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We explore a novel data representation scheme for multilevel flash memory cells, in which a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The only allowed charge-placement mechanism is a "push-to-the-top" operation, which takes a single cell of the set and makes it the top-charged cell. The resulting scheme eliminates the need for discrete cell levels, as well as overshoot errors, when programming cells. We present unrestricted Gray codes spanning all possible n-cell states and using only "push-to-the-top" operations, and also construct balanced Gray codes. One important application of the Gray codes is the realization of logic multilevel cells, which is useful in conventional storage solutions. We also investigate rewriting schemes for random data modification. We present both an optimal scheme for the worst case rewrite performance and an approximation scheme for the average-case rewrite performance. Index Terms--Asymmetric channel, flash memory, Gray codes, permutations, rank modulation.

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. In this paper, we explore $[n,k,d]$ systematic error-correcting codes for rank modulation. Such codes have length $n$ , $k$ information symbols, and minimum distance $d$ . Systematic codes have the benefits of enabling efficient information retrieval in conjunction with memory-scrubbing schemes. We study systematic codes for rank modulation under Kendall’s $\tau $ -metric as well as under the $\ell _\infty $ -metric. In Kendall’s $\tau $ -metric, we present $[k+2,k,3]$ systematic codes for correcting a single error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multierror-correcting codes, and provide a construction of $[k+t+1,k,2t+1]$ systematic codes, for large-enough $k$ . We use nonconstructive arguments to show that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes. Finally, in the $\ell _\infty $ -metric, we construct two $[n,k,d]$ systematic multierror-correcting codes, the first for the case of $d=O(1)$ and the second for $d=\Theta (n)$ . In the latter case, the codes have the same asymptotic rate as the best codes currently known in this metric. [ABSTRACT FROM PUBLISHER]

Flash memories are currently the most widely used type of nonvolatile memories. A flash memory consists of floating-gate cells as its storage elements, where the charge level stored in a cell is used to represent data. Compared to magnetic recording and optical recording, flash memories have the unique property that the cells are programmed using an iterative procedure that monotonically shifts each cell's charge level upward toward its target value. In this paper, we model the cell as a monotonic storage channel, and explore its capacity and optimal programming. We present two optimal programming algorithms based on a few different noise models and optimization objectives. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We investigate error-correcting codes for a the rank-modulation scheme with an application to flash memory devices. In this scheme, a set of n cells stores information in the permutation induced by the different charge levels of the individual cells. The resulting scheme eliminates the need for discrete cell levels, overcomes overshoot errors when programming cells (a serious problem that reduces the writing speed), and mitigates the problem of asymmetric errors. In this paper, we study the properties of error-correcting codes for charge-constrained errors in the rank-modulation scheme. In this error model the number of errors corresponds to the minimal number of adjacent transpositions required to change a given stored permutation to another erroneous one--a distance measure known as Kendall's τ-distance. We show bounds on the size of such codes, and use metric-embedding techniques to give constructions which translate a wealth of knowledge of codes in the Lee metric to codes over permutations in Kendall's τ-metric. Specifically, the one-error-correcting codes we construct are at least half the ball-packing upper bound. [ABSTRACT FROM AUTHOR]