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Errorcorrecting codes for rank modulation
 in Proc. IEEE ISIT
, 2008
"... Abstract—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 nee ..."
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Cited by 23 (12 self)
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Abstract—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. I.
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 17 (2 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.
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 15 (1 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.
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 12 (10 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.
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 12 (2 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.
Position Modulation Code for Rewriting WriteOnce Memories,” IEEE Trans on Information Theory, vol 57, no.6, pp 36923697, 2010. A. The proof of theorem 1 VII. APPENDIX Proof: An = (B − C) n = n∑ ( ) n n−i i i i
 0 Q i (q−i(1+D))×(q−i(1+D)) where Q i (q−i(1+D))×(q−i(1+D)) = ⎛ ⎜ ⎝ ( i−1 ) ( ) i ... ( ) i−1
"... Abstract—A writeonce memory (wom) is a storage medium formed by a number of “writeonce ” bit positions (wits), where each wit initially is in a “0 ” state and can be changed to a “1 ” state irreversibly. Examples of writeonce memories include SLC flash memories and optical disks. This paper prese ..."
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Cited by 12 (1 self)
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Abstract—A writeonce memory (wom) is a storage medium formed by a number of “writeonce ” bit positions (wits), where each wit initially is in a “0 ” state and can be changed to a “1 ” state irreversibly. Examples of writeonce memories include SLC flash memories and optical disks. This paper presents a low complexity coding scheme for rewriting such writeonce 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 stateoftheart 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. Index Terms—Flash memories, position modulation, writeonce memories. I.
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 9 (4 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.
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 8 (8 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.
Storage Coding for Wear Leveling in Flash Memories
, 2009
"... NAND flash memories are currently the most widely used flash memories. In a NAND flash memory, although a cell block consists of many pages, to rewrite one page, the whole block needs to be erased and reprogrammed. Block erasures determine the longevity and efficiency of flash memories. So when dat ..."
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Cited by 6 (5 self)
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NAND flash memories are currently the most widely used flash memories. In a NAND flash memory, although a cell block consists of many pages, to rewrite one page, the whole block needs to be erased and reprogrammed. Block erasures determine the longevity and efficiency of flash memories. So when data is frequently reorganized, which can be characterized as a data movement process, how to minimize block erasures becomes an important challenge. In this paper, we show that coding can significantly reduce block erasures for data movement, and present several optimal or nearly optimal algorithms. While the sortingbased noncoding schemes require O(n log n) erasures to move data among n blocks, codingbased schemes use only O(n) erasures and also optimize the utilization of storage space.
LDPC Codes for Rank Modulation in Flash Memories
"... Abstract—An LDPC code is proposed for flash memories based on rank modulation. In contrast to previous approaches, this enables the use of long ECCs with fixedlength modulation codes. For ECC design, the rank modulation scheme is treated as part of an equivalent channel. A probabilistic model of th ..."
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Cited by 4 (3 self)
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Abstract—An LDPC code is proposed for flash memories based on rank modulation. In contrast to previous approaches, this enables the use of long ECCs with fixedlength modulation codes. For ECC design, the rank modulation scheme is treated as part of an equivalent channel. A probabilistic model of the equivalent channel is derived and a simple highSNR approximation is given. LDPC codes over integer rings and finite fields are designed for the approximate channel and a lowcomplexity symbolflipping verificationbased (SFVB) messagepassing decoding algorithm is proposed to take advantage of the channel structure. Density evolution (DE) is used to calculate decoding thresholds and simulations are used to compare the lowcomplexity decoder with sumproduct decoding. I.