Limited-magnitude Errors Research Articles

Perfect Codes Correcting a Single Burst of Limited-Magnitude Errors

Motivated by applications to DNA-storage, flash memory, and magnetic recording, we study perfect burst-correcting codes for the limited-magnitude error channel. These codes are lattices that tile the integer grid with the appropriate error ball. We construct two classes of such perfect codes correcting a single burst of length 2, where each error affects the corresponding position by increasing it by one, both for cyclic and non-cyclic bursts. We also present a generic construction that requires a primitive element in a finite field with specific properties. We then show that in various parameter regimes such primitive elements exist, and hence, infinitely many perfect burst-correcting codes exist.

Sequence Reconstruction for Limited-Magnitude Errors

Motivated by applications to DNA storage, we study reconstruction and list-reconstruction schemes for integer vectors that suffer from limited-magnitude errors. We characterize the asymptotic size of the intersection of error balls in relation to the code’s minimum distance. We also devise efficient reconstruction algorithms for various limited-magnitude error parameter ranges. We then extend these algorithms to the list-reconstruction scheme, and show the trade-off between the asymptotic list size and the number of required channel outputs. These results apply to all codes, without any assumptions on the code structure. Finally, we also study linear reconstruction codes with small intersection, as well as show a connection to list-reconstruction codes for the tandem-duplication channel.

Improved Coding Over Sets for DNA-Based Data Storage

Error-correcting codes over sets, with applications to DNA storage, are studied. The DNA-storage channel receives a set of sequences, and produces a corrupted version of the set, including sequence loss, symbol substitution, symbol insertion/deletion, and limited-magnitude errors in symbols. Various parameter regimes are studied. New bounds on code parameters are provided, which improve upon known bounds. New codes are constructed, at times matching the bounds up to lower-order terms or small constant factors.

Detection of Limited Magnitude Errors in Emerging Multilevel Cell Memories by One-Bit Parity (OBP) or Two-Bit Parity (TBP)

Emerging memory technologies rely on Multilevel Cells (MLC) to achieve high density; the use of multiple levels per cell allows storage of multiple bits, but it also reduces the margins and makes it error prone. Error control codes (including error correction and detection codes) can be used to protect MLC memories from errors; however, most existing coding schemes have been designed for traditional binary memories (so storing a single bit). In MLC memories, errors cause a change from a level to an adjacent level or to the next one (depending on the employed technology), so they are often referred to as limited magnitude errors. For a binary coding of levels to bits, these limited magnitude errors can corrupt several bits making traditional coding schemes inefficient. In this paper, error detection of MLC memories is considered when a binary encoding of levels to bits is used and two new schemes are proposed: One-Bit Parity (OBP) and Two-Bit Parity (TBP). The first scheme targets errors of magnitude-1 for detection using a single parity bit that checks only one bit per cell. The second scheme detects both magnitude-1 and -2 errors using only two parity bits. Both schemes are compared to existing alternatives, namely Gray coding combined with a single parity bit (GP) for OBP and Interleaved Parity (IP) for TBP. The results show that OBP reduces the encoding and error detection circuitry complexity and delay, while TBP additionally reduces the number of parity bits for some configurations. Therefore, OBP and TBP can be efficient alternatives for detection of limited magnitude errors in MLC memories that use a binary encoding of levels to bits.

On Lattice Packings and Coverings of Asymmetric Limited-Magnitude Balls

We construct integer error-correcting codes and covering codes for the limited-magnitude error channel with more than one error. The codes are lattices that pack or cover the space with the appropriate error ball. Some of the constructions attain an asymptotic packing/covering density that is constant. The results are obtained via various methods, including the use of codes in the Hamming metric, modular B <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">t</sub> -sequences, 2-fold Sidon sets, and sets avoiding arithmetic progression.

Codes for Limited Magnitude Error Correction in Multilevel Cell Memories

Multilevel cell (MLC) memories have been advocated for increasing density at low cost in next generation memories. However, the feature of several bits in a cell reduces the distance between levels; this reduced margin makes such memories more vulnerable to defective phenomena and parameter variations, leading to an error in stored data. These errors typically are of limited magnitude, because the induced change causes the stored value to exceed only a few of the level boundaries. To protect these memories from such errors and ensure that the stored data is not corrupted, Error Correction Codes (ECCs) are commonly used. However, most existing codes have been designed to protect memories in which each cell stores a bit and thus, they are not efficient to protect MLC memories. In this paper, an efficient scheme that can correct up to magnitude-3 errors is presented and evaluated. The scheme is based by combining ECCs that are commonly used to protect traditional memories. In particular, Interleaved Parity (IP) bits and Single Error Correction and Double Adjacent Error Correction (SEC-DAEC) codes are utilized; both these codes are combined in the proposed IP-DAEC scheme to efficiently provide a strong coding function for correction, thus exceeding the capabilities of most existing coding schemes for limited magnitude errors. The SEC-DAEC code is used to detect the cell in error and correct some bits, while the IP bits identify the remaining erroneous bits in the memory cell. The use of these simple codes results in an efficient implementation of the decoder compared to existing techniques as shown by the evaluation results presented in this paper. The proposed scheme is also competitive in terms of number of parity check bits and memory redundancy. Therefore, the proposed IP-DAEC scheme is a very efficient alternative to protect and correct MLC memories from limited magnitude errors.

Efficient One-Step Decodable Limited Magnitude Error Correcting Codes for Multilevel Cell Main Memories

As technology scales further, dynamic random-access memory (DRAM) scaling faces numerous challenges. Emerging non-volatile main memories (e.g., phase change memories) provide an attractive solution to the challenges faced by DRAM scaling due to their high density and low cost. But phase change memories suffer from the problem of resistance drifts, which are of limited magnitude and causes read reliability degradation. For single error correction, Hamming codes and beyond single-error correction, Orthogonal Latin Square codes have been used due to their one-step decoding procedure. But with the limited magnitude error model, these codes are inefficient since they incur considerable redundancy. This paper proposes a new systematic limited magnitude error correcting non-binary Hamming code for a single-error correction specifically to address limited magnitude errors in multilevel cell memories storing multiple bits per cell. For memory architectures involving multiple error correction, a new decimal arithmetic based Orthogonal Latin Square code is also proposed. A general construction methodology is presented to correct errors of limited magnitude and is compared to existing schemes addressing limited magnitude errors in phase change memories. For non-binary Hamming codes, it is shown that the proposed codes provide better latency and complexity compared to existing limited magnitude error correcting non-binary Hamming codes as well as better redundancy compared to the symbol extended version of binary Hamming codes. The proposed limited magnitude Orthogonal Latin Square codes are shown to achieve better redundancy and better area compared to existing schemes with the tradeoff of slight increase in decoding and encoding latency.

Enhanced Limited Magnitude Error Correcting Codes for Multilevel Cell Main Memories

Technological advancements in emerging non-volatile memory (e.g., phase change memories) has led to viable alternative solutions to problems faced by technology scaling of conventional memories. Phase change memories suffer from the issue of resistance drifts wherein the value stored changes by a limited magnitude. Decimal arithmetic based limited magnitude Orthogonal Latin Square codes have been recently proposed to correct multiple errors based on asymmetric limited magnitude error models. This research letter proposes an additional technique which is combined with the decimal arithmetic based Orthogonal Latin Square codes previously proposed. This combined approach is shown to further reduce the data redundancy without incurring any significant penalty in terms of area and latency of the decoding and encoding circuit.

On Codes Achieving Zero Error Capacities in Limited Magnitude Error Channels

Shannon in his 1956 seminal paper introduced the concept of the zero error capacity, $C_{0}$ , of a noisy channel. This is defined as the least upper bound of rates, at which, it is possible to transmit information with zero probability of error. At present not many codes are known to achieve the zero error capacity. In this paper, some codes which achieve zero error capacities in limited magnitude error channels are described. The code lengths of these zero error capacity achieving codes can be of any finite length $n=1,2,\ldots$ , in contrast to the long lengths required for the known regular capacity achieving codes, such as turbo codes, LDPC codes, and polar codes. Both wrap around and non-wrap around limited magnitude error models are considered in this paper. For non-wrap around error model, the exact value of zero error capacities is derived, and optimal non-systematic and systematic codes are designed. The non-systematic codes achieve the zero error capacity with any finite length. The optimal systematic codes achieve the systematic zero error capacity of the channel, which is defined as the zero error capacity with the additional requirements that the communication must be carried out with a systematic code. It is also shown that the rates of the proposed systematic codes are equal to or approximately equal to the zero error capacity of the channel. For the wrap around model bounds are derived for the zero error capacity and in many cases the bounds give the exact value. In addition, optimal wrap around non-systematic and systematic codes are developed which either achieve or are close to achieving the zero error capacity with finite length.

On Zero Error Capacity of Nearest Neighbor Error Channels with Multilevel Alphabet

This paper studies the zero error capacity of the Nearest Neighbor Error (NNE) channels with a multilevel alphabet. In the NNE channels, a transmitted symbol is a $d$-tuple of elements in $\{0,1,2,\dots, n-1 \}$. It is assumed that only one element error to a nearest neighbor element in a transmitted symbol can occur. The NNE channels can be considered as a special type of limited magnitude error channels, and it is closely related to error models for flash memories. In this paper, we derive a lower bound of the zero error capacity of the NNE channels based on a result of the perfect Lee codes. An upper bound of the zero error capacity of the NNE channels is also derived from a feasible solution of a linear programming problem defined based on the confusion graphs of the NNE channels. As a result, a concise formula of the zero error capacity is obtained using the lower and upper bounds.

New Results on Codes Correcting Single Error of Limited Magnitude for Flash Memory

Some physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This motivated the application of the asymmetric limited magnitude error model in flash memory. In this paper, we present a new construction of quasi-perfect codes for such errors, and we also study the perfect codes with symmetric errors. Moreover, we show some nonexistence results on perfect codes for correcting single error of limited magnitude.

Two Constructions of Covering Sets for Limited-Magnitude Errors

Linear covering codes and covering sets for the limited-magnitude-error channel are studied. Two new general covering set constructions are given.

On covering sets for limited-magnitude errors

The concepts of a linear covering code and a covering set for the limited-magnitude-error channel were introduced in KlOve and Schwartz (2013). Here, a number of new covering set constructions are given, in particular for sets of r-tuples of large r. The main emphasis is on errors of size up to three.

Covering Sets for Limited-Magnitude Errors

For a set M = {-μ, -μ + 1, ... , λ} \ {0} with nonnegative integers λ, μ <; q not both 0, a subset S of the residue class ring Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> modulo an integer q > 1 is called a (λ, μ; q)-covering set if MS = {ms mod q : m ∈ M, s ∈ S} = Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</sub> . Small covering sets play an important role in codes correcting limited-magnitude errors. We give an explicit construction of a (λ, μ; q)-covering set S, which is of the size q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+o(1)</sup> max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> for almost all integers q ≥ 1 and optimal order of magnitude (that is up to a multiplicative constant) p max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> if q = p is prime. Furthermore, using a bound on the fourth moment of character sums of Cochrane and Shi that there is a (λ, μ; q)-covering set of size at most q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+o(1)</sup> max{λ, μ} <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1/2</sup> for any integer q ≥ 1, however the proof of this bound is not constructive.

Erratum to: Linear covering codes and error-correcting codes for limited-magnitude errors

Erratum to: Linear covering codes and error-correcting codes for limited-magnitude errors

Limited Magnitude Error Detecting Codes over Z_{q}

The error detecting problem for limited magnitude errors over high radix channels is studied. In this error model, the error magnitude does not exceed a certain limited value and it is known beforehand. For asymmetric, unidirectional, and symmetric channels, both all and t error detecting codes are studied. In all these cases, close-to-optimal codes are proposed.

Bidirectional Limited-Magnitude Error Correction Codes for Flash Memories

NAND multi-level cell (MLC) flash memories are widely used due to low cost and high capacity. However, the increased number of levels in MLC results in larger interference and errors. The errors in MLC flash memories tend to be directional and limited-magnitude. Many related works focus on asymmetric errors, but bidirectional errors also occur because of the bidirectional interference and the adjustment of the hard-decision reference voltages. To take advantage of the characteristics, we propose t bidirectional (lu,ld) limited-magnitude error correction codes, which can reduce errors more effectively. The proposed code is systematic, and can correct t bidirectional errors with upward and downward magnitude of lu and ld, respectively. The proposed method is advantageous in that the parity size is reduced, and it has lower bit error rate than conventional error correction codes with the same code rate.

Codes Correcting Single Errors of Limited Magnitude

An error model with symmetric errors of limited magnitude is considered. Limited magnitude means that the size of any error is limited by a number smaller (usually much smaller) than the alphabet size. Several constructions of codes correcting a single error are given. In some cases, the codes are perfect or quasi- perfect.

Quasi-Cross Lattice Tilings With Applications to Flash Memory

We consider lattice tilings of ℝn by a shape we call a (k+, k-, n)-quasi-cross. Such lattices form perfect error-correcting codes which correct a single limited-magnitude error with prescribed maximal-magnitudes k+ and k- of positive error and negative error respectively (the ratio of which, β = k-/k+, is called the balance ratio). These codes can be used to correct both disturb and retention errors in flash memories, which are characterized by having limited magnitudes and different signs. For any rational 0 <; β <; 1 we construct an infinite family of (k+, k-, n)-quasi-cross lattice tilings with balance ratio k-/k+ = β. We also provide a specific construction for an infinite family of (2,1, n) -quasi-cross lattice tilings. The constructions are related to group splitting and modular B1 sequences. In addition, we study bounds on the parameters of lattice-tilings by quasi-crosses, and express them in terms of the arm lengths of the quasi-crosses and the dimension. We also prove constraints on group splitting, a specific case of which shows that the parameters of the lattice tiling by (2, 1, n)-quasi-crosses are the only ones possible for these quasi-crosses.

Some Codes Correcting Asymmetric Errors of Limited Magnitude

An error model with asymmetric errors of limited magnitude is a good model for some multilevel flash memories. This paper is about constructions of codes correcting such errors. The main results are about codes correcting a single such error and codes of length <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> correcting all errors in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> -1 or less positions.