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A collection of sets displays a proximity gap with respect to some property if for every set in the collection, either (i) all members are δ-close to the property in relative Hamming distance or (ii) only a tiny fraction of members are δ-close to the property. In particular, no set in the collection has roughly half of its members δ-close to the property and the others δ-far from it. We show that the collection of affine spaces displays a proximity gap with respect to Reed--Solomon (RS) codes, even over small fields, of size polynomial in the dimension of the code, and the gap applies to any δ smaller than the Johnson/Guruswami--Sudan list-decoding bound of the RS code.We also show near-optimal gap results, over fields of (at least) linear size in the RS code dimension, for δ smaller than the unique decoding radius. Concretely, if δ is smaller than half the minimal distance of an RS code V ⊂ Fnq, then every affine space is either entirely δ-close to the code or, alternatively, at most an (n/q)-fraction of it is δ-close to the code. Finally, we discuss several applications of our proximity gap results to distributed storage, multi-party cryptographic protocols, and concretely efficient proof systems. We prove the proximity gap results by analyzing the execution of classical algebraic decoding algorithms for Reed--Solomon codes (due to Berlekamp--Welch and Guruswami--Sudan) on a formal element of an affine space. This involves working with Reed--Solomon codes whose base field is an (infinite) rational function field. Our proofs are obtained by developing an extension (to function fields) of a strategy of Arora and Sudan for analyzing low-degree tests. [ABSTRACT FROM AUTHOR]

Low-hit-zone frequency hopping sequence (LHZ-FHS) sets having optimal Hamming correlation are desirable in quasi-synchronous communication systems. In this paper, we first derive a new bound on maximum nontrivial Hamming correlation of LHZ-FHS sets from the famous Singleton bound in error correcting code literature, then obtain a general construction of LHZ-FHS sets from cyclic codes. Especially, two classes of LHZ-FHS sets meeting the new bound are constructed from punctured Reed-Solomon codes. [ABSTRACT FROM AUTHOR]

DNA sequences are prone to creating secondary structures by folding back on themselves by non-specific hybridization of its nucleotides. The formation of large stem-length secondary structures makes the sequences chemically inactive towards synthesis and sequencing processes. Furthermore, in DNA computing, other constraints like homopolymer run length also introduce complications. In this paper, our goal is to tackle the problems due to the creation of secondary structures in DNA sequences along with constraints such as not having a large homopolymer run length. This paper presents families of DNA codes with secondary structures of stem length at most two and homopolymer run length at most four. We identified Z 11 as an ideal structure to construct DNA codes to avoid the above problems. By mapping the error-correcting codes over Z 11 to DNA nucleotides, we obtained DNA codes with rates 0.5765 times the corresponding code rate over Z 11 , including some new secondary structure-free and better-performing codes for DNA-based data storage and DNA computing purposes. [ABSTRACT FROM AUTHOR]

Martínez-Peñas and Kschischang (IEEE Trans. Inf. Theory 65(8):4785–4803, 2019) proposed lifted linearized Reed–Solomon codes as suitable codes for error control in multishot network coding. We show how to construct and decode lifted interleaved linearized Reed–Solomon (LILRS) codes. Compared to the construction by Martínez-Peñas–Kschischang, interleaving allows to increase the decoding region significantly and decreases the overhead due to the lifting (i.e., increases the code rate), at the cost of an increased packet size. We propose two decoding schemes for LILRS that are both capable of correcting insertions and deletions beyond half the minimum distance of the code by either allowing a list or a small decoding failure probability. We propose a probabilistic unique Loidreau–Overbeck-like decoder for LILRS codes and an efficient interpolation-based decoding scheme that can be either used as a list decoder (with exponential worst-case list size) or as a probabilistic unique decoder. We derive upper bounds on the decoding failure probability of the probabilistic-unique decoders which show that the decoding failure probability is very small for most channel realizations up to the maximal decoding radius. The tightness of the bounds is verified by Monte Carlo simulations. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extensions, we focus on MDS linear codes. Recently, He and Liao showed that for an MDS linear code 풞 with minimum distance 2ℓ + 2, if it has an ℓ-error-correcting pair, then the parameters of the pair have three possibilities. Moreover, for the first case, they gave a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and for the other two cases, they only gave some counterexamples. For the second case, in this paper, we give a necessary condition for an MDS linear code 풞 with minimum distance 2ℓ + 2 to have an ℓ-error-correcting pair, and then basing on the Product Singleton Bound, we prove that there are two cases for such pairs, and then give some counterexamples basing on twisted generalized Reed–Solomon codes for these cases. [ABSTRACT FROM AUTHOR]

Quantum error-correcting codes and entanglement-assisted quantum error-correcting codes have important applications in quantum computing and quantum communication. In this paper, we construct several classes of quantum MDS codes and entanglement-assisted quantum MDS codes by using generalized Reed-Solomon codes. The parameters of most of the codes constructed are new. [ABSTRACT FROM AUTHOR]

The hull of a linear code is defined to be the intersection of the code and its dual, and was originally introduced to classify finite projective planes. The objective of this paper is to construct some MDS codes with l-Galois hulls of arbitrary dimensions by using the generalized Reed–Solomon codes over finite fields with regard to l-Galois inner product. We give a general construction theorem and some construction ideas of MDS with l-Galois hulls of arbitrary dimensions. Our approach provides a general framework that effectively unifies similar known techniques for constructing MDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we provide the method of constructing asymmetric quantum codes by means of the Euclidean sums of matrix-product codes over finite fields. We construct asymmetric quantum codes of length 2n by using the Euclidean sums of matrix-product codes whose constituent codes are Reed–Solomon codes, and asymmetric quantum codes of length 3n by using the Euclidean sums of matrix-product codes whose constituent codes are generated by Fourier matrices. Using these constructions, concrete examples are presented to construct new asymmetric quantum codes. In addition, our obtained asymmetric quantum codes have better parameters than the ones available in the literature. [ABSTRACT FROM AUTHOR]

The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than $ ({q}/{2})+1 $ (q / 2) + 1. [ABSTRACT FROM AUTHOR]

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Let GRSk(α, υ) be a k-dimensional generalized Reed-Solomon (GRS) code over Fq associated with α = (α1,. . ., αn) and υ = (υ1, . . ., υn). In this paper, we determined the dimension of the Euclidean hull GRSk(α, υ) ∩ GRSk(α, υ)⊥, which addresses an open problem posed in [Chen et al., IEEE-TIT, 2023]. We also presentd a new approach to generating all self-dual RS codes. [ABSTRACT FROM AUTHOR]

In communication technologies such as digital watermarking, wireless sensor networks (WSNs), and visual light communication (VLC), error-correcting codes are crucial. The Enhanced Hadamard Error-Correcting Code (EHC), which is based on 2D Hadamard Basis Images, is a novel error correction technique that is presented in this study. This technique is used to evaluate the effectiveness of the video watermarking scheme. Even with highly sophisticated embedding techniques, watermarks usually fail to resist such comprehensive attacks because of the extraordinarily high compression rate of approximately 1:200 that is frequently employed in video dissemination. It can only be used in conjunction with a sufficient error-correcting coding method. This study compares the efficacy of the well-known Reed–Solomon Code with this novel technique, the Enhanced Hadamard Error-Correcting Code (EHC), in maintaining watermarks in embedded videos. The main idea behind this newly created multidimensional Enhanced Hadamard Error-Correcting Code is to use a 1D Hadamard decoding approach on the 2D base pictures after they have been transformed into a collection of one-dimensional rows. Following that, the image is rebuilt, allowing for a more effective 2D decoding procedure. Using this technique, it is possible to exceed the theoretical error-correcting capacity threshold of ⌊ d m i n − 1 2 ⌋ bits, where d m i n is the Hamming distance. It may be possible to achieve better results by converting the 2D EHC into a 3D format. The new Enhanced Hadamard Code is used in a video watermarking coding scheme to show its viability and efficacy. The original video is broken down using a multi-level interframe wavelet transform during the video watermarking embedding process. Low-pass filtering is applied to the video stream in order to extract a certain frequency range. The watermark is subsequently incorporated using this filtered section. Either the Reed–Solomon Correcting Code or the Enhanced Hadamard Code is used to protect the watermarks. The experimental results show that EHC far outperforms the RS Code and is very resilient against severe MPEG compression. [ABSTRACT FROM AUTHOR]

In this paper, we find a necessary and sufficient condition for multi-twisted Reed-Solomon codes to be MDS. In particular, we introduce a new class of MDS double-twisted Reed-Solomon codes C α , t , h , η with twists t = (1 , 2) and hooks h = (0 , 1) over the finite field F q , providing a non-trivial example over F 16 and enumeration over the finite fields of size up to 17. Moreover, we obtain necessary conditions for the existence of multi-twisted Reed-Solomon codes with small dimensional hull. Consequently, we derive conditions for the existence of MDS multi-twisted Reed-Solomon codes with small dimensional hull. [ABSTRACT FROM AUTHOR]

This paper describes generalized Reed-Solomon (GRS) codes over number fields that are invariant under certain permutations. We call these codes generalized quasi-cyclic (GQC) GRS codes. Moreover, we describe an application of GQC GRS codes over number fields to exact gradient coding. [ABSTRACT FROM AUTHOR]

Collision-resistant hash functions (CRH ) are a fundamental and ubiquitous cryptographic primitive. Several recent works have studied a relaxation of CRH called t-way multi-collision-resistant hash functions ( t - MCRH ). These are families of functions for which it is computationally hard to find a t-way collision, even though such collisions are abundant (and even (t - 1) -way collisions may be easy to find). The case of t = 2 corresponds to standard CRH , but it is natural to study t- MCRH for larger values of t. Multi-collision resistance seems to be a qualitatively weaker property than standard collision resistance. Nevertheless, in this work we show a non-blackbox transformation of any moderately shrinking t- MCRH , for t ∈ { 3 , 4 } , into an (infinitely often secure) CRH . This transformation is non-constructive—we can prove the existence of a CRH but cannot explicitly point out a construction. Our result partially extends to larger values of t. In particular, we show that for suitable values of t > t ′ , we can transform a t- MCRH into a t ′ - MCRH , at the cost of reducing the shrinkage of the resulting hash function family and settling for infinitely often security. This result utilizes the list-decodability properties of Reed–Solomon codes. [ABSTRACT FROM AUTHOR]

In this paper we present a concrete algebraic construction of a novel class of convolutional codes. These codes are built upon generalized Vandermonde matrices and therefore can be seen as a natural extension of Reed–Solomon block codes to the context of convolutional codes. For this reason we call them weighted Reed–Solomon (WRS) convolutional codes. We show that under some constraints on the defining parameters these codes are Maximum Distance Profile (MDP), which means that they have the maximal possible growth in their column distance profile. We study the size of the field needed to obtain WRS convolutional codes which are MDP and compare it with the existing general constructions of MDP convolutional codes in the literature, showing that in many cases WRS convolutional codes require significantly smaller fields. [ABSTRACT FROM AUTHOR]

Motivation DNA as a novel storage medium is considered an effective solution to the world's growing demand for information due to its high density and long-lasting reliability. However, early coding schemes ignored the biologically constrained nature of DNA sequences in pursuit of high density, leading to DNA synthesis and sequencing difficulties. This article proposes a novel DNA storage coding scheme. The system encodes half of the binary data using each of the two GC-content complementary encoding rules to obtain a DNA sequence. Results After simulating the encoding of representative document and image file formats, a DNA sequence strictly conforming to biological constraints was obtained, reaching a coding potential of 1.66 bit/nt. In the decoding process, a mechanism to prevent error propagation was introduced. The simulation results demonstrate that by adding Reed-Solomon code, 90% of the data can still be recovered after introducing a 2% error, proving that the proposed DNA storage scheme has high robustness and reliability. Availability and implementation : The source code for the codec scheme of this paper is available at https://github.com/Mooreniah/DNA-dual-rule-rotary-encoding-storage-system-DRRC. [ABSTRACT FROM AUTHOR]

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered. [ABSTRACT FROM AUTHOR]

This paper shows that there exist Reed-Solomon (RS) codes, over large finite fields, that are combinatorially list-decodable well beyond the Johnson radius, in fact almost achieving list-decoding capacity. In particular, we show that for any ε E (0,1] there exist RS codes with rate Ω(ε(log(1/∈)+1) that are list-decodable from radius of 1-ε. We generalize this result to list-recovery, showing that there exist (1-ε,l,O(l/ε)) -list-recoverable RS codes with rate Ω(εl√(log(1/ε)+1)) . Along the way we use our techniques to give a new proof of a result of Blackburn on optimal linear perfect hash matrices, and strengthen it to obtain a construction of strongly perfect hash matrices. To derive the results in this paper we show a surprising connection of the above problems to graph theory, and in particular to the tree packing theorem of Nash-Williams and Tutte. We also state a new conjecture that generalizes the tree-packing theorem to hypergraphs, and show that if this conjecture holds, then there would exist RS codes that are optimally (non-asymptotically) list-decodable. [ABSTRACT FROM AUTHOR]

By generalizing the stabilizer quantum error-correcting codes, entanglement-assisted quantum error-correcting (EAQEC) codes were introduced, which could be derived from any classical linear codes via the relaxation of self-orthogonality conditions with the aid of pre-shared entanglement between the sender and the receiver. In this paper, three classes of entanglement-assisted quantum error-correcting maximum-distance-separable (EAQMDS) codes are constructed through generalized Reed–Solomon codes. Under our constructions, the minimum distances of our EAQMDS codes are much larger than those of the known EAQMDS codes of the same lengths that consume the same number of bits. Furthermore, some of the lengths of the EAQMDS codes are not divisors of q 2 - 1 , which are completely new and unlike all those known lengths existed before. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) quantum error-correcting codes (QECCs) and MDS entanglement-assisted QECCs (EAQECCs) play significant roles in quantum information theory. In this paper, we construct several new families of MDS QECCs and MDS EAQECCs by utilizing Hermitian self-orthogonal generalized Reed–Solomon codes. These newly obtained MDS QECCs contain some known classes of MDS QECCs as subclasses and some of them have larger minimum distance. In addition, many q-ary MDS QECCs and MDS EAQECCs in our constructions have length exceeding q + 1 and minimum distance surpassing q 2 + 1 . [ABSTRACT FROM AUTHOR]

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm. [ABSTRACT FROM AUTHOR]

In order to achieve the change rule of the induced residual stress (RS) field after multipass ultrasonic surface rolling (USR), a mathematical model of the induced residual stress (RS) field after multipass ultrasonic surface rolling is first established. Then, the coupling mechanisms of the RS field after dual-pass USR and multipass USR are analyzed, respectively. Subsequently, a finite element (FE) model is established, and the influence of the interval between two adjacent rolling paths L S is investigated. Finally, both the mathematical model and the FE model are experimentally verified. The results show that both the mathematical model and the FE model can predict the RS field after multipass USR. Two adjacent RS fields will couple with each other in their overlapping regions. For a relatively small interval L S , the RS field after multipass USR can be fully coupled, so as to form a uniform compressive RS layer. In this study, when L S = 0.05 mm, the values of the surface compressive RS, the maximum compressive RS, the depth of the maximum compressive RS, and the depth of the compressive RS layer reach 426.71 MPa, 676.54 MPa, 0.05 mm, and 0.54 mm, respectively. [ABSTRACT FROM AUTHOR]

Skew polynomials are a class of non-commutative polynomials that have several applications in computer science, coding theory and cryptography. In particular, skew polynomials can be used to construct and decode evaluation codes in several metrics, like e.g. the Hamming, rank, sum-rank and skew metric. We propose a fast divide-and-conquer variant of Kötter–Nielsen–Høholdt (KNH) interpolation algorithm: it inputs a list of linear functionals on skew polynomial vectors, and outputs a reduced Gröbner basis of their kernel intersection. We show, that the proposed KNH interpolation can be used to solve the interpolation step of interpolation-based decoding of interleaved Gabidulin codes in the rank-metric, linearized Reed–Solomon codes in the sum-rank metric and skew Reed–Solomon codes in the skew metric requiring at most O ~ s ω M (n) operations in F q m , where n is the length of the code, s the interleaving order, M (n) the complexity for multiplying two skew polynomials of degree at most n, ω the matrix multiplication exponent and O ~ · the soft-O notation which neglects log factors. This matches the previous best speeds for these tasks, which were obtained by top–down minimal approximant bases techniques, and complements the theory of efficient interpolation over free skew polynomial modules by the bottom-up KNH approach. In contrast to the top–down approach the bottom-up KNH algorithm has no requirements on the interpolation points and thus does not require any pre-processing. [ABSTRACT FROM AUTHOR]

In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum errorcorrecting codes constructed in this paper are different from the ones in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we construct three classes of Clifford subsystem maximum distance separable (MDS) codes based on Reed-Solomon codes and extended generalized Reed-Solomon codes over finite fields Fq for specific code lengths. Moreover, our Clifford subsystem MDS codes are new because their parameters differ from the previously known ones. [ABSTRACT FROM AUTHOR]

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes such as generalized Reed-Solomon codes. In this paper, we define a new extended generalized Reed-Solomon code, i.e., lengthened generalized Reed-Solomon code, which has good algebraic structure and many excellent properties, thus we extend the error-correcting pair to the case for lengthened generalized Reed-Solomon codes. Firstly, we give some sufficient conditions for which an LGRS code is non-GRS, and a necessary and sufficient condition for an LGRS code to be MDS or AMDS, respectively. And then, we constructively determine the existence of the error-correcting pair for lengthened generalized Reed-Solomon codes and give several examples to support our main results. [ABSTRACT FROM AUTHOR]

We give new constructions of two classes of algebraic code families that are efficiently list decodable with small output list size from a fraction 1 - R - ε of adversarial errors, where R is the rate of the code, for any desired positive constant ε. The alphabet size depends only ε and is nearly optimal. The first class of codes are obtained by folding algebraic-geometric codes using automorphisms of the underlying function field. The second class of codes are obtained by restricting evaluation points of an algebraicgeometric code to rational points from a subfield. In both cases, we develop a linear-algebraic approach to perform list decoding, which pins down the candidate messages to a subspace with a nice "periodic" structure. To prune this subspace and obtain a good bound on the list size, we pick subcodes of these codes by pre-coding into certain subspace-evasive sets that are guaranteed to have small intersection with the sort of periodic subspaces that arise in our list decoding. We develop two approaches for constructing such subspaceevasive sets. The first is a Monte Carlo construction of hierearchical subspace-evasive (h.s.e.) sets that leads to excellent list size but is not explicit. The second approach exploits a further ultra-periodicity of our subspaces and uses a novel construct called subspace designs, which were subsequently constructed explicitly and also found further applications in pseudorandomness. To get a family of codes over a fixed alphabet size, we instantiate our approach with algebraic-geometric codes based on the Garcia-Stichtenoth tower of function fields. Combining this with pruning via h.s.e. sets yields codes list-decodable up to a 1 - R - ε error fraction with list size bounded by O(1/ε), matching the existential bound for random codes up to constant factors. Further, the alphabet size can be made exp(˜O (1/ε2)), which is not much worse than the lower bound of exp(Ω(1/ε)). The parameters we achieve are thus quite close to the existential bounds in all three aspects (error-correction radius, alphabet size, and list size) simultaneously. This construction is, however, Monte Carlo and the claimed list-decoding property only holds with high probability. Once the code is (efficiently) sampled, the encoding/decoding algorithms are deterministic with a running time Oε (Nc) for an absolute constant c, where N is the code's block length. Using subspace designs instead for the pruning, our approach yields the first deterministic construction of an algebraic code family of rate R with efficient list decoding from 1 - R - ε fraction of errors over an alphabet of constant size exp(˜O (1/ε2)). The list-size bound is upper bounded by a very slowly growing function of the block length N; in particular, it is at most O(log(r) N) (the r th iterated logarithm) for any fixed integer r. The explicit construction avoids the shortcoming of the Monte Carlo sampling at the expense of a slightly worse list size. [ABSTRACT FROM AUTHOR]

The e-book services we use today have a serious drawback in that we will no longer be able to read the books we have purchased when the service is terminated. One way to solve this problem is to build a decentralized system that does not depend on a specific company or organization by combining smart contracts running on the Ethereum blockchain and distributed storage such as an IPFS. However, a simple combination of existing technologies does not make the stored e-book data persistent, so the risk of purchased e-books becoming unreadable remains. In this paper, we propose a decentralized distributed storage called d-book-repository, which has both access management function and data durability for purchased e-books. This system uses NFTs as access rights to realize strict access control by preventing clients who do not have NFTs from downloading e-book data. In addition, e-book data stored on storage nodes in the distributed storage is divided into shards using Reed–Solomon codes, and each storage node stores only a single shard, thereby preventing the creation of nodes that can restore the entire content from locally stored data. The storage of each shard is not handled by a single node but by a group of nodes, and the shard is propagated to all nodes in the group using the gossip protocol, where erasure codes are utilized to increase the resilience against node departure. Furthermore, an incentive mechanism to encourage participation as a storage node is implemented using smart contracts. We built a prototype of the proposed system on AWS and evaluated its performance. The results showed that both downloading and uploading 100 MB of e-book data (equivalent to one comic book) were completed within 10 s using an instance type of m5.xlarge. This value is only 1.3 s longer for downloading and 2.2 s longer for uploading than the time required for a simple download/upload without access control, confirming that the overhead associated with the proposed method is sufficiently small. [ABSTRACT FROM AUTHOR]

Maximum distance separable (MDS) codes are optimal in the sense that the minimum distance cannot be improved for a given length and code size. The most prominent MDS codes are generalized Reed–Solomon (GRS) codes. The square C 2 of a linear code C is the linear code spanned by the component-wise products of every pair of codewords in C . For an MDS code C , it is convenient to determine whether C is a GRS code by determining the dimension of C 2 . In this paper, we investigate under what conditions that MDS constacyclic codes are GRS. For this purpose, we first study the square of constacyclic codes. Then, we give a sufficient condition that a constacyclic code is GRS. In particular, we provide a necessary and sufficient condition that a constacyclic code of a prime length is GRS. [ABSTRACT FROM AUTHOR]

Backscatter communication relies on passive reflections of the existing radio frequency (RF) signals, making it suitable for low-power and low-complexity communication in IoT applications. However, the performance of existing systems severely degrades in real-life environments, due to irregular "on" and "off" states of ambient signals like WiFi, which are not controllable. In this paper, we propose a joint coding and framing scheme for the backscattering physical layer to fight against the off-state in the excitation signal. We first design transmission schemes including both the Reed-Solomon (RS) codes and the frame structure, to correct the burst error caused by the off states. In order to implement the codes at the resource-constrained tag, we design a look-up table for the encoding process. We prototype our system NuWa that could efficiently backscatter with uncontrolled traffics generated randomly. We demonstrate that NuWa could achieve a 1 Mbps transmission throughput when the tag is over 1 m away from the receiver in high traffic load and 150 kbps in low traffic load. Finally, we evaluate the throughput with respect to the distance change between the tag and the receiver, and 950 kbps is achieved at a distance of 6 m . [ABSTRACT FROM AUTHOR]

With the rapid development of online social networks recently, more and more online users have participated in social network activities and rich social relationships are formed accordingly. These social relationships provide a rich data source and research basis for in-depth study on recommender systems (RSs), while also promoting the development of RSs based on social networks. To solve the problems of cold start and sparsity in RSs, many recommendation algorithms are constantly being proposed. Motivated by the availability of rich social connections in today's RSs, a large number of recommendation techniques based on social relationships have been proposed recently, achieving good recommendation results, and have become the mainstream research direction in the field of RSs, attracting more and more researchers to engage in this research. In this study, we mainly review and summarize the social relationship-based recommendation methods and techniques in RSs, and study some recent deep social relationship recommendation methods and techniques based on deep learning (DL), including the latest social matrix factorization (MF)-based recommendation methods and graph neural network (GNN)-based recommendation methods. Finally, we discuss the potential impact that may improve the RS and future direction. In this article, we aim to introduce the recent recommendation techniques integrating social relationships to solve data sparsity and cold start, and provide a new perspective for improving the performance of RSs, thereby providing useful resources in the state-of-the-art research results for future researchers. [ABSTRACT FROM AUTHOR]

In this letter, a novel single parity check‐aided minimum sum (SPC‐aided MS) algorithm is proposed. A cascaded compiled code scheme is used to approximate the adaptive belief propagation (ABP) algorithm for Reed–Solomon (RS) codes using the min sum (MS) algorithm. The computational complexity is greatly reduced compared with the previous algorithm. Simulation results show that the SPC‐aided MS algorithm can achieve coding gains of up to 1.1, 0.7, and 0.6 dB for RS (255,239) code compared with the conventional ABP, LRBP, and HD‐P‐ABP algorithms, respectively under the condition of FER=10−3$\text{FER}=10^{-3}$. [ABSTRACT FROM AUTHOR]

Reactive species (RS) play significant roles in many disease contexts. Despite their crucial roles in diseases including cancer, the RS are not adequately modeled in the genome-scale metabolic (GSM) models, which are used to understand cell metabolism in disease contexts. We have developed a scalable RS reactions module that can be integrated with any Recon 3D-derived human metabolic model, or after fine-tuning, with any metabolic model. With RS-integration, the GSM models of three cancers (basal-like triple negative breast cancer (TNBC), high grade serous ovarian carcinoma (HGSOC) and colorectal cancer (CRC)) built from Recon 3D, precisely highlighted the increases/decreases in fluxes (dysregulation) occurring in important pathways of these cancers. These dysregulations were not prominent in the standard cancer models without the RS module. Further, the results from these RS-integrated cancer GSM models suggest the following decreasing order in the ease of ferroptosis-targeting to treat the cancers: TNBC > HGSOC > CRC. • Developed a novel, scalable module of reactive species (RS) reactions. • RS module can be integrated with any genome-scale metabolic model. • RS module integration better predicts metabolic dysregulations in cancer. • RS module integration better predicts ferroptosis regulation. • Different effects by different RS levels can be achieved with suitable constraints. [ABSTRACT FROM AUTHOR]

We design and implement a simple zero-knowledge argument protocol for NP whose communication complexity is proportional to the square-root of the verification circuit size. The protocol can be based on any collision-resistant hash function. Alternatively, it can be made non-interactive in the random oracle model, yielding concretely efficient zk-SNARKs that do not require a trusted setup or public-key cryptography. Our protocol is obtained by applying an optimized version of the general transformation of Ishai et al. (in: STOC, pp. 21–30, 2007) to a variant of the protocol for secure multiparty computation of Damgård and Ishai (in: CRYPTO, pp. 501–520, 2006). It can be viewed as a simple zero-knowledge interactive PCP based on "interleaved" Reed-Solomon codes. This paper is an extended version of the paper published in CCS 2017 and features a tighter analysis, better implementation along with formal proofs. For large verification circuits, the Ligero prover remains competitive against subsequent works with respect to the prover's running time, where our efficiency advantages become even bigger in an amortized setting, where several instances need to be proven simultaneously. Our protocol is attractive not only for very large verification circuits but also for moderately large circuits that arise in applications. For instance, for verifying a SHA-256 preimage with 2 - 40 soundness error, the communication complexity is roughly 35KB. The communication complexity of our protocol is independent of the circuit structure and depends only on the number of gates. For 2 - 40 soundness error, the communication becomes smaller than the circuit size for circuits containing roughly 3 million gates or more. With our refined analysis the Ligero system's proof lengths and prover's running times are better than subsequent post-quantum ZK-SNARKs for small to moderately large circuits. [ABSTRACT FROM AUTHOR]

Abstract In this letter, a novel single parity check‐aided minimum sum (SPC‐aided MS) algorithm is proposed. A cascaded compiled code scheme is used to approximate the adaptive belief propagation (ABP) algorithm for Reed–Solomon (RS) codes using the min sum (MS) algorithm. The computational complexity is greatly reduced compared with the previous algorithm. Simulation results show that the SPC‐aided MS algorithm can achieve coding gains of up to 1.1, 0.7, and 0.6 dB for RS (255,239) code compared with the conventional ABP, LRBP, and HD‐P‐ABP algorithms, respectively under the condition of FER=10−3.

A Coxeter group of classical type A n , B n or D n contains a chain of subgroups of the same type. We show that intersections of conjugates of these subgroups are again of the same type, and make precise in which sense and to what extent this property is exclusive to the classical types of Coxeter groups. As the main tool for the proof, we use Solomon's descent algebra. Using Stirling numbers, we express certain basis elements of the descent algebra as polynomials and derive explicit multiplication formulas for a commutative subalgebra of the descent algebra. [ABSTRACT FROM AUTHOR]

The novel distributed Reed-Solomon coded cooperative (DRSCC) scheme is presented in this manuscript to construct a promising resultant code with an improved code weight distribution at the destination terminal. To optimize relay cooperation, an exhausted search (ES) algorithm is proposed to select the optimal code. Based on this algorithm, the more practical reduced-complexity search (RCS) algorithm is then proposed. The results of Monte-Carlo simulations indicate that the bit error rate (BER) performance of DRSCC schemes utilizing the two algorithms is almost equivalent. Moreover, a novel joint decoding algorithm, namely Critical-SNR-Aided (CSA), is suggested and implemented to enhance overall performance at the destination. To generalize the proposed DRSCC scheme, multiple RS codes with varying lengths at the source and relay are investigated. Numerical simulations show that the proposed DRSCC scheme clearly outperforms the corresponding point-to-point RS-coded scheme by a BER gain of 3.6–3.8 dBs. Furthermore, the proposed DRSCC scheme outperforms some existing RS-coded cooperative schemes over the fast Rayleigh fading channel. [ABSTRACT FROM AUTHOR]

Quantum maximum-distance-separable (MDS) codes play an important role in the quantum codes. The previous quantum MDS codes had been constructed according to q is odd or even. However, such classification omits to consider some special categories of quantum MDS codes. Because of this, we will discuss the other classifications of q. In this paper, we construct some new q -ary quantum MDS codes from generalized Reed–Solomon codes by using Hermitian construction, and prove that these quantum MDS codes have minimum distance greater than q 2 , where q ≡ − 1 (mod 3). [ABSTRACT FROM AUTHOR]

This paper presents the implementation of a low-complex iterative symbol-level decoding scheme for Reed-Solomon codes. Most soft-decision iterative decoders for Reed-Solomon codes work on a bit level due to the efficient passing of soft information due to the sparsity of the binary parity check matrix. These decoders yield a good error correction performance, but this comes at the cost of an increased computational complexity resulting from working at a bit-level. This study aims to lower the computational complexity of iterative decoding of Reed-Solomon codes by introducing a high-performance soft-decision iterative Reed-Solomon decoder that works on a symbol-level, in contrast to the bit-level implementation used in most iterative Reed-Solomon decoders. The proposed algorithm utilises soft information to decode while applying information set decoding techniques to an extended parity check matrix. The use of the extended parity check matrix provides additional parity check equations which assist the algorithm in determining the correct information set of symbols used in the decoding process. For a given (n , k) Reed-Solomon code, the algorithm converges to a valid codeword by correctly decoding a specified information set of k symbols from the received vector. Simulations run show the proposed algorithm performs favourably when compared to other symbol-level iterative decoders while working at a relatively lower complexity. [ABSTRACT FROM AUTHOR]

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The revolution in the field of information processing systems has created a huge demand for reliable and enhanced data storage capabilities. This demand is being met by advances in channel coding algorithms along with upward scaling of the capacities of hardware devices. NAND Flash memory is a type of non-volatile memory. Scaling of the size of flash memories from Single Level Cell (SLC) devices to Multilevel cell (MLC) devices has increased the storage capacity. However, these multi-bit per cell architectures are characterized by significantly higher Raw Bit Error Rate (RBER) values when compared with SLC architectures. The requirement of low Undetected Bit Error Rate (UBER) values has motivated us to synthesize powerful channel codes for enhancing the integrity of information Storage in multi-level NAND Flash Memory devices. This paper describes the synthesis of novel Subfield Subcodes of Reed Solomon Codes (SSRS) and Reed-Solomon (RS) codes which are matched to multi-bit per cell architectures. UBER values have been calculated for each of the synthesized codes described in this paper. This allows the determination of the performance and the improvement in data storage integrity brought by using these codes. We have shown that the synthesized SSRS and RS codes can provide very low UBER even when the corresponding RBER values are appreciable. As RS codes permit the detection and correction of a greater number of errors for a given code length, their performance is superior to that of SSRS codes. This improved performance is obtained at the cost of greater complexity of encoding and decoding processes.

The problem of private information retrieval with graph-based replicated storage was recently introduced by Raviv, Tamo and Yaakobi. Its capacity remains open in almost all cases. In this work the asymptotic (large number of messages) capacity of this problem is studied along with its generalizations to include arbitrary T-privacy and X-security constraints, where the privacy of the user must be protected against any set of up to T colluding servers and the security of the stored data must be protected against any set of up to X colluding servers. A general achievable scheme for arbitrary storage patterns is presented that achieves the rate (ρ -X-T)/N, where N is the total number of servers, and each message is replicated at leastρ times. Notably, the scheme makes use of a special structure inspired by dual Generalized Reed Solomon (GRS) codes. A general converse is also presented. The two bounds are shown to match for many settings, including symmetric storage patterns. Finally, the asymptotic capacity is fully characterized for the case without security constraints (X=0) for arbitrary storage patterns provided that each message is replicated no more than T+2 times. As an example of this result, consider PIR with arbitrary graph based storage (T=1, X=0) where every message is replicated at exactly 3 servers. For this 3-replicated storage setting, the asymptotic capacity is equal to 2/ν 2(G) where ν 2(G) is the maximum size of a 2-matching in a storage graph G[V,E]. In this undirected graph, the vertices V correspond to the set of servers, and there is an edge uv\in E between vertices u,v only if a subset of messages is replicated at both servers u and v.

We propose a novel slot-pattern-control based coded compressed sensing for unsourced random access with an outer A-channel code capable of correcting t errors. Specifically, an RM extension code called patterned Reed–Muller (PRM) code is proposed. We demonstrate the high spectral efficiency due to its enormous sequence space and prove the geometry property in the complex domain that enhances the reliability and efficiency of detection. Accordingly, a projective decoder based on its geometry theorem is also proposed. Next, the "patterned" property of the PRM code, which partitions the binary vector space into several subspaces, is further extended as the primary principle for designing a slot control criterion that reduces the number of simultaneous transmissions in each slot. The factors affecting the chance of sequence collisions are identified. Finally, the proposed scheme is implemented in two practical outer A-channel codes: (i) the t-tree code and (ii) the Reed–Solomon code with Guruswami–Sudan list decoding, and the optimal setups are determined to minimize SNR by optimizing the inner and outer codes jointly. In comparison with the existing counterpart, our simulation results confirm that the proposed scheme compares favorably with benchmark schemes regarding the energy-per-bit requirement to meet a target error probability as well as the number of accommodated active users in the system. [ABSTRACT FROM AUTHOR]

This paper proposed a novel method for constructing quasi‐cyclic low‐density parity‐check (QC‐LDPC) codes of medium to high code rates that can be applied in cloud data storage systems, requiring better error correction capabilities. The novelty of this method lies in the construction of sparse base matrices, using a girth greater than 4 that can then be expanded with a lift factor to produce high code rate QC‐LDPC codes. Investigations revealed that the proposed large‐sized QC‐LDPC codes with high code rates displayed low encoding complexities and provided a low bit error rate (BER) of 10−10 at 3.5 dB Eb/N0 than conventional LDPC codes, which showed a BER of 10−7 at 3 dB Eb/N0. Subsequently, implementation of the proposed QC‐LDPC code in a software‐defined radio, using the NI USRP 2920 hardware platform, was conducted. As a result, a BER of 10−6 at 4.2 dB Eb/N0 was achieved. Then, the performance of the proposed codes based on their encoding–decoding speeds and storage overhead was investigated when applied to a cloud data storage (GCP). Our results revealed that the proposed codes required much less time for encoding and decoding (of data files having a 10 MB size) and produced less storage overhead than the conventional LDPC and Reed–Solomon codes. [ABSTRACT FROM AUTHOR]

The structural parameters, electronic, optical, and elastic properties of Niobium Carbide (NbC) compound within four different structures: rock-salt (RS), wurtzite (WZ), cesium chloride (CsCl), and Nickel Arsenide (NiAs) are examined using the Full-Potential Linearized-Augmented Plane Wave (FP-LAPW). The exchange–correlation potential (VXC) has been treated by Perdew, Burke, and Ernzerhof's generalized gradient approximation (PBE-GGA) when structural properties, transition pressure, and elastic properties are estimated. For electronic properties, in addition to (PBE-GGA), the modified Becke–Johnson (mBJ-GGA) was used for increased accuracy. Present results show that the RS of NbC is the most stable among the four examined structures with the lowest equilibrium energy. The calculated lattice constants are in good agreement with the former calculations. The elasticity and the formation energy calculations show that NbC is stable within the four studied structures. The electronic band structure calculations of NbC show that it has a metallic nature in the four considered structures. [ABSTRACT FROM AUTHOR]

We show new and improved list decoding properties of folded Reed--Solomon (RS) codes and multiplicity codes. Both of these families of codes are based on polynomials over finite fields, and both have been the source of recent advances in coding theory: folded RS codes were the first known explicit construction of capacity-achieving list decodable codes [V. Guruswami and A. Rudra, IEEE Trans. Inform. Theory, 54 (2008), pp. 135--150], and multiplicity codes were the first construction of high-rate locally decodable codes [S. Kopparty, S. Saraf, and S. Yekhanin, J. ACM, 61 (2014), 28]. In this work, we show that folded RS codes and multiplicity codes are in fact better than previously known in the context of list decoding and local list decoding. Our first main result shows that folded RS codes achieve list decoding capacity with constant list sizes, independent of the block length. Prior work with constant list sizes first obtained list sizes that are polynomial in the block length and relied on pre-encoding with subspace evasive sets to reduce the list sizes to a constant [V. Guruswami and C. Wang, IEEE Trans. Inform. Theory, 59 (2013), pp. 3257--3268], [Z. Dvir and S. Lovett, Proc. 44th STOC, ACM, 2012, 351--358]. The list size we obtain is (1/\varepsilon)O(1/\varepsilon) where \varepsilon is the gap to capacity, which matches the list size obtained by pre-encoding with subspace evasive sets. For our second main result, we observe that univariate multiplicity codes exhibit similar behavior, and we use this, together with additional ideas, to show that multivariate multiplicity codes are locally list decodable up to their minimum distance. By known reductions, this gives, in turn, capacity-achieving locally list decodable codes with query complexity exp(\~ O((logN)5/6)). This improves on the tensor-based construction of [B. Hemenway, N. Ron-Zewi, and M. Wootters, SIAM J. Comput., 49 (2019), pp. 157--195], which gave capacity-achieving locally list decodable codes of query complexity N \~ O(1/\mathrm{l}\mathrm{o}\mathrm{g} \mathrm{l}\mathrm{o}\mathrm{g}N), and is close to the best known query complexity of exp(\~ O(\surd logN)) for high-rate locally (uniquely) decodable codes [S. Kopparty et al., J. ACM, 64 (2017), 11]. [ABSTRACT FROM AUTHOR]