being given with their supports and their algebraic normal forms being mathematically derived.&lt;/p&gt;&lt;/abstract&gt;

Coset leaders of the first-order Reed-Muller codes in four new classes of Boolean functions

Coset leaders of the first-order Reed-Muller codes in four new classes of Boolean functions

Reduced-Complexity Projection-Aggregation List Decoder for Reed-Muller Codes

Reduced-Complexity Projection-Aggregation List Decoder for Reed-Muller Codes

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

On the minimum distance, minimum weight codewords, and the dimension of projective Reed-Muller codes

Approximate symmetries and quantum error correction

Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.

Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

Jacobi polynomials and harmonic weight enumerators of the first-order Reed–Muller codes and the extended Hamming codes

On the codewords of generalized Reed-Muller codes reaching the fourth weight

Recently, the fourth weight of generalized Reed-Muller codes Rq(a(q−1)+b,m) have been characterized for 3≤b<q+43 and 0≤a≤m−3. In this paper, we study the codewords reaching the fourth weight of generalized Reed-Muller codes. We will determine the fourth weight codewords of generalized Reed-Muller codes of order r=a(q−1)+b with 3≤b<q+43 and 0≤a≤m−3. Also, we characterize the number of fourth weight codewords of Rq(b,m) for 3≤b<q+43.

Smoothing of Binary Codes, Uniform Distributions, and Applications.

The action of a noise operator on a code transforms it into a distribution on the respective space. Some common examples from information theory include Bernoulli noise acting on a code in the Hamming space and Gaussian noise acting on a lattice in the Euclidean space. We aim to characterize the cases when the output distribution is close to the uniform distribution on the space, as measured by the Rényi divergence of order α∈(1,∞]. A version of this question is known as the channel resolvability problem in information theory, and it has implications for security guarantees in wiretap channels, error correction, discrepancy, worst-to-average case complexity reductions, and many other problems. Our work quantifies the requirements for asymptotic uniformity (perfect smoothing) and identifies explicit code families that achieve it under the action of the Bernoulli and ball noise operators on the code. We derive expressions for the minimum rate of codes required to attain asymptotically perfect smoothing. In proving our results, we leverage recent results from harmonic analysis of functions on the Hamming space. Another result pertains to the use of code families in Wyner's transmission scheme on the binary wiretap channel. We identify explicit families that guarantee strong secrecy when applied in this scheme, showing that nested Reed-Muller codes can transmit messages reliably and securely over a binary symmetric wiretap channel with a positive rate. Finally, we establish a connection between smoothing and error correction in the binary symmetric channel.

Berman Codes: A Generalization of Reed–Muller Codes That Achieve BEC Capacity

Berman Codes: A Generalization of Reed–Muller Codes That Achieve BEC Capacity

On the generalized Hamming weights of hyperbolic codes

[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.

On binary linear codes and binary pseudo-random sequences

In this paper, we study the relation between the linear subspace of the pseudo-noise (PN)-sequences generated by a primitive polynomial and the simplex code. This family of sequences can be also seen as an Maximum Distance Separable (MDS) [Formula: see text]-linear code over [Formula: see text]. Furthermore, we see how to compute the family of generalized sequences produced by a primitive polynomial by means of a first-order Reed–Muller code.