Write-once Memory Research Articles

Using Short Synchronous WOM Codes to Make WOM Codes Decodable

In the framework of write-once memory (WOM) codes, it is important to distinguish between codes that can be decoded directly and those that require the decoder to know the current generation so as to successfully decode the state of the memory. A widely used approach to constructing WOM codes is to design first nondecodable codes that approach the boundaries of the capacity region and then make them decodable by appending additional cells that store the current generation, at an expense of rate loss. In this paper, we propose an alternative method to making nondecodable WOM codes decodable by appending cells that also store some additional data. The key idea is to append to the original (nondecodable) code a short synchronous WOM code and write generations of the original code and the synchronous code simultaneously. We consider both the binary and the nonbinary case. Furthermore, we propose a construction of synchronous WOM codes, which are then used to make nondecodable codes decodable. For short-to-moderate block lengths, the proposed method significantly reduces the rate loss as compared to the standard method.

Short <inline-formula> <tex-math notation="TeX">\(Q\) </tex-math></inline-formula>-Ary Fixed-Rate WOM Codes for Guaranteed Rewrites and With Hot/Cold Write Differentiation

To the body of works on rewrite codes for constrained memories, we add a comprehensive study in a direction that is especially relevant to practical storage. The subject of this paper is codes for the q-ary extension of the write-once memories model, with input sizes that are fixed throughout the write sequence. Seven code constructions are given with guarantees on the number of writes they can support. For the parameters addressed by the constructions, we also prove upper bounds on the number of writes, which prove the optimality of three of the constructions. We concentrate on codes with short block lengths to keep the complexity of decoding and updates within the feasibility of practical implementation. Even with these short blocks the constructed codes are shown to be within a small additive constant from capacity for an arbitrarily large number of input bits. Part of the study addresses a new rewrite model where some of the input bits can be updated multiple times in a write sequence (hot bits), while other are updated at most once (cold bits). We refer to this new model as hot/cold rewrite codes. It is shown that adding cold bits to a rewrite code has a negligible effect on the total number of writes, while adding an important feature of leveling the physical wear of memory cells between hot and cold input data.

Constrained Codes that Mitigate Inter-Cell Interference in Read/Write Cycles for Flash Memories

Inter-cell interference (ICI) is one of the main obstacles to precise programming (i.e., writing) of a flash memory. In the presence of ICI, the voltage level of a cell might increase unexpectedly if its neighboring cells are programmed to high levels. For q-ary cells, the most severe ICI arises when three consecutive cells are programmed to levels high - low - high, represented as (q-1)0(q-1), resulting in an unintended increase in the level of the middle cell and the possibility of decoding it incorrectly as a nonzero value. ICI-free codes are used to mitigate this phenomenon by preventing the programming of any three consecutive cells as (q-1)0(q-1). In this work, we extend ICI-free codes in two directions. First, we consider binary balanced ICI-free codes which, in addition to forbidding the 101 pattern, require the number of 0 symbols and 1 symbols to be the same. Using combinatorial methods, we determine the asymptotic information rate of these codes and show that the asymptotic rate loss due to the imposition of the balanced property is approximately 2%. Extensions to q-ary cells, for q>2 are also discussed. Next, we consider q-ary ICI-free write-once-memory (WOM) codes that support multiple writes of a WOM while mitigating ICI effects. These codes forbid the appearance of the (q-1)0(q-1) pattern in any codeword used in any writing step. Using properties of two-dimensional constrained codes and generalized WOMs, we characterize the maximum sum-rate of t-write ICI-free WOM codes or, equivalently, the t-write sum-capacity of an ICI-free WOM.

Lattice-Based WOM Codes for Multilevel Flash Memories

We consider t-write codes for write-once memories with n cells that can store multiple levels. Assuming an underlying lattice-based construction and using the continuous approximation, we derive upper bounds on the worst-case sum-rate optimal and fixed-rate optimal n-cell t-write write-regions for the asymptotic case of continuous levels. These are achieved using hyperbolic shaping regions that have a gain of 1 bit/cell over cubic shaping regions. Motivated by these hyperbolic write-regions, we discuss construction and encoding of codebooks for cells with discrete support. We present a polynomial-time algorithm to assign messages to the codebooks and show that it achieves the optimal sum-rate for any given codebook when n = 2. Using this approach, we construct codes that achieve high sum-rate. We describe an alternative formulation of the message assignment problem for n≥ 3, a problem which remains open.

Polar Write Once Memory Codes

A coding scheme for write once memory (WOM) using polar codes is presented. It is shown that the scheme achieves the capacity region of noiseless WOMs when an arbitrary number of multiple writes is permitted. The encoding and decoding complexities scale as O(N log N), where N is the blocklength. For N sufficiently large, the error probability decreases subexponentially in N. The results can be generalized from binary to generalized WOMs, described by an arbitrary directed acyclic graph, using nonbinary polar codes. In the derivation, we also obtain results on the typical distortion of polar codes for lossy source coding. Some simulation results with finite length codes are presented.

New Constructions of WOM Codes Using the Wozencraft Ensemble

In this paper, we give several new constructions of write-once-memory (WOM) codes. The novelty in our constructions is the use of the so-called Wozencraft ensemble of linear codes. Specifically, we obtain the following results. We give an explicit construction of a two-write WOM code that approaches capacity, over the binary alphabet. More formally, for every ϵ > 0, 0 <; p <; 1, and n=(1/ϵ)O(1/pϵ), we give a construction of a two-write WOM code of length n and capacity H(p)+1-p-ϵ. Since the capacity of a two-write WOM code is maxp (H(p)+1-p), we get a code that is ϵ-close to capacity. Furthermore, encoding and decoding can be done in time O(n2 ·poly (logn)) and time O(n ·poly (logn)), respectively, and in logarithmic space. In addition, we exhibit an explicit randomized encoding scheme of a two-write capacity-achieving WOM code of block length polynomial in 1/ϵ (again, ϵ is the gap to capacity), with a polynomial time encoding and decoding. We obtain a new encoding scheme for three-write WOM codes over the binary alphabet. Our scheme achieves rate 1.809-ϵ, when the block length is exp(1/ϵ). This gives a better rate than what could be achieved using previous techniques. We highlight a connection to linear seeded extractors for bit-fixing sources. In particular, we show that obtaining such an extractor with seed length O(logn) can lead to improved parameters for two-write WOM codes. We then give an application of existing constructions of extractors to the problem of designing encoding schemes for memory with defects.

Tilings With $n$-Dimensional Chairs and Their Applications to Asymmetric Codes

An <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair consists of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional box from which a smaller <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional box is removed. A tiling of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair has two nice applications in some memories using asymmetric codes. The first one is in the design of codes that correct asymmetric errors with limited magnitude. The second one is in the design of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> cells <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary write-once memory codes. We show an equivalence between the design of a tiling with an integer lattice and the design of a tiling from a generalization of splitting (or of Sidon sequences). A tiling of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional chair can define a perfect code for correcting asymmetric errors with limited magnitude. We present constructions for such tilings and prove cases where perfect codes for these type of errors do not exist.

Codes for Write-Once Memories

A write-once memory (WOM) is a storage device that consists of cells that can take on $q$ values, with the added constraint that rewrites can only increase a cell's value. A length- $n$ , $t$ -write WOM-code is a coding scheme that allows $t$ messages to be stored in $n$ cells. If on the $i$ th write we write one of $M_{i}$ messages, then the rate of this write is the ratio of the number of written bits to the total number of cells, i.e., $\log_{2}M_{i}/n$ . The sum-rate of the WOM-code is the sum of all individual rates on all writes. A WOM-code is called a fixed-rate WOM-code if the rates on all writes are the same, and otherwise, it is called a variable-rate WOM-code. We address two different problems when analyzing the sum-rate of WOM-codes. In the first one, called the fixed-rate WOM-code problem, the sum-rate is analyzed over all fixed-rate WOM-codes, and in the second problem, called the unrestricted-rate WOM-code problem, the sum-rate is analyzed over all fixed-rate and variable-rate WOM-codes. In this paper, we first present a family of two-write WOM-codes. The construction is inspired by the coset coding scheme, which was used to construct multiple-write WOM-codes by Cohen and recently by Wu, in order to construct from each linear code a two-write WOM-code. This construction improves the best known sum-rates for the fixed- and unrestricted-rate WOM-code problems. We also show how to take advantage of two-write WOM-codes in order to construct codes for the Blackwell channel. The two-write construction is generalized for two-write WOM-codes with $q$ levels per cell, which is used with ternary cells to construct three- and four-write binary WOM-codes. This construction is used recursively in order to generate a family of $t$ -write WOM-codes for all $t$ . A further generalization of these $t$ -write WOM-codes yields additional families of efficient WOM-codes. Finally, we show a recursive method that uses the previously constructed WOM-codes in order to construct fixed-rate WOM-codes. We conclude and show that the WOM-codes constructed here outperform all previously known WOM-codes for $2\leqslant t\leqslant 10$ for both the fixed- and unrestricted-rate WOM-code problems.

Geometric WOM codes and coding strategies for multilevel flash memories

This paper investigates the design and application of write-once memory (WOM) codes for flash memory storage. Using ideas from Merkx (1984) we present a construction of WOM codes based on finite Euclidean geometries over \({\mathbb{F}_2}\). This construction yields WOM codes with new parameters and provides insight into the criterion that incidence structures should satisfy to give rise to good codes. We also analyze methods of adapting binary WOM codes for use on multilevel flash cells. In particular, we give two strategies based on different rewrite objectives. A brief discussion of the average-write performance of these strategies, as well as concatenation methods for WOM codes is also provided.

Multiple Error-Correcting WOM-Codes

A Write Once Memory (WOM) is a storage medium with binary memory elements, called cells, that can change from the zero state to the one state only once. Examples of WOMs include punch cards and optical disks. WOM-codes, introduced by Rivest and Shamir, permit the reuse of a WOM by taking into account the location of cells that have already been changed to the one state. The objective in designing WOM-codes is to use the fewest number of cells to store a specified number of information bits in each of several reuses of the memory. An [n,k,t] WOM-code C is a coding scheme for storing k information bits in n cells t times. At each write, the state of each cell can be changed, provided that the cell is changed from the zero state to the one state. The rate of C, defined by R(C) = kt/n, indicates the total amount of information that is possible to store in a cell in t writes. Two WOM-code constructions correcting a single cell-error were presented by Zemor and Cohen. In this paper, we present another construction of a single-error-correcting WOM-code with a better rate. Our construction can be adapted also for single-error-detection, double-error-correction, and triple-error-correction. For the last case, we use triple-error-correcting BCH-like codes, which were presented by Kasami and more recently described again by Bracken and Helleseth. Finally, we show two constructions that can be combined for the correction of an arbitrary number of errors.

Position Modulation Code for Rewriting Write-Once Memories

A write-once memory (wom) is a storage medium formed by a number of “write-once” bit positions (wits), where each wit initially is in a “0” state and can be changed to a “1” state irreversibly. Examples of write-once memories include SLC flash memories and optical disks. This paper presents a low complexity coding scheme for rewriting such write-once memories, which is applicable to general problem configurations. The proposed scheme is called the position modulation code, as it uses the positions of the zero symbols to encode some information. The proposed technique can achieve code rates higher than state-of-the-art practical solutions for some configurations. For instance, there is a position modulation code that can write 56 bits 10 times on 278 wits, achieving rate 2.01. In addition, the position modulation code is shown to achieve a rate at least half of the optimal rate.

Deterministic History-Independent Strategies for Storing Information on Write-Once Memories

Motivated by the challenging task of designing “secure” vote storage mechanisms, we study information storage mechanisms that operate in extremely hostile environments. In such environments, the majority of existing techniques for information storage and for security are susceptible to powerful adversarial attacks. We propose a mechanism for storing a set of at most $K$ elements from a large universe of size $N$ on write-once memories in a manner that does not reveal the insertion order of the elements. We consider a standard model for write-once memories, in which the memory is initialized to the all-zero state, and the only operation allowed is flipping bits from $0$ to $1$. Whereas previously known constructions were either inefficient (required $\Theta(K^2)$ memory), randomized, or employed cryptographic techniques which are unlikely to be available in hostile environments, we eliminate each of these undesirable properties. The total amount of memory used by the mechanism is linear in the number of stored elements and poly-logarithmic in the size of the universe of elements. We also demonstrate a connection between secure vote storage mechanisms and one of the classical distributed computing problems: conflict resolution in multiple-access channels. By establishing a tight connection with the basic building block of our mechanism, we construct the first deterministic and non-adaptive conflict resolution algorithm whose running time is optimal up to poly-logarithmic factors.

Two-Dimensional Ultrahigh-Density X-ray Optical Memory

Most important aspect of nanotechnology applications in the information ultrahigh storage is the miniaturization of data carrier elements of the storage media with emphasis on the long-term stability. Proposed two-dimensional ultrahigh-density X-ray optical memory, named X-ROM, with long-term stability is an information carrier basically destined for digital data archiving. X-ROM is a semiconductor wafer, in which the high-reflectivity nanosized X-ray mirrors are embedded. Data are encoded due to certain positions of the mirrors. Ultrahigh-density data recording procedure can e.g., be performed via mask-less zone-plate-array lithography (ZPAL), spatial-phase-locked electron-beam lithography (SPLEBL), or focused ion-beam lithography (FIB). X-ROM manufactured by nanolithography technique is a write-once memory useful for terabit-scale memory applications, if the surface area of the smallest recording pits is less than 100 nm2. In this case the X-ROM surface-storage capacity of a square centimetre becomes by two orders of magnitude higher than the volumetric data density really achieved for three-dimensional optical data storage medium. Digital data read-out procedure from proposed X-ROM can e.g., be performed via glancing-angle incident X-ray micro beam (GIX) using the well-developed X-ray reflectometry technique. In presented theoretical paper the crystal-analyser operating like an image magnifier is added to the set-up of X-ROM data handling system for the purpose analogous to case of application the higher numerical aperture objective in optical data read-out system. We also propose the set-up of the X-ROM readout system based on more the one incident X-ray micro beam. Presented scheme of two-beam data handling system, which operates on two mutually perpendicular well-collimated monochromatic incident X-ray micro beams, essentially increases the reliability of the digital information read-out procedure. According the graphs of characteristic functions presented in paper, one may choose optimally the incident radiation wavelength, as well as the angle of incidence of X-ray micro beams, appropriate for proposed digital data read-out procedure.

On the Capacity of Write-Unidirectional Memories With Nonperiodic Codes

Write-unidirectional memories (WUMs) were introduced by Willems, Vinck, and Borden as an information-theoretic model for storing and updating information on a rewritable medium with the writing constraints: During the odd (resp., even) cycles of updating information, the encoder can only write 1's (resp., 0's) in selected bit positions of WUMs, and not change the contents of other positions. In this correspondence, motivated by the research works of Wolf, Wyner, Ziv, and Ko/spl uml/rner on write-once memories (WOMs), we study the problem of how to reuse a WUM for fixed T successive cycles with nonperiodic codes (i.e., all coding strategies are permitted for every cycle). For the situation where the encoder knows and the decoder does not know the previous content of the memory, we determine the zero-error capacity region, the average capacity, and the maximum total number of information bits stored in the WUM for fixed T successive cycles. Motivated by the research works of Heegard on WOMs with symmetric input noise, we introduce two models of WUMs with symmetric or asymmetric input noise. By using /spl epsiv/-error as performance criterion, we extend the above results for WUMs to the two models of WUMs with symmetric or asymmetric input noise.

Photo-Generation of Acids and its Fluorescence Detection in a Small Area: a Near-Field Write-Once Memory

A polymer film containing the photo-acid generator, 1,2-naphthoquinone-2-diazido-5-sulfonic acid sodium salt (SNQD), and a fluorescence pH indicator, seminaphthofluorescein (SNAFL-1), has been used as a near-field write-once memory medium. Near-field 455 nm light from a optical fiber tip with a nano-size (∼ 100 nm) aperture produced acids in a small area and the acids were detected as the fluorescence intensity increased with the near-field 488 nm excitation light. The fluorescent spot-size was as small as 160 nm and could be read many times without destruction.

On the capacity of generalized write-once memory with state transitions described by an arbitrary directed acyclic graph

The generalized write-once memory introduced by Fiat and Shamir (1984) is a q-ary information storage medium. Each storage cell is expected to store one of q symbols, and the legal state transitions are described by an arbitrary directed acyclic graph. This memory model can be understood as a generalization of the binary write-once memory which was introduced by Rivest and Shamir (1982). During the process of updating information, the contents of a cell can be changed from a 0-state to a 1-state but not vice versa. We study the problem of reusing a generalized write-once memory for T successive cycles (generations). We determine the zero-error capacity region and the maximum total number of information hits stored in the memory for T consecutive cycles for the situation where the encoder knows and the decoder does not know the previous state of the memory. These results extend the results of Wolf, Wyner, Ziv, and Korner (1984) for the binary write-once memory.

On the general defective channel with informed encoder and capacities of some constrained memories

From an information-theoretical point of view the write once memory (WOM), the unidirectional memory (WUM), the write isolated memory (WIM), the memory with address faults (MAF), Blackwell's broadcast channel, and some other constrained memories and channels with an informed encoder can be considered as particular cases of the general defective channel (GDC) introduced by Kuznetsov (1983) as a generalization of a memory with defects. Using the concept of the GDC we consider a unified approach to the investigation of different types of natural and artificial channels with a finite number of states known to the encoder, but unknown to the decoder. To illustrate the usefulness of this approach we derive the capacities of the above-mentioned constrained memories (WOM, WUM, WIM, MAF) as corollaries of lower and upper bounds for the number of messages transmitted over the GDC. >

Laser writing of submicrometre lines on copper island films

Laser writing on copper island films is demonstrated for the first time. Very fine lines with widths of less than 1μm have been written using a commercially available 0.82 μm 20 mW laser diode. The reflectance of the 8nm-thick copper island film is 49% at a wavelength of 0.82μm and it has been reduced to 15% by the irradiation of the focused laser beam due to the oxidisation of copper islands. The Cu island film is considered as a new medium for an efficient heat-mode write-once memory.

On the time and space complexity of computation using write-once memory or is pen really much worse than pencil?

We introduce a model of computation based on the use of write-once memory. Write-once memory has the property that bits may be set but not reset. Our model consists of a RAM with a small amount of regular memory (such as logarithmic orn α for α<1, wheren is the size of the problem) and a polynomial amount of write-once memory. Bounds are given on the time required to simulate on write-once memory algorithms which originally run on a RAM with a polynomial amount of regular memory. We attempt to characterize algorithms that can be simulated on our write-once memory model with very little slow-down. A persistent computation is one in which, at all times, the memory state of the computation at any previous point in time can be reconstructed. We show that any data structure or computation implemented on this write-once memory model can be made persistent without sacrificing much in the way of running time or space. The space requirements of algorithms running on the write-once model are studied. We show that general simulations of algorithms originally running on a RAM with regular memory by algorithms running on our write-once memory model require space proportional to the number of steps simulated. In order to study the space complexity further, we define an analogue of the pebbling game, called the pebble-sticker game. A sticker is different from a pebble in that it cannot be removed once placed on a node of the computation graph. As placing pebbles correspond to writes to regular memory, placing stickers correspond to writes to the write-once memory. Bounds are shown on pebble-sticker tradeoffs required to evaluate trees and planar graphs. Finally, we define the complexity class WO-PSPACE as the class of problems which can be solved with a polynomial amount of write-once memory, and show that it is equal toP.

Coding for a degraded memory under a partial modification of records

The coding problem for the binary degraded memory (DM) or write-once-memory (WOM) under a partial modification of records is considered. For this memory, if '1' is recorded in any cell then the content of this cell cannot be changed during following recordings. Double recording into such a memory with specified constraints is considered. Upper and lower bounds for the capacity of the DM under this kind of recording are derived. The lower bound is derived only for the case where the size of input messages alphabet is going to infinity.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>