Systematic Error-Correcting Codes for Rank Modulation

The rank-modulation scheme has been recently proposed for efficiently storing data in nonvolatile memories. Error-correcting codes are essential for rank modulation, however, existing results have been limited. In this work we explore a new approach, \emph{systematic error-correcting codes for rank modulation}. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. 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 one error, which have optimal rates, unless systematic perfect codes exist. We also study the design of multi-error-correcting codes, and provide two explicit constructions, one resulting in $[n+1,k+1,2t+2]$ systematic codes with redundancy at most $2t+1$. We use non-constructive arguments to show the existence of $[n,k,n-k]$-systematic codes for general parameters. Furthermore, we prove 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 multi-error-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

Error-Correction in Flash Memories via Codes in the Ulam Metric

We consider rank modulation codes for flash memories that allow for handling arbitrary charge-drop errors. Unlike classical rank modulation codes used for correcting errors that manifest themselves as swaps of two adjacently ranked elements, the proposed \emph{translocation rank codes} account for more general forms of errors that arise in storage systems. Translocations represent a natural extension of the notion of adjacent transpositions and as such may be analyzed using related concepts in combinatorics and rank modulation coding. Our results include derivation of the asymptotic capacity of translocation rank codes, construction techniques for asymptotically good codes, as well as simple decoding methods for one class of constructed codes. As part of our exposition, we also highlight the close connections between the new code family and permutations with short common subsequences, deletion and insertion error-correcting codes for permutations, and permutation codes in the Hamming distance

Correcting Charge-Constrained Errors in the Rank-Modulation Scheme

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

Systematic Codes for Rank Modulation

The goal of this paper is to construct systematic error-correcting codes for permutations and multi-permutations in the Kendall's $\tau$-metric. These codes are important in new applications such as rank modulation for flash memories. The construction is based on error-correcting codes for multi-permutations and a partition of the set of permutations into error-correcting codes. For a given large enough number of information symbols $k$, and for any integer $t$, we present a construction for ${(k+r,k)}$ systematic $t$-error-correcting codes, for permutations from $S_{k+r}$, with less redundancy symbols than the number of redundancy symbols in the codes of the known constructions. In particular, for a given $t$ and for sufficiently large $k$ we can obtain $r=t+1$. The same construction is also applied to obtain related systematic error-correcting codes for multi-permutations.Comment: to be presented ISIT201

Limited-Magnitude Error-Correcting Gray Codes for Rank Modulation

We construct Gray codes over permutations for the rank-modulation scheme, which are also capable of correcting errors under the infinity-metric. 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 (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.Comment: Revised version for journal submission. Additional results include more tight auxiliary constructions, a decoding shcema, ranking/unranking procedures, and application to snake-in-the-box codes under the Kendall tau-metri

Systematic Error-Correcting Codes for Rank Modulation

The rank modulation scheme has been proposed recently for efficiently writing and storing data in nonvolatile memories. Error-correcting codes are very important for rank modulation, and they have attracted interest among researchers. In this work, we explore a new approach, systematic error-correcting codes for rank modulation. In an (n,k) systematic code, we use the permutation induced by the levels of n cells to store data, and the permutation induced by the first k cells (k < n) has a one-to-one mapping to information bits. Systematic codes have the benefits of enabling efficient information retrieval and potentially supporting more efficient encoding and decoding procedures. We study systematic codes for rank modulation equipped with the Kendall's τ-distance. We present (k + 2, k) systematic codes for correcting one error, which have optimal sizes unless perfect codes exist. We also study the design of multi-error-correcting codes, and prove that for any 2 ≤ k < n, there always exists an (n, k) systematic code of minimum distance n-k. Furthermore, we prove that for rank modulation, systematic codes achieve the same capacity as general error-correcting codes

Constructions of Rank Modulation Codes

Rank modulation is a way of encoding information to correct errors in flash memory devices as well as impulse noise in transmission lines. Modeling rank modulation involves construction of packings of the space of permutations equipped with the Kendall tau distance. We present several general constructions of codes in permutations that cover a broad range of code parameters. In particular, we show a number of ways in which conventional error-correcting codes can be modified to correct errors in the Kendall space. Codes that we construct afford simple encoding and decoding algorithms of essentially the same complexity as required to correct errors in the Hamming metric. For instance, from binary BCH codes we obtain codes correcting $t$ Kendall errors in $n$ memory cells that support the order of $n!/(\log_2n!)^t$ messages, for any constant $t= 1,2,...$ We also construct families of codes that correct a number of errors that grows with $n$ at varying rates, from $\Theta(n)$ to $\Theta(n^{2})$. One of our constructions gives rise to a family of rank modulation codes for which the trade-off between the number of messages and the number of correctable Kendall errors approaches the optimal scaling rate. Finally, we list a number of possibilities for constructing codes of finite length, and give examples of rank modulation codes with specific parameters.Comment: Submitted to IEEE Transactions on Information Theor

Coding Techniques for Error Correction and Rewriting in Flash Memories

Flash memories have become the main type of non-volatile memories. They are widely used in mobile, embedded and mass-storage devices. Flash memories store data in floating-gate cells, where the amount of charge stored in cells – called cell levels – is used to represent data. To reduce the level of any cell, a whole cell block (about 106 cells) must be erased together and then reprogrammed. This operation, called block erasure, is very costly and brings significant challenges to cell programming and rewriting of data. To address these challenges, rank modulation and rewriting codes have been proposed for reliably storing and modifying data. However, for these new schemes, many problems still remain open. In this work, we study error-correcting rank-modulation codes and rewriting codes for flash memories. For the rank modulation scheme, we study a family of one- error-correcting codes, and present efficient encoding and decoding algorithms. For rewriting, we study a family of linear write-once memory (WOM) codes, and present an effective algorithm for rewriting using the codes. We analyze the performance of our solutions for both schemes