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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]

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 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 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]

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]

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]

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 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]

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]

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]