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103
Codes in Permutations and Error Correction for Rank Modulation
, 2009
"... Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations of n elements equipped with the Kendall tau distance. We derive several lo ..."
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Cited by 53 (5 self)
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Codes for rank modulation have been recently proposed as a means of protecting flash memory devices from errors. We study basic coding theoretic problems for such codes, representing them as subsets of the set of permutations of n elements equipped with the Kendall tau distance. We derive several lower and upper bounds on the size of codes. These bounds enable us to establish the exact scaling of the size of optimal codes for large values of n. We also show the existence of codes whose size is within a constant factor of the sphere packing bound for any fixed number of errors. Index terms—BoseChowla theorem, flash memory, inversion, Kendall tau distance, rank permutation codes.
Errorcorrecting codes for rank modulation
 IN PROC. IEEE ISIT
, 2008
"... We investigate errorcorrecting codes for a novel storage technology for flash memories, the rankmodulation scheme. 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 di ..."
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Cited by 45 (16 self)
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We investigate errorcorrecting codes for a novel storage technology for flash memories, the rankmodulation scheme. 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 correction in rank modulation codes. We show that the adjacency graph of permutations is a subgraph of a multidimensional array of a special size, a property that enables code designs based on Leemetric codes. We present a oneerrorcorrecting code whose size is at least half of the optimal size. We also present additional errorcorrecting codes and some related bounds.
Correcting limitedmagnitude errors in the rankmodulation scheme
, 2010
"... Abstract—We study errorcorrecting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories called rank modulation. In this scheme, a set of n flash cells are combined to create a single virtual multilevel cell. Information is stored in the permutation i ..."
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Cited by 41 (17 self)
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Abstract—We study errorcorrecting codes for permutations under the infinity norm, motivated by a novel storage scheme for flash memories called rank modulation. In this scheme, a set of n flash cells are combined to create a single virtual multilevel cell. Information is stored in the permutation induced by the cell charge levels. Spike errors, which are characterized by a limitedmagnitude change in cell charge levels, correspond to a lowdistance change under the infinity norm. We define codes protecting against spike errors, called limitedmagnitude rankmodulation codes (LMRM codes), and present several constructions for these codes, some resulting in optimal codes. These codes admit simple recursive, and sometimes direct, encoding and decoding procedures. We also provide lower and upper bounds on the maximal size of LMRM codes both in the general case, and in the case where the codes form a subgroup of the symmetric group. In the asymptotic analysis, the codes we construct outperform the GilbertVarshamovlike bound estimate. Index Terms—flash memory, rank modulation, asymmetric channel, permutation arrays, subgroup codes, infinity norm I.
Correcting ChargeConstrained Errors in the RankModulation Scheme
"... We investigate errorcorrecting codes for a novel storage technology for flash memories, the rankmodulation scheme. 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 dis ..."
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Cited by 40 (24 self)
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We investigate errorcorrecting codes for a novel storage technology for flash memories, the rankmodulation scheme. 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 errorcorrecting codes for chargeconstrained errors in the rankmodulation 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 τdistance. We show bounds on the size of such codes, and use metricembedding techniques to give constructions which translate a wealth of knowledge of binary codes in the Hamming metric as well as qary codes in the Lee metric, to codes over permutations in Kendall’s τmetric. Specifically, the oneerrorcorrecting codes we construct are at least half the ballpacking upper bound.
Permutation arrays under the Chebyshev distance
, 2009
"... An (n, d) permutation array (PA) is a subset of Sn with the property that the distance (under some metric) between any two permutations in the array is at least d. They became popular recently for communication over power lines. Motivated by an application to flash memories, in this paper the metric ..."
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Cited by 24 (5 self)
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An (n, d) permutation array (PA) is a subset of Sn with the property that the distance (under some metric) between any two permutations in the array is at least d. They became popular recently for communication over power lines. Motivated by an application to flash memories, in this paper the metric used is the Chebyshev metric. A number of different constructions are given as well as bounds on the size of such PA.
Multidimensional Flash Codes
, 2009
"... Flash memory is a nonvolatile computer memory comprised of blocks of cells, wherein each cell can take on q different levels corresponding to the number of electrons it contains. Increasing the cell level is easy; however, reducing a cell level forces all the other cells in the same block to be er ..."
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Cited by 24 (5 self)
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Flash memory is a nonvolatile computer memory comprised of blocks of cells, wherein each cell can take on q different levels corresponding to the number of electrons it contains. Increasing the cell level is easy; however, reducing a cell level forces all the other cells in the same block to be erased. This erasing operation is undesirable and therefore has to be used as infrequently as possible. We consider the problem of designing codes for this purpose, where k bits are stored using a block of n cells with q levels each. The goal is to maximize the number of bit writes before an erase operation is required. We present an efficient construction of codes that can store an arbitrary number of bits. Our construction can be viewed as an extension to multiple dimensions of the earlier work of Jiang and Bruck, where singledimensional codes that can store only 2 bits were proposed.
A nearly optimal construction of flash codes
 Proc. IEEE International Symposium on Inform. Theory
, 2009
"... Abstract — Flash memory is a nonvolatile computer memory comprised of blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are high ..."
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Cited by 18 (5 self)
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Abstract — Flash memory is a nonvolatile computer memory comprised of blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes — known as floating codes or flash codes — have been designed in order to maximize the number of times that information stored in a flash memory can be written (and rewritten) prior to incurring a block erasure. An (n, k, t)q flash code C is a coding scheme for storing k information bits in n cells in such a way that any sequence of up to t writes (where a write is a transition 0 → 1 or 1 → 0 in any one of the k bits) can be accommodated without a block erasure. The total number of available level transitions in n cells is n(q−1), and the write deficiency of C, defined as δ(C) = n(q−1) − t, is a measure of how close the code comes to perfectly utilizing all these transitions. For k> 6 and large n, the best previously known construction of flash codes achieves a write deficiency of O(qk 2). On the other hand, the best known lower bound on write deficiency is Ω(qk). In this paper, we present a new construction of flash codes that approaches this lower bound to within a factor logarithmic in k. To this end, we first improve upon the socalled “indexed ” flash codes, due to Jiang and Bruck, by eliminating the need for index cells in the JiangBruck construction. Next, we further increase the number of writes by introducing a new multistage (recursive) indexing scheme. We then show that the write deficiency of the resulting flash codes is O(qk log k) if q � log 2 k, and at most O(k log 2 k) otherwise. I.
ConstantWeight Gray Codes for Local Rank Modulation
, 2010
"... We consider the local rankmodulation scheme in which a sliding window going over a sequence of realvalued variables induces a sequence of permutations. Local rankmodulation is a generalization of the rankmodulation scheme, which has been recently suggested as a way of storing information in flas ..."
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Cited by 18 (12 self)
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We consider the local rankmodulation scheme in which a sliding window going over a sequence of realvalued variables induces a sequence of permutations. Local rankmodulation is a generalization of the rankmodulation scheme, which has been recently suggested as a way of storing information in flash memory. We study constantweight Gray codes for the local rankmodulation scheme in order to simulate conventional multilevel flash cells while retaining the benefits of rank modulation. We provide necessary conditions for the existence of cyclic and cyclic optimal Gray codes. We then specifically study codes of weight 2 and upper bound their efficiency, thus proving that there are no such asymptoticallyoptimal cyclic codes. In contrast, we study codes of weight 3 and efficiently construct codes which are asymptoticallyoptimal. We conclude with a construction of codes with asymptoticallyoptimal rate and weight asymptotically half the length, thus having an asymptoticallyoptimal charge difference between adjacent cells.
On the Capacity of Bounded Rank Modulation for Flash Memories
"... Abstract—Rank modulation has been recently introduced as a new information representation scheme for flash memories. Given the charge levels of a group of flash cells, sorting is used to induce a permutation, which in turn represents data. Motivated by the lower sorting complexity of smaller cell gr ..."
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Cited by 15 (10 self)
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Abstract—Rank modulation has been recently introduced as a new information representation scheme for flash memories. Given the charge levels of a group of flash cells, sorting is used to induce a permutation, which in turn represents data. Motivated by the lower sorting complexity of smaller cell groups, we consider bounded rank modulation, where a sequence of permutations of given sizes are used to represent data. We study the capacity of bounded rank modulation under the condition that permutations can overlap for higher capacity. I.
Rewriting codes for joint information storage in flash memories
 IEEE TRANSACTIONS ON INFORMATION THEORY
, 2010
"... Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories have become a very important family of such memories. A flash memory cell has q states—state 0; 1;...;q 0 1—and can only transit from a l ..."
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Cited by 15 (6 self)
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Memories whose storage cells transit irreversibly between states have been common since the start of the data storage technology. In recent years, flash memories have become a very important family of such memories. A flash memory cell has q states—state 0; 1;...;q 0 1—and can only transit from a lower state to a higher state before the expensive erasure operation takes place. We study rewriting codes that enable the data stored in a group of cells to be rewritten by only shifting the cells to higher states. Since the considered state transitions are irreversible, the number of rewrites is bounded. Our objective is to maximize the number of times the data can be rewritten. We focus on the joint storage of data in flash memories, and study two rewriting codes for two different scenarios. The first code, called floating code, is for the joint storage of multiple variables, where every rewrite changes one variable. The second code, called buffer code, is for remembering the most recent data in a data stream. Many of the codes presented here are either optimal or asymptotically optimal. We also present bounds to the performance of general codes. The results show that rewriting codes can integrate a flash memory’s rewriting capabilities for different variables to a high degree.