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Chinese Spelling Error Correction is a hot subject in the field of natural language processing. Researchers have already produced many great solutions, from the initial rule-based solution to the current deep learning method. At present, SpellGCN, proposed by Alibaba's team, achieves the best results of which character level precision over SIGHAN2013 is 98.4%. However, when we apply this algorithm to practical error correction tasks, it produces many false error correction results. We believe that this is because the corpus used for model training contains significantly more errors than the text used for model correcting. In response to this problem, we propose performing a post-processing operation on the error correction tasks. We employ the initial model's output as a candidate character, obtain various features of the character itself and its context, and then use a classification model to filter the initial model's false error correction results. The post-processing idea introduced in this paper can apply to most Chinese Spelling Error Correction models to improve their performance over practical error correction tasks. [ABSTRACT FROM AUTHOR]

The process of DNA-based data storage (DNA storage for short) can be mathematically modelled as a communication channel, termed DNA storage channel, whose inputs and outputs are sets of unordered sequences. To design error correcting codes for DNA storage channel, a new metric, termed the sequence-subset distance, is introduced, which generalizes the Hamming distance to a distance function defined between any two sets of unordered vectors and helps to establish a uniform framework to design error correcting codes for DNA storage channel. We further introduce a family of error correcting codes, referred to as sequence-subset codes, for DNA storage and show that the error-correcting ability of such codes is completely determined by their minimum distance. We derive some upper bounds on the size of the sequence-subset codes including a tight bound for a special case, a Singleton-like bound and a Plotkin-like bound. We also propose some constructions, including an optimal construction for that special case, which imply lower bounds on the size of such codes. [ABSTRACT FROM AUTHOR]

Limited magnitude asymmetric error model is well suited for flash memory. In this paper, we consider the construction of asymmetric codes correcting single error over $\mathbb {Z}_{2^{k}r}$ which is based on so called $B_{1}[{4}](2^{k}r)$ set. In fact, we reduce the construction of a maximal size $B_{1}[{4}](2^{k}r)$ set for $k\geq 3$ to the construction of a maximal size $B_{1}[{4}](2^{k-3}r)$ set. Finally, we give an explicit formula of a maximal size $B_{1}[{4}](4r)$ set and some lower bounds of a maximal size $B_{1}[{4}](2r)$ set. By computer searching up to $q\leq 106$ , we conjecture that those lower bounds are tight. [ABSTRACT FROM AUTHOR]

To equip DNA-based data storage with random-access capabilities, Yazdi et al. (2018) prepended DNA strands with specially chosen address sequences called primers and provided certain design criteria for these primers. We provide explicit constructions of error-correcting codes that are suitable as primer addresses and equip these constructions with efficient encoding algorithms. Specifically, our constructions take cyclic or linear codes as inputs and produce sets of primers with similar error-correcting capabilities. Using certain classes of BCH codes, we obtain infinite families of primer sets of length $n$ , minimum distance $d$ with $(d+1) \log _{4}\,\,n +O(1)$ redundant symbols. Our techniques involve reversible cyclic codes (1964), an encoding method of Tavares et al. (1971) and Knuth’s balancing technique (1986). In our investigation, we also construct efficient and explicit binary balanced error-correcting codes and codes for DNA computing. [ABSTRACT FROM AUTHOR]

Because of their capacity-approaching performance and their complexity/latency advantages, spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern dense data storage systems. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for additive white Gaussian noise (AWGN) and Flash channels. The OO stage operates on the protograph of the SC code to derive the optimal partitioning that minimizes the number of graphical objects that undermine the performance of SC codes under iterative decoding. Then, the CPO optimizes the circulant powers to further reduce this number. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the proposed OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by up to around 3 orders of magnitude in frame error rate (FER) and 1.1 dB in signal-to-noise ratio (SNR). More intriguingly, our SC codes outperform structured block codes of the same length and rate by up to around 1.8 orders of magnitude in FER and 0.4 dB in SNR. The performance advantage of SC codes designed using the devised OO-CPO approach over block codes of the same parameters is not only pronounced in the error floor region, but also in the waterfall region. [ABSTRACT FROM AUTHOR]

Error-correcting codes add redundancy to transmitted data to ensure reliable communication over noisy channels. Since they form the foundations of digital communication, their correctness is a matter of concern. To enable trustful verification of linear error-correcting codes, we have been carrying out a systematic formalization in the Coq proof-assistant. This formalization includes the material that one can expect of a university class on the topic: the formalization of well-known codes (Hamming, Reed–Solomon, Bose–Chaudhuri–Hocquenghem) and also a glimpse at modern coding theory. We demonstrate the usefulness of our formalization by extracting a verified decoder for low-density parity-check codes based on the sum-product algorithm. To achieve this formalization, we needed to develop a number of libraries on top of Coq's Mathematical Components. Special care was taken to make them as reusable as possible so as to help implementers and researchers dealing with error-correcting codes in the future. [ABSTRACT FROM AUTHOR]

We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining ${p}_{0}^{(adv)}$ , the zero-rate threshold of adversarial deletions, defined to be the supremum of all $p$ for which there exists a code family with rate bounded away from 0 capable of correcting a fraction $p$ of adversarial deletions. A recent construction of deletion-correcting codes shows that ${p}_{0}^{(adv)} \ge \sqrt {2}-1$ , and the trivial upper bound, ${p}_{0}^{(adv)}\le {1}/{2}$ , is the best known. Perhaps surprisingly, we do not know whether or not ${p}_{0}^{(adv)} = 1/2$. In this work, to gain further insight into deletion codes, we explore two related error models: oblivious deletions and online deletions, which are in between random and adversarial deletions in power. In the oblivious model, the channel can inflict an arbitrary pattern of ${pn}$ deletions, picked without knowledge of the codeword. We prove the existence of binary codes of positive rate that can correct any fraction ${p} < 1$ of oblivious deletions, establishing that the associated zero-rate threshold ${p}_{0}^{(obliv)}$ equals 1. For online deletions, where the channel decides whether to delete bit ${x}_{{i}}$ based only on knowledge of bits ${x}_{1}{x}_{2} {\dots }{x}_{{i}}$ , define the deterministic zero-rate threshold for online deletions ${p}_{0}^{(on,{d})}$ to be the supremum of $p$ for which there exist deterministic codes against an online channel causing ${pn}$ deletions with low average probability of error. That is, the probability that a randomly chosen codeword is decoded incorrectly is small. We prove ${p}_{0}^{(adv)}={1}/{2}$ if and only if ${p}_{0}^{(on,{d})}={1}/{2}$. [ABSTRACT FROM AUTHOR]

We consider the decoding of linear and array codes from errors when we are only allowed to download a part of the codeword. More specifically, suppose that we have encoded $k$ data symbols using an $(n,k)$ code with code length $n$ and dimension $k$. During storage, some of the codeword coordinates might be corrupted by errors. We aim to recover the original data by reading the corrupted codeword with a limit on the transmission bandwidth, namely, we can only download an $\alpha $ proportion of the corrupted codeword. For a given $\alpha $ , our objective is to design a code and a decoding scheme such that we can recover the original data from the largest possible number of errors. A naive scheme is to read $\alpha n$ coordinates of the codeword. This method used in conjunction with MDS codes guarantees recovery from any $\lfloor (\alpha n-k)/2\rfloor $ errors. In this paper we show that we can instead download an $\alpha $ proportion from each of the codeword’s coordinates. For a well-designed MDS code, this method can guarantee recovery from $\lfloor (n-k/\alpha)/2 \rfloor $ errors, which is $1/\alpha $ times more than the naive method, and is also the maximum number of errors that an $(n,k)$ code can correct by downloading only an $\alpha $ proportion of the codeword. We present two families of such optimal constructions and decoding schemes of which one is based on Interleaved Reed-Solomon codes and the other on Folded Reed-Solomon codes. We further show that both code constructions attain asymptotically optimal list decoding radius when downloading only a part of the corrupted codeword. We also construct an ensemble of random codes that with high probability approaches the upper bound on the number of correctable errors when the decoder downloads an $\alpha $ proportion of the corrupted codeword. [ABSTRACT FROM AUTHOR]

With technology scaling, burst errors or clustered errors are becoming increasingly common in different types of memories. Multiple bit upsets due to particle strikes, write disturbance errors, and magnetic field coupling are a few of the mechanisms which cause clustered errors. In this article, a new class of single burst error correcting codes are presented which correct a single burst of any size b within a codeword. A code construction methodology is presented which enables us to construct the proposed scheme from existing codes, e.g., Hamming codes. A new single step decoding methodology for the proposed class of codes is also presented which enables faster decoding. Different code constructions using Hamming codes, and BCH codes have been presented in this paper and a comparison is made with existing schemes in terms of decoding complexity and data redundancy. The proposed scheme in all cases reduces the decoder complexity for little to no increase in data redundancy, specifically for higher burst error sizes. [ABSTRACT FROM AUTHOR]

We first show that a class of operators acting on a given bipartite pure state on ℋA ⊗ ℋB can shrink its supports on ℋA ⊗ ℋB to only ℋA or ℋB while keeping its mappings. Using this result, we show how to systematically construct the decoders of the quantum error-correcting codes against erasure errors. The implications of the results for the operator dictionary in the AdS/CFT correspondence are also discussed. The "sub- algebra code with complementary recovery" introduced in the recent work of Harlow is a quantum error-correcting code that shares many common features with the AdS/CFT correspondence. We consider it under the restriction of the bulk (logical) Hilbert space to a subspace that generally has no tensor factorization into subsystems. In this code, the central operators of the reconstructed algebra on the boundary subregion can emerge as a consequence of the restriction of the bulk Hilbert space. Finally, we show a theorem in this code which implies the validity of not only the entanglement wedge reconstruction but also its converse statement with the central operators. [ABSTRACT FROM AUTHOR]

We present a new class of linear error-correcting codes taking numerical data into codewords with numerical symbols. These codes can correct large random numerical errors added to codewords. The goal is for numerical data to be protected directly as numbers whether in storage or transmission. The coding structures are based on discrete Fourier transform (DFT) codes, defined by parity-check matrices where rows are consecutively indexed DFT vectors. The new tensor codes use the Kronecker product of two parity-check DFT matrices of shorter length codes. We construct all necessary processing matrices. Error correction methods involve two levels of syndromes and use several stochastic Berlekamp-Massey Algorithms (sBMA) to find big syndromes, locate and evaluate large errors. Decoding is done in two stages with stage one producing corrected syndromes. Stage two determines errors within a well-defined segment for the codewords. Tensor codes have simpler and more efficient decoding operations. The characteristics of the tensor codes are compared to standard DFT codes of equal length. The processing operations are significantly less for the tensor codes but the standard DFT codes sometimes have better correcting performance. Nevertheless, tensor product codes can have acceptable levels of correction, more efficiently. [ABSTRACT FROM AUTHOR]

We construct error-correcting write-once memory (WOM) codes that guarantee correction of any specified number of errors in $q$ -level memories. The constructions use suitably designed short $q$ -ary WOM codes and concatenate them with outer error-correcting codes over different alphabets using suitably designed mappings. With a new storage-efficiency measure, we call EC-rate and show that for common error types the codes save redundancy and implementation complexity over straightforward concatenation. In addition to constructions for guaranteed error correction, we extend the error-correcting WOM scheme to binary multi-level coding for random errors, and toward soft-decision decoding provide an efficient way to extract reliability information without using higher-precision readout. [ABSTRACT FROM AUTHOR]

In the conventional index coding problem, the messages transmitted by the source to satisfy the demands of the receivers are error-free. However, in practice, the messages may be error prone which led to the introduction of error-correcting index codes. An error-correcting index code is an encoding scheme which enables every receiver to decode its demanded message even in the presence of certain number of transmission errors. Formally, an index code capable of correcting at most $\delta $ errors at all its receivers is defined as a $\delta $ -error-correcting index code. In this paper, we explore the connections between vector linear error-correcting index codes and discrete polymatroids. This is motivated from the connections between linear network error-correcting codes and matroids. It is shown that a vector linear error-correcting index code exists if and only if there exists a representable discrete polymatroid satisfying certain conditions. Differential error-correcting index codes which allow different receivers to have different error-correcting capabilities are also studied. Note that the $\delta $ -error-correcting code is a special case of a differential error-correcting index code. Error correction at a subset of receivers is another special case which is discussed. [ABSTRACT FROM AUTHOR]

In recent years, flash memory has been widely used in embedded systems, portable devices, and high-performance storage products due to its nonvolatility, shock resistance, low power consumption, and high performance natures. To reduce the product cost, multi-level-cell (MLC) flash memory has been proposed; compared with the traditional single-level-cell (SLC) flash memory that only stores one bit of data per cell, each MLC cell can store two or more bits of data. Thus MLC can achieve a larger capacity and reduce the cost per unit. However, MLC also suffers from the degradation in both performance and reliability. In this paper, we try to enhance the reliability and reduce the product cost of flash-memory-based solid-state drive (SSD) from a totally different perspective. We propose the half-level-cell (HLC) scheme to manage and reuse the worn-out space in SSD; through our management scheme, the system can treat two bad pages as a normal page without sacrificing performance and reliability. The proposed scheme is purely on software/firmware-level, thus there is no need to change the hardware. The experiment results show that the lifetime of SSD with our proposed HLC scheme can be extended to 50.56% under the Windows workload and up to 65.45% under the multimedia workload. When we apply the HLC scheme to flash-memory cache of hybrid storage systems, the response time can be improved up to 20.57%. [ABSTRACT FROM AUTHOR]

Cache vulnerability due to soft errors is one of the reliability concerns in embedded systems. Dynamic reconfiguration techniques are widely studied for improving cache energy without considering the implications of cache vulnerability. Maintaining a useful data longer in the cache can be beneficial for energy improvement due to reduction in miss rates, however, longer data retention negatively impacts the vulnerability due to soft errors. This paper studies the tradeoff between energy efficiency improvement and reduction in cache vulnerability during cache reconfiguration. We propose a heuristic approach for both intertask and intratask cache reconfiguration in multitasking systems. Experimental results demonstrate that our proposed approach can significantly improve both vulnerability (25% on average) and energy efficiency (21% on average) for data cache without violating real-time constraints. [ABSTRACT FROM AUTHOR]

In this paper, we present linear systematic error-correcting codes Lk and L+k which are the results of our research on the sub-exceeding functions. Given an integer k such that k ≥ 3, these two codes are re-spectively [2k, k] and [3k, k] linear codes. The minimum distance of L3 is 3 and for k ≥ 4 the minimum distance of Lk is 4. The code L+k, the minimum distances are respectively 5 and 6 for k = 4 and k ≥ 5. By calculating the complexity of the algorithms, our codes have fast and efficient decoding. Then, for a short and medium distance data transmission (wifi network, bluetooth, cable, ...), we see that the codes mentioned above present many advantages. [ABSTRACT FROM AUTHOR]

Additional check bits, which are commonly attached to the message’s input data, are normally used to minimize the error during data transmission. The receiver system implements a checking algorithm to determine if an error was occurred in the received data. This algorithm will correct a corrupted bit and recover the original message. An enhanced error detection correction code was presented to better detect and correct the corrupted conveyed bits. It improves the existing limitations of utilizing cyclic redundancy checking (CRC), Hamming code, and other checksum techniques. Also, it reduced the length of the redundancy bits which exists in CRC, the overhead of interspersing of the redundancy bits in a typical Hamming code, and the system resources such as processor time and bandwidth in checksum techniques. This paper was synthesized and simulated using the Xilinx Spartan 6 (XC7Z020-2CLG4841) FPGA. Results show that the resource utilization of the designed memory architecture using EEDC is lower compared to CRC, Hamming, and Checksum algorithms. [ABSTRACT FROM AUTHOR]

In this work, we propose constructions that correct duplications of multiple consecutive symbols. These errors are known as tandem duplications, where a sequence of symbols is repeated; respectively as palindromic duplications, where a sequence is repeated in reversed order. We compare the redundancies of these constructions with code size upper bounds that are obtained from sphere packing arguments. Proving that an upper bound on the code cardinality for tandem deletions is also an upper bound for inserting tandem duplications, we derive the bounds based on this special tandem deletion error as this results in tighter bounds. Our upper bounds on the cardinality directly imply lower bounds on the redundancy which we compare with the redundancy of the best known construction correcting arbitrary burst insertions. Our results indicate that the correction of palindromic duplications requires more redundancy than the correction of tandem duplications and both significantly less than arbitrary burst insertions. [ABSTRACT FROM AUTHOR]

Abstract Phase change memory (PCM) is a promising alternative to conventional DRAM main memories, due to its read performance, density, and nonvolatility and resulting low static energy. Unfortunately, reliability is still a significant challenge as limited write endurance, exacerbated by process variation, leads to increasing numbers of stuck-at faults over the memory's lifetime. This includes a significant number of stuck-at faults that appear early in the memory's service. Error-correcting Pointers (ECP) is a popular proposal to mitigate stuck-at faults in PCM by recording the addresses and the values of faulty bits in order to extend the lifetime of the memory. We propose a method to extend the effectiveness of ECP coverage called Yoda, which utilizes a small number of additional encoding bits in order to dramatically improve the effectiveness and fault correction capability of ECP. By adding one additional bit to ECP which corrects f faults, Yoda can correct 2 f +1 faults. Further improvements are possible introducing small numbers additional bits. Our simulation results demonstrate that Yoda has a 3.0× improvement in fault coverage compared to a fault-aware ECP with a similar overhead, while also providing a 2.5–3.0× improvement over state-of-the-art schemes with comparable complexity. Furthermore, Yoda provides a method to protect the auxiliary bits, also with a small overhead. By adding one auxiliary bit to protect the auxiliary bits, Yoda can achieve extra improvement. Highlights • Extending pointers with partition and flip methods improve guaranteed correction capabilities. • Using small additional computational overhead decreases the number of required encoding bits. • Applying partition and flip to the auxiliary bits can allow them to operate reliably in faulty memory. • Yoda improves tolerated faults by 2.5–3.0× compared to state-of-the-art designs. [ABSTRACT FROM AUTHOR]

The Error Correction Code (ECC) is utilized to reduce the probability of error in digital systems. The binary Golay code is an ECC that can correct any combination of three or fewer random errors over a block of 23 digits. This code can be extended by appending a parity check bit to each codeword. There are several algorithms for constructing of Golay code, but more of them are not comfortable for hardware implementation. In this paper, an efficient hardware architecture is presented for the encoder of both binary Golay code and extended Golay code based on CRC. The proposed Golay code encoder is constructed of three units, which are designed carefully: data path, control unit and conversion unit. The proposed architecture is implemented on FPGA using Xilinx ISE 14.2. The implementation results demonstrate that low latency, high throughput, low area and less complexity are the advantages of this architecture compared to previous architectures. Thus, this hardware module can be used for high-speed digital systems. [ABSTRACT FROM AUTHOR]

We propose a new structure of a self-checking combinational device where, based on the properties of parity and Berger codes, as well as a code with the detection of all double errors in information vectors, the problem of detecting all single faults of logical elements can be solved without transforming the structure of the source device. The properties of binary codes with the detection of all double errors that can be used in constructing the proposed structure are considered. We give an example of constructing a new structure. [ABSTRACT FROM AUTHOR]

Motivated by communication channels in which the transmitted sequences are subjected to random permutations, as well as by certain DNA storage systems, we study the error control problem in settings where the information is stored/transmitted in the form of multisets of symbols from a given finite alphabet. A general channel model is assumed in which the transmitted multisets are potentially impaired by insertions, deletions, substitutions, and erasures of symbols. Several constructions of error-correcting codes for this channel are described, and bounds on the size of optimal codes correcting any given number of errors are derived. The construction based on the notion of Sidon sets in finite Abelian groups is shown to be optimal, in the sense of the asymptotic scaling of code redundancy, for any error radius and alphabet size. It is also shown to be optimal in the stronger sense of maximal code cardinality in various cases. [ABSTRACT FROM AUTHOR]

Quantum metrology has many important applications in science and technology, ranging from frequency spectroscopy to gravitational wave detection. Quantum mechanics imposes a fundamental limit on measurement precision, called the Heisenberg limit, which can be achieved for noiseless quantum systems, but is not achievable in general for systems subject to noise. Here we study how measurement precision can be enhanced through quantum error correction, a general method for protecting a quantum system from the damaging effects of noise. We find a necessary and sufficient condition for achieving the Heisenberg limit using quantum probes subject to Markovian noise, assuming that noiseless ancilla systems are available, and that fast, accurate quantum processing can be performed. When the sufficient condition is satisfied, a quantum error-correcting code can be constructed that suppresses the noise without obscuring the signal; the optimal code, achieving the best possible precision, can be found by solving a semidefinite program. [ABSTRACT FROM AUTHOR]

In this paper, new index coding problems are studied, where each receiver has erroneous side information. Although side information is a crucial part of index coding, the existence of erroneous side information has not been considered yet. We study an index code with receivers that have erroneous side information symbols in the error-free broadcast channel, which is called an index code with side information errors (ICSIE). The encoding and decoding procedures of the ICSIE are proposed, based on the syndrome decoding. Then, we derive the bounds on the optimal codelength of the proposed index code with erroneous side information. Furthermore, we introduce a special graph for the proposed index coding problem, called a \delta s -cycle whose properties are similar to those of the cycle in the conventional index coding problem. Properties of the ICSIE are also discussed in the \delta s -cycle and clique. Finally, the proposed ICSIE is generalized to an index code for the scenario having both additive channel errors and side information errors, called a generalized error correcting index code. [ABSTRACT FROM AUTHOR]

We construct error-correcting codes over permutations under the infinity-metric, which are also Gray codes in the context of rank modulation, i.e., are generated as simple circuits in the rotator graph. These errors model limited-magnitude or spike errors, for which only single-error-detecting Gray codes are currently known. Surprisingly, the error-correcting codes we construct achieve a better asymptotic rate than that of presently known constructions not having the Gray property, and exceed the Gilbert-Varshamov bound. Additionally, we present efficient ranking and unranking procedures, as well as a decoding procedure that runs in linear time. Finally, we also apply our methods to solve an outstanding issue with error-detecting rank-modulation Gray codes (also known in this context as snake-in-the-box codes) under a different metric, the Kendall $\tau $ -metric, in the group of permutations over an even number of elements S_{2n} , where we provide asymptotically optimal codes. [ABSTRACT FROM AUTHOR]

Error correction coding is extensively used to achieve reliability in digital communication especially in Physical Layer. In this paper, construction of concatenated kernel code defined over algebraic structure group is discussed. Minimal trellis representation is given for constructed concatenated kernel codeword. Constructed code is tested in cognitive radio network environment subjected to continuous interference due to the behavior of primary users. Proposed code is tested through simulations and its performance is analyzed in terms of Bit Error Rate (BER) in mitigating the effect of continuous interference. Maximum likelihood graph Viterbi decoding technique is used to decode the concatenated kernel code. [ABSTRACT FROM AUTHOR]

We consider the problem of computing a binary linear transformation when all circuit components are unreliable. Two models of unreliable components are considered: probabilistic errors and permanent errors. We introduce the “ENCODED” technique that ensures that the error probability of the computation of the linear transformation is kept bounded below a small constant independent of the size of the linear transformation even when all logic gates in the computation are noisy. By deriving a lower bound, we show that in some cases, the computational complexity of the ENCODED technique achieves the optimal scaling in error probability. Further, we examine the gain in energy-efficiency from the use of a “voltage-scaling” scheme, where gate-energy is reduced by lowering the supply voltage. We use a gate energy-reliability model to show that tuning gate-energy appropriately at different stages of the computation (“dynamic” voltage scaling), in conjunction with ENCODED, can lead to orders of magnitude energy-savings over the classical “uncoded” approach. Finally, we also examine the problem of computing a linear transformation when noiseless decoders can be used, providing upper and lower bounds to the problem. [ABSTRACT FROM AUTHOR]

We describe a general method for turning quantum circuits into sparse quantum subsystem codes. The idea is to turn each circuit element into a set of low-weight gauge generators that enforce the input–output relations of that circuit element. Using this prescription, we can map an arbitrary stabilizer code into a new subsystem code with the same distance and number of encoded qubits but where all the generators have constant weight, at the cost of adding some ancilla qubits. With an additional overhead of ancilla qubits, the new code can also be made spatially local. Applying our construction to certain concatenated stabilizer codes yields families of subsystem codes with constant-weight generators and with minimum distance d = n^1- \epsilon , where \epsilon = O(1/\sqrt \log n) . For spatially local codes in D dimensions, we nearly saturate a bound due to Bravyi and Terhal and achieve d = n^{1- \epsilon -1/D} . Previously the best code distance achievable with constant-weight generators in any dimension, due to Freedman, Meyer, and Luo, was O(\sqrt {n\log n})$ for a stabilizer code. [ABSTRACT FROM PUBLISHER]

We give an infinite family of maximum rank distance (MRD) codes, which covers properly the largest known linear MRD code family. Our family contains infinite families of non-linear MRD codes, which are the first non-linear examples for most of the parameters. We also give explicit examples and a table that demonstrates the proportion of linear and non-linear families for some small parameters. [ABSTRACT FROM PUBLISHER]

In this paper, list decoding of crisscross errors in arrays over finite fields is considered. For this purpose, the so-called cover metric is used, where the cover of a matrix is a set of rows and columns which contains all non-zero elements of the matrix. A Johnson-like upper bound on the maximum list size in the cover metric is derived, showing that the list of codewords has polynomial size up to a certain radius. Furthermore, a simple list decoding algorithm for a known optimal code construction is presented, which decodes errors in the cover metric up to our upper bound. These results reveal significant differences between the cover metric and the rank metric and show that the cover metric is more suitable for correcting crisscross errors. [ABSTRACT FROM PUBLISHER]

We consider codes over fixed alphabets against worst case symbol deletions. For any fixed k \ge 2 , we construct a family of codes over alphabet of size k with positive rate, which allow efficient recovery from a worst case deletion fraction approaching 1-(2/(k+\sqrt k)) . In particular, for binary codes, we are able to recover a fraction of deletions approaching 1/(\sqrt 2 +1)=\sqrt 2-1 \approx 0.414 . Previously, even non-constructively, the largest deletion fraction known to be correctable with positive rate was 1-\Theta (1/\sqrt k) , and around 0.17 for the binary case. Our result pins down the largest fraction of correctable deletions for $k$ -ary codes as 1-\Theta (1/k)$ , since 1-1/k$ is an upper bound even for the simpler model of erasures where the locations of the missing symbols are known. Closing the gap between and 1/2 for the limit of worst case deletions correctable by binary codes remains a tantalizing open question. [ABSTRACT FROM PUBLISHER]

We study error correction codes for multimedia streaming applications where a stream of source packets must be transmitted in real-time, with in-order decoding, and strict delay constraints. In our setup, the encoder observes a stream of source packets in a sequential fashion, and $M$ channel packets must be transmitted between the arrival of successive source packets. Each channel packet can depend on all the source packets observed up to and including that time, but not on any future source packets. The decoder must reconstruct the source stream with a delay of $T$ packets. We consider a class of packet erasure channels with burst and isolated erasures, where the erasure patterns are locally constrained. Our proposed model provides a tractable approximation to statistical models, such as the Gilbert–Elliott channel, for capacity analysis. When $M=1$ , i.e., when the source-packet arrival and channel-packet transmission rates are equal, we establish upper and lower bounds on the capacity, that are within one unit of the decoding delay $T$ . We also establish necessary and sufficient conditions on the column distance and column span of a convolutional code to be feasible, and in turn establish a fundamental tradeoff between these. Our proposed codes—maximum distance and span codes—achieve a near-optimal tradeoff between the column distance and column span, and involve a layered construction. When $M>1$ , we establish the capacity for the burst-erasure channel and an achievable rate in the general case. Extensive numerical simulations over Gilbert–Elliott and Fritchman channel models suggest that our codes also achieve significant gains in the residual loss probability over statistical channel models. [ABSTRACT FROM PUBLISHER]

Error-correcting output coding (ECOC) is one of the most widely used strategies for dealing with multi-class problems by decomposing the original multi-class problem into a series of binary sub-problems. In traditional ECOC-based methods, binary classifiers corresponding to those sub-problems are usually trained separately without considering the relationships among these classifiers. However, as these classifiers are established on the same training data, there may be some inherent relationships among them. Exploiting such relationships can potentially improve the generalization performances of individual classifiers, and, thus, boost ECOC learning algorithms. In this paper, we explore to mine and utilize such relationship through a joint classifier learning method, by integrating the training of binary classifiers and the learning of the relationship among them into a unified objective function. We also develop an efficient alternating optimization algorithm to solve the objective function. To evaluate the proposed method, we perform a series of experiments on eleven datasets from the UCI machine learning repository as well as two datasets from real-world image recognition tasks. The experimental results demonstrate the efficacy of the proposed method, compared with state-of-the-art methods for ECOC-based multi-class classification. [ABSTRACT FROM PUBLISHER]

We propose binary discrete parametric channel models for multi-level cell (MLC) flash memories that provide accurate error-correcting code (ECC) performance estimation by modeling the empirically observed error characteristics under program/erase cycling stress. Through a detailed empirical error characterization of 1X-nm and 2Y-nm MLC flash memory chips from two different vendors, we observe and characterize the overdispersion phenomenon in the number of bit errors per ECC frame. A well-studied channel model, such as the binary asymmetric channel model, is unable to provide accurate ECC performance estimation. Hence, we propose a channel model based on the beta-binomial probability distribution [2-beta-binomial (2-BBM) channel model], which is a good fit for the overdispersed empirical error characteristics, and show through statistical tests and simulation results for BCH, low density parity check, and polar codes, that the 2-BBM channel model provides accurate ECC performance estimation in MLC flash memories. [ABSTRACT FROM AUTHOR]

We present the first model of optimal voting under adversarial noise. From this viewpoint, voting rules are seen as error-correcting codes: their goal is to correct errors in the input rankings and recover a ranking that is close to the ground truth. We derive worst-case bounds on the relation between the average accuracy of the input votes, and the accuracy of the output ranking. Empirical results from real data show that our approach produces significantly more accurate rankings than alternative approaches. [ABSTRACT FROM AUTHOR]

In pebbling problems, pebbles are placed on the vertices of a graph. A pebbling move consists of removing two pebbles from one vertex, throwing one away, and moving the other pebble to an adjacent vertex. We say a distribution D is solvable if starting from D , we can move a pebble to any vertex by a sequence of pebbling moves. The optimal pebbling number of a graph G is the smallest number of pebbles in a solvable distribution on G . It is known that every solvable distribution on the n -dimensional hypercube Q n contains at least ( 4 3 ) n pebbles. Fu, Huang, and Shiue, building on the work of Moews, used probabilistic methods to show that there are solvable distributions where the number of pebbles is in O ( ( 4 3 ) n n 3 2 ) , but hitherto, the number of pebbles in the best constructed distributions was in O ( 1.37 7 n ) . We use error-correcting codes to construct solvable distributions of pebbles on Q n in which the number of pebbles is in O ( 1.3 4 n ) . [ABSTRACT FROM AUTHOR]

The code design of non-binary low-density parity-check codes for the erasure channel, under maximum a posteriori decoding, is addressed. In particular, a partially structured ensemble of codes, characterized by a careful control of the amount and of the connectivity of the variable nodes of small degrees, is proposed. The identified ensemble of codes is analyzed in terms of asymptotic thresholds and weight distribution and it is shown that codes from the ensemble provide a remarkable trade-off between waterfall performance, error floor, and decoding complexity. As an example, the performance curve of a short (256,128) code on the memoryless 16-ary erasure channel tightly approaches the Singleton bound at least down to a codeword error rate of 10^-9, at low decoding complexity. [ABSTRACT FROM AUTHOR]

Improved lower and upper bounds on the size and the rate of grain-correcting codes are presented. The lower bound is Gilbert–Varshamov-like combined with a construction by Gabrys et al., and it improves on the previously best known lower bounds on the asymptotic rate of \lceil \tau n \rceil -grain-correcting codes of length n on the interval [0,0.0668] . One of the two newly presented upper bounds improves on the best known upper bounds on the asymptotic rate of \lceil \tau n \rceil -grain-correcting codes of length n on the interval \tau \in (0,{1}/{8}] and meets the lower bound of {1}/{2} for \tau \geq 1/8 . Moreover, in a nonasymptotic regime, both upper bounds improve on the previously best known results on the largest size of t -grain-correcting codes of length $n$ , for certain values of n$ and t$ . Constructions of 1-grain-correcting codes based on a partitioning technique are presented for lengths up to 18. Finally, a lower bound of on the minimum redundancy of $\infty $ -grain-detecting codes of length $n$ is presented. [ABSTRACT FROM PUBLISHER]

Random network coding recently attracts attention as a technique to disseminate information in a network. This paper considers a noncoherent multishot network, where the unknown and time-variant network is used several times. In order to create dependence between the different shots, particular convolutional codes in rank metric are used. These codes are so-called (partial) unit memory ((P)UM) codes, i.e., convolutional codes with memory one. First, distance measures for convolutional codes in rank metric are shown and two constructions of (P)UM codes in rank metric based on the generator matrices of maximum rank distance codes are presented. Second, an efficient error-erasure decoding algorithm for these codes is presented. Its guaranteed decoding radius is derived and its complexity is bounded. Finally, it is shown how to apply these codes for error correction in random linear and affine network coding. [ABSTRACT FROM AUTHOR]

Incorporating information security and error correction in network coding, which has various applications in communication theory, for example, secret key sharing through a network, is of special interest and has been widely studied. To make a more intensive analysis of secure error-correcting network codes, we investigate how to secure k source symbols transmission in a multicast network against an adversary that can obtain k1 source symbols as side information, eavesdrop µ channels and contaminate d channels. We introduce relative network generalised Hamming weight (RNGHW) with network error correction (NEC), or briefly NEC-RNGHW, to measure the equivocation to the adversary. Network generalised singleton bound on NEC-RNGHW is obtained and code constructions achieving the bound are provided. By these constructions, the maximum rate of a secure linear multicast is n - 2d - &#µ; - k1, where n is the minimum value of the maxflows from a source node to sink nodes. We also characterise the equivocation by the relative profiles of a linear code and a subcode, and tighten the singleton bound on equivocation by the generalised Griesmer bound on relative profiles. [ABSTRACT FROM AUTHOR]

Several bounds on the size of ( n + l , M , d , ( m , 0 ) ) q codes in attenuated space over finite fields are provided in this paper. Then, we prove that codes in attenuated space attain the Wang–Xing–Safavi-Naini bound if and only if they are certain Steiner structures. [ABSTRACT FROM AUTHOR]

We present new constructions of codes for asymmetric channels for both binary and nonbinary alphabets, based on methods of generalized code concatenation. For the binary asymmetric channel, our methods construct nonlinear single-error-correcting codes from ternary outer codes. We show that some of the Varshamov–Tenengol’ts–Constantin–Rao codes, a class of binary nonlinear codes for this channel, have a nice structure when viewed as ternary codes. In many cases, our ternary construction yields even better codes. For the nonbinary asymmetric channel, our methods construct linear codes for many lengths and distances which are superior to the linear codes of the same length capable of correcting the same number of symmetric errors. [ABSTRACT FROM AUTHOR]

This work presents a study of the complexity of the Blum-Kalai-Wasserman (BKW) algorithm when applied to the Learning with Errors (LWE) problem, by providing refined estimates for the data and computational effort requirements for solving concrete instances of the LWE problem. We apply this refined analysis to suggested parameters for various LWE-based cryptographic schemes from the literature and compare with alternative approaches based on lattice reduction. As a result, we provide new upper bounds for the concrete hardness of these LWE-based schemes. Rather surprisingly, it appears that BKW algorithm outperforms known estimates for lattice reduction algorithms starting in dimension $$n \approx 250$$ when LWE is reduced to SIS. However, this assumes access to an unbounded number of LWE samples. [ABSTRACT FROM AUTHOR]

Quantum synchronizable error-correcting codes are special quantum error-correcting codes that are designed to correct both the effect of quantum noise on qubits and misalignment in block synchronization. It is known that, in principle, such a code can be constructed through a combination of a classical linear code and its subcode if the two are both cyclic and dual-containing. However, finding such classical codes that lead to promising quantum synchronizable error-correcting codes is not a trivial task. In fact, although there are two families of classical codes that are proved to produce quantum synchronizable codes with good minimum distances and highest possible tolerance against misalignment, their code lengths have been restricted to primes and Mersenne numbers. In this paper, examining the incidence vectors of projective spaces over the finite fields of characteristic 2, we give quantum synchronizable codes from cyclic codes whose lengths are not primes or Mersenne numbers. These projective geometric codes achieve good performance in quantum error correction and possess the best possible ability to recover synchronization, thereby enriching the variety of good quantum synchronizable codes. We also extend the current knowledge of cyclic codes in classical coding theory by explicitly giving generator polynomials of the finite geometric codes and completely characterizing the minimum weight nonzero codewords. In addition to the codes based on projective spaces, we carry out a similar analysis on the well-known cyclic codes from Euclidean spaces that are known to be majority logic decodable and determine their exact minimum distances. [ABSTRACT FROM PUBLISHER]

ABSTRACT Consider a signed file that cannot be verified because of some fraction loss or noise, which might not affect the business of users. It would be desirable for the verifier to locate the damaged fractions so that the authenticity of other components of the file can be ensured. Thereafter, the verifier can decide whether or not to accept it. To address this issue, in this paper, we present a loss-tolerant authentication scheme, which allows the verifier to locate the damaged elements in the signed file stream. Our idea is to allow the signed file to be fragmented, and each fragment is appended a short authentication tag, which enables the verifier to find the modified fragments in the signed file and verify the signature even if some fragments are lost. The proposed scheme adopts a combination of Reed-Solomon error correction and erasure codes, aiming to allow verifiers to locate the modified fractions and thereafter reconstruct the signature when sufficient original fractions of the file are obtained. Our scheme offers many potential applications such as loss-tolerant transmission and storage. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

As the technology scales down, shrinking geometry and layout dimension, on- chip interconnects are exposed to different noise sources such as crosstalk coupling, supply voltage fluctuation and temperature variation that cause random and burst errors. These errors affect the reliability of the on-chip interconnects. Hence, error correction codes integrated with noise reduction techniques are incorporated to make the on-chip interconnects robust against errors. The proposed error correction code uses triplication error correction scheme as crosstalk avoidance code (CAC) and a parity bit is added to it to enhance the error correction capability. The proposed error correction code corrects all the error patterns of one bit error, two bit errors. The proposed code also corrects 7 out of 10 possible three bit error patterns and detects burst errors of three. Hybrid Automatic Repeat Request (HARQ) system is employed when burst errors of three occurs. The performance of the proposed codec is evaluated for residual flit error rate, codec area, power, delay, average flit latency and link energy consumption. The proposed codec achieves four magnitude order of low residual flit error rate and link energy minimization of over 53 % compared to other existing error correction schemes. Besides the low residual flit error rate, and link energy minimization, the proposed codec also achieves up to 4.2 % less area and up to 6 % less codec power consumption compared to other error correction codes. The less codec area, codec power consumption, low link energy and low residual flit error rate make the proposed code appropriate for on chip interconnection link. [ABSTRACT FROM AUTHOR]

In this paper, a new class of (N,K) near-optimal partial Hadamard codebooks is proposed. The construction of the proposed codebooks from Hadamard matrices is based on binary row selection sequences, which are generated by quadratic residue mapping of p -ary m -sequences. The proposed codebooks have parameters and K=({p-1}/{2p})(N+\sqrt{N})+1 for an odd prime p$ and an even positive integer $n$ . We prove that the maximum magnitude of inner products between the code vectors of the proposed codebooks asymptotically achieves the Welch bound equality for sufficiently large $p$ and derive their inner product distribution. [ABSTRACT FROM PUBLISHER]

Two error correction schemes are proposed for word-oriented binary memories that can be affected by erasures, i.e. errors with known location but unknown value. The erasures considered here are due to the drifting of the electrical parameter used to encode information outside the normal ranges associated to a logic 0 or a logic 1 value. For example, a dielectric breakdown in a magnetic memory cell may reduce its electrical resistance sensibly below the levels which correspond to logic 0 and logic 1 values stored in healthy memory cells. Such deviations can be sensed during memory read operations and the acquired information can be used to boost the correction capability of an error-correcting code (ECC). The proposed schemes enable the correction of double-bit errors based on the combination of erasure information with single-bit error correction and double-bit error detection (SEC-DED) codes or shortened (SEC) codes. The correction of single-bit errors is always guaranteed. Ways to increase the number of double-bit and triple-bit errors that can be detected by shortened SEC and SEC-DED codes are considered in order to augment the error correction capability of the proposed solutions. [ABSTRACT FROM AUTHOR]

In this work, we consider high-rate error-control systems for storage devices using multi-level per cell (MLC) NAND flash memories. Aiming at achieving a strong error-correcting capability, we propose error-control systems using block-wise parallel/serial concatenations of short Bose-Chaudhuri-Hocquenghem (BCH) codes with two iterative decoding strategies, namely, iterative hard-decision decoding (IHDD) and iterative reliability based decoding (IRBD). It will be shown that a simple but very efficient IRBD is possible by taking advantage of a unique feature of the block-wise concatenation. For tractable performance analysis and design of IHDD and IRBD at very low error rates, we derive semi-analytic approaches. The proposed error-control systems are compared with various error-control systems with well-known coding schemes such as a product code, multiple BCH codes, a single long BCH code, and low-density parity-check codes in terms of page error rates, which confirms our claim: the proposed error-control systems achieve good tradeoffs between error-performance and complexity as compared to the traditional schemes and is also very favorable for implementation. [ABSTRACT FROM PUBLISHER]