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Secure distributed matrix multiplication (SDMM) schemes are crucial for distributed learning algorithms where extensive data computation is distributed across multiple servers. Inspired by the application of repairing Reed–Solomon (RS) codes in distributed storage and secret sharing, we propose SDMM schemes with reduced communication overhead through the use of trace polynomials. Specifically, these schemes are designed to address three critical concerns: (i) ensuring information-theoretic privacy against collusion among servers; (ii) providing security against Byzantine servers; and (iii) offering resiliency against stragglers to mitigate computing delays. To the best of our knowledge, security and resiliency are being considered for the first time within trace polynomial-based approaches. Furthermore, our schemes offer the advantage of reduced sub-packetization and a lower server-count requirement, which diminish the computational complexity and download cost for the user. [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]

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]

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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]

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]

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]

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]

NAND flash memory is a non-volatile storage device that is extensively used in personal electronic gadgets, digital television, digital cameras, and many consumer/ professional electronics devices. Error control coding techniques have been incorporated to improve the integrity of information stored in these devices. We have synthesized the Subfield Subcodes of Reed Solomon codes (SSRS) for use on Multi-Level cell (MLC), Triple Level Cell (TLC), and Quadruple Level Cell (QLC) NAND flash devices. The primary advantage of these codes is that the codeword symbols can be correctly matched to the number of bits that can be stored in these multilevel cells. Deployment of these codes improves the integrity of information storage and useful life. This paper describes the implementation of the encoder and decoder of SSRS codes synthesized for MLC, TLC, and QLC NAND flash devices. The encoder circuit is designed using addition and multiplication tables derived from elements of synthesized SSRS codes. The Non-binary decoding procedure consists of the syndrome computation, Berlekamp -Massey algorithm, Chein search, and Forney’s algorithm. The designed encoder requires 16% resources for MLC, 18% of resources for TLC, and 18% of resources for QLC. This research work has reported the design of very high rate ( $R \geq 0.97$ ) codes that can bring about significant improvements to the Undetected Bit Error Rate (UBER) even when the Raw Bit Error rate (RBER) values are significant ( $\gt 10^{-3}$ ).

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]

Secure distributed matrix multiplication (SDMM) schemes are crucial for distributed learning algorithms where extensive data computation is distributed across multiple servers. Inspired by the application of repairing Reed–Solomon (RS) codes in distributed storage and secret sharing, we propose SDMM schemes with reduced communication overhead through the use of trace polynomials. Specifically, these schemes are designed to address three critical concerns: (i) ensuring information-theoretic privacy against collusion among servers; (ii) providing security against Byzantine servers; and (iii) offering resiliency against stragglers to mitigate computing delays. To the best of our knowledge, security and resiliency are being considered for the first time within trace polynomial-based approaches. Furthermore, our schemes offer the advantage of reduced sub-packetization and a lower server-count requirement, which diminish the computational complexity and download cost for the user.

We study the problem of solving a system of m polynomials in n variables over the ring of integers modulo a prime-power p k. The problem over finite fields is well studied in varied parameter settings. For small characteristic p = 2 , Lokshtanov et al. (SODA'17) initiated the study, for degree d = 2 systems, to improve the exhaustive search complexity of O (2 n) ⋅ poly (m , n) to O (2 0.8765 n) ⋅ poly (m , n) ; which currently is improved to O (2 0.6943 n) ⋅ poly (m , n) in Dinur (SODA'21). For large p but constant n , Huang and Wong (FOCS'96) gave a randomized poly (d , m , log ⁡ p) time algorithm. Note that for growing n , system-solving is known to be intractable even with p = 2 and degree d = 2. We devise a randomized poly (d , m , log ⁡ p) -time algorithm to find a root of a given system of m integral polynomials of degrees bounded by d , in n variables, modulo a prime power p k ; when n + k is constant. In a way, we extend the efficient algorithm of Huang and Wong (FOCS'96) for system-solving over Galois fields (i.e., characteristic p) to system-solving over Galois rings (i.e., characteristic p k); when k > 1 is constant. The challenge here is to find a lift of singular F p -roots (exponentially many); as there is no efficient general way known in algebraic-geometry for resolving singularities. Our algorithm has applications to factoring univariate polynomials over Galois rings. Given f ∈ Z [ x ] and a prime-power p k (k ≥ 2), finding factors of f mod p k has a curious state-of-the-art. It is solved for large k by p -adic factoring algorithms (von zur Gathen, Hartlieb, ISSAC'96); but unsolved for small k. In particular, no nontrivial factoring method is known for k ≥ 5 (Dwivedi, Mittal, Saxena, ISSAC'19). One issue is that degree- δ factors of f (x) mod p k could be exponentially many, as soon as k ≥ 2. We give the first randomized poly (deg ⁡ (f) , log ⁡ p) -time algorithm to find a degree- δ factor of f (x) mod p k , when k + δ is constant. Our method has potential application in algebraic coding theory. In particular, extending algebraic geometric and Reed-Solomon codes to Galois rings could enable new and improved bounds on their underlying efficiency parameters. [ABSTRACT FROM AUTHOR]

Let F q denote the finite field of size q and F q n denote the set of n -tuples of elements from F q. A family of k -dimensional subspaces of F q n , which forms a partial spread, is called L -almost affinely disjoint (or briefly [ n , k , L ] q -AAD) if each affine coset of a member of this family intersects with only at most L subspaces from the family. Polyanskii and Vorobyev introduced AAD subspace families in [23] (2019) in order to construct some types of primitive batch codes. For this purpose, they made use of Reed-Solomon codes and hence they presented [ n = 2 k + 1 , k , L = k ] -AAD subspace families of size ⌊ q / k ⌋. Later on, AAD spaces received a notable attention and have been studied for various parameters, see Liu et al. [18] (2021) and Otal and Arıkan [21] (2022) for example. In Arıkan et al. (2023) [1] , we presented a construction of [ 2 k + 1 , k , k ] -AAD subspace families of size q , which improves the lower bound ⌊ q / k ⌋ given by Polyanskii and Vorobyev in [23] (2019) k times. We also proved that our construction yields AAD subspace families for k = 2 and k = 3 , and left the proof of the case k ≥ 4 as a conjecture. In this paper, we now prove this conjecture by a newer and more abstract approach. For the proof, we carefully utilize several sophisticated arguments coming from coding theory, algebraic geometry, and linear algebra. In particular, we interpret each subspace as a suitable generator matrix (the lifting idea in random network coding), then algebraically manipulate their AAD-ness relation and produce suitable Van der Monde matrices (the fundamental tools for the BCH bound in coding theory), later apply Cramer's rule (a well-known key tool in linear algebra), and lastly obtain low-degree polynomials as contradictions (the trick coming from algebraic geometry utilized for Goppa codes). We present our constructive proof step-by-step to make it more easy to understand. [ABSTRACT FROM AUTHOR]

We used modulating retro-reflector (MRR), hybrid frequency-phase-keying (FPK) modulation, and Reed-Solomon (RS) code technology to improve the performance of the terrestrial-satellite laser communication link. A system using FPK modulation and RS codes based on the Fisher-Snedecor (F) distribution atmospheric turbulence model is proposed for the MRR-assisted terrestrial-high-altitude-platform (HAP)-satellite laser communication, in which MRR is installed on the HAP. The laser beam without signal is transmitted to HAP through the forward path. The MRR loads the optical signal from the terrestrial-HAP link onto the optical carrier, which is then reflected back to the satellite terminal along the backward path. The combined effects of light intensity scintillation of atmospheric turbulence, beam wander of uplink, atmospheric attenuation, pointing error, and HAP random jitter on the terrestrial-HAP (equipped with MRR)-satellite (T-HMRR-S) laser communication link are considered. The expression of the BER of RS codes and FPK modulation (RS-FPK) system under weak atmospheric turbulence is derived and compared with terrestrial-HAP-satellite (T-H-S) system with different modulation and without MRR. At the same time, the effects of modulation coding parameters, zenith angle, the field-of-view angle, and the coverage area of the MRR on the performance of the T-HMRR-S system are simulated and analyzed. The simulation results show that the T-HMRR-S system using RS-FPK technology has better performance. These works provide a theoretical basis for the research of modulation coding technology of T-HMRR-S link systems. [ABSTRACT FROM AUTHOR]

Entanglement-assisted quantum error-correcting codes (EAQECCs) extend the usual structure of quantum error-correcting codes (QECCs) by introducing the pre-shared entanglement. This new ingredient increases quantum error-correcting capability in some cases. The nonexistence of general constructions of q -ary quantum maximum distance separable (MDS) codes with minimum distances d > q + 1 motivates us to establish entanglement-assisted quantum MDS codes whose minimum distances are greater than q + 1. In this paper, we propose new constructions of EAQECCs achieving the quantum Singleton bound. Our constructions are based on Reed-Solomon codes over the finite field F q 2 . The constructed q -ary EAQECCs have new parameters and minimum distances d > q + 1. [ABSTRACT FROM AUTHOR]

Error correction and erasure codes and steganographic channels use related methods, but are investigated separately. We detail an idea from literature for a steganographic channel in a transmission with error correction code and experimentally investigate it with respect to bandwidth, robustness and detectability. We expand this construction to provide an example of multi-level steganography, i.e., a steganographic channel within a steganographic channel. Furthermore, we investigate the advantages on bandwidth and stealthyness that reversibility of such a steganographic channel brings, together with a new proposal for a covert channel in error-corrected data. [ABSTRACT FROM AUTHOR]

Secret image sharing (SIS) is a promising image protection technology boasting high security and loss tolerance. However, shadow images generated by SIS are susceptible to noise contamination during real-world transmission, storage, and processing, resulting in data loss when recovering the secret image. Therefore, improving the robustness of SIS systems against noise is crucial. Unfortunately, there is currently limited research on this topic, and existing schemes require improvement in the anti-noise performance. Under the premise of no pixel expansion, this paper proposes a robust SIS scheme that can effectively resist many types of shadow image noise, single or compound. We have achieved the organic fusion of SIS and Reed–Solomon coding technology by elaborately designing many critical factors, such as the shortened Reed–Solomon codes, the construction and utilization method of pixel-blocks, and other prerequisites. The proposed scheme is particularly effective in resisting burst noise, such as salt-and-pepper noise, with lossless recovery possible when the noise density d ≤ 0. 01 and high-quality recovery when d < 0. 06. Compared with existing schemes, it has many advantages, such as stronger anti-noise capability, various applicable noise types, stable anti-noise performance, and no pixel expansion. Analyses and illustrations are conducted to validate the proposed scheme. • A robust secret image sharing scheme with improved anti-noise ability is proposed. • It can resist various high-density single or compound shadow image noise. • For the first time, SIS and RS codes are organically fused without pixel expansion. • Robustness stems from the secret sharing technology itself. [ABSTRACT FROM AUTHOR]

We consider information-theoretical private information retrieval (PIR) from a coded database with colluding servers. We target, for the first time, locally repairable storage codes (LRCs). We consider any number of local groups g , locality r , local distance δ and dimension k. Our main contribution is a PIR scheme for maximally recoverable (MR) LRCs based on linearized Reed–Solomon codes, which achieve the smallest field sizes among MR-LRCs for many parameter regimes. In our scheme, nodes are identified with codeword symbols and servers are identified with local groups of nodes. Only locally non-redundant information is downloaded from each server, that is, only r nodes (out of r + δ − 1) are downloaded per server. The PIR scheme achieves the (download) rate R = (N − k − r t + 1) / N , where N = g r is the length of the MDS code obtained after removing the local parities, and for any t colluding servers such that k + r t ≤ N. For an unbounded number of stored files, the obtained rate is strictly larger than those of known PIR schemes that work for any MDS code. Finally, the obtained PIR scheme can also be adapted when communication between the user and each server is performed via linear network coding, achieving the same rate as previous PIR schemes for this scenario but with polynomial finite field sizes, instead of exponential. Our rates are equal to those of PIR schemes for Reed–Solomon codes, but Reed–Solomon codes are incompatible with the MR-LRC property or linear network coding, thus our PIR scheme is less restrictive in its applications. [ABSTRACT FROM AUTHOR]

BCH codes are a subclass of cyclic codes with many interesting properties. In many cases BCH codes are the best linear codes. For example, among all binary cyclic codes of odd length n with n ≤ 125 the best cyclic code is always a BCH code except for two special cases. Reed-Solomon codes are also BCH codes and are widely used in communication devices and consumer electronics. Binary BCH codes were discovered by Hocquenghem and independently by Bose and Ray-Chaudhuri. They were extended to BCH codes over finite fields by Gorenstein and Zierler in 1961. In fact, BCH codes were introduced by Bush in 1952 in the context of orthogonal arrays. In the past sixty years, a lot of progress on the study of BCH codes has been made. The objective of this survey is to present the state-of-the-art of BCH cyclic codes and their applications. [ABSTRACT FROM AUTHOR]

In this paper, let q be a prime power, we focus on a class of double-twisted generalized Reed-Solomon code C over F q. We give a sufficient and necessary condition for C to be MDS or AMDS, and prove that C is non-GRS by calculating the Schur square of its dual code. Furthermore, we present a sufficient and necessary condition for C to be self-dual, and then construct several classes of self-dual NMDS or non-GRS MDS codes. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study two new classes of additive codes over finite fields, viz. additive generalized Reed-Solomon (additive GRS) codes and additive generalized twisted Reed-Solomon (additive GTRS) codes, which are extensions of linear generalized Reed-Solomon (GRS) codes and twisted Reed-Solomon (GTRS) codes, respectively. Unlike linear GRS codes, additive GRS codes are not maximum distance separable (MDS) codes and the dual of an additive GRS code need not be an additive GRS code in general. We derive necessary and sufficient conditions under which an additive GRS code is MDS. We further apply this result to identify several new classes of additive MDS codes and a class of additive MDS codes whose dual codes are also MDS within the family of additive GRS codes. We also identify several new classes of additive codes that are either MDS or almost MDS within the family of additive GTRS codes. We also obtain several classes of additive TRS codes that are not monomially equivalent to additive RS codes. Besides this, we identify classes of monomially inequivalent additive MDS TRS codes and additive MDS RS codes, whose dual codes are also MDS. We also provide methods to construct additive MDS self-orthogonal, self-dual, and ACD codes through additive GRS and GTRS codes. Based on additive MDS codes whose dual codes are also MDS, we present a perfect threshold secret-sharing scheme that can detect cheating, identify a certain number of cheaters among the participants, and correctly recover the secret. [ABSTRACT FROM AUTHOR]

In this paper, we give a parity check matrix of the (+) -extended twisted generalized Reed Solomon code, and then not only prove that it is MDS or NMDS, but also determine the weight distribution. Especially, based on Schur method, we show that the (+) -extended twisted generalized Reed Solomon code is neither the generalized Reed Solomon code nor extended generalized Reed Solomon code. Furthermore, we present necessary and sufficient conditions for the (+) -twisted generalized Reed Solomon code and the (+) -extended twisted generalized Reed Solomon code to be self-orthogonal, respectively. Finally, several classes of self-dual (+) -twisted generalized Reed Solomon code codes and almost self-dual (+) -extended twisted generalized Reed Solomon codes are constructed. [ABSTRACT FROM AUTHOR]

Insertion and deletion (insdel in short) codes are designed to deal with synchronization errors in communication channels caused by insertions and deletions of message symbols. These codes have received a lot of attention due to their applications in diverse areas such as computational biology, DNA data storage, race-track memory error corrections, language processing, and synchronous digital communication networks. In the present work, we study constructions and limitations of insdel codes from rank metric and subspace codes. This paper studies and improves the idea of the work [4] by Hao Chen on the connection between insdel codes and subspace codes. We discuss why subspace code is a natural choice for constructing insdel codes and show that the interleaved Gabidulin codes can be used to construct nonlinear insdel codes approaching the Singleton bound. Then we show that the indexing scheme of transforming efficient Hamming metric codes to efficient insdel codes can be adapted for the class of rank metric codes. And that improves the base field size of the construction of insdel codes from lifted rank-metric codes. It is also shown that the size of the insdel code from a subspace code can be improved significantly than in the previously proposed construction. We give an algebraic condition for a linear Gabidulin rank metric code to be optimal insdel code adapting the condition proved for Reed-Solomon codes. Moreover, we give constructions of linear and nonlinear insdel codes from Sidon spaces. [ABSTRACT FROM AUTHOR]

According to the well-known CSS construction, constructing quantum MDS codes are extensively investigated via Hermitian self-orthogonal generalized Reed-Solomon (GRS) codes. In this paper, given two Hermitian self-orthogonal GRS codes G R S k 1 (A , v A) and G R S k 2 (B , v B) , we propose a sufficient condition to ensure that G R S k (A ∪ B , v A ∪ B) is still a Hermitian self-orthogonal code. Consequently, we first present a new general construction of infinitely families of quantum MDS codes from known ones. Moreover, applying the trace function and norm function over finite fields, we give another two new constructions of quantum MDS codes with flexible parameters. It turns out that the forms of the lengths of our quantum MDS codes are quite different from previous known results in the literature. Meanwhile, the minimum distances of all the q -ary quantum MDS codes are bigger than q / 2 + 1. [ABSTRACT FROM AUTHOR]