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Coding schemes for multi-level Flash memories that are intrinsically resistant against unknown gain and/or offset using reference symbols

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Abstract

Coding schemes for storage channels, such as optical recording and non-volatile memory (Flash), with unknown gain and offset are presented. In its simplest case, the coding schemes guarantee that a symbol with a minimum value (floor) and a symbol with a maximum (ceiling) value are always present in a codeword so that the detection system can estimate the momentary gain and the offset. The results of the computer simulations show the performance of the new coding and detection methods in the presence of additive noise.

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... It has been shown in [19] that the min-max detector operates satisfactorily over a large range of unknown parameters a and b. However, sinceâ andb are biased estimates of a and b, the above dynamic threshold detector loses error performance with respect to the ideal matched case, especially for larger codeword length n. ...
... Clearly, it is impossible to distinguish between the two choices, where the sent codeword is (2,4,4) and a = 1 or where (1, 2, 2) and a = 2. Let S be the adopted codebook, then we can cope with the above ambiguity if (2, 4, 4) ∈ S then (1, 2, 2) / ∈ S, or vice versa. The name Pearson code was coined [19] for a set of codewords that can be uniquely decoded by a detector immune to large uncertainties in both gain a, a > 0, and word offset b. Pearson code design can be found in [22]. ...
... , c) / ∈ S for all c ∈ R. We adopt a Pearson code that has codewords with at least one '0' symbol and at least one 'q−1' symbol. We may easily verify that such codewords satisfy Properties A and B. The number of allowable n-symbol codewords equals [19] |S| = q n − 2(q − 1) n + (q − 2) n , q > 1. ...
Article
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We report on the feasibility of k-means clustering techniques for the dynamic threshold detection of encoded q-ary symbols transmitted over a noisy channel with partially unknown channel parameters. We first assess the performance of k-means clustering technique without dedicated constrained coding. We apply constrained codes which allows a wider range of channel uncertainties so improving the detection reliability.
... whereâ andb denote the estimates of the actual channel gain and offset [13]. The dynamic thresholds, denoted byθ i , are scaled in a similar fashion as the received codeword, that is, ...
... It has been shown [13] that the min-max detector operates over a large range of unknown parameters a and b. However, since the estimates,â andb, are biased, the above dynamic threshold detector loses error performance with respect to the matched case, especially for larger codeword length n. ...
... Clearly, it is impossible to distinguish between the two choices, where the sent codeword is (2, 4, 4) and a = 1 or where (1, 2, 2) and a = 2. Let S be the adopted codebook, then we can cope with the above ambiguity if (2, 4, 4) ∈ S then (1, 2, 2) / ∈ S, or vice versa. The name Pearson code was coined for a set of codewords that can be uniquely decoded by a detector immune to large uncertainties in both a > 0 and b [13]. Codewords in a Pearson code, S, satisfy two conditions, namely ∈ S for all c ∈ R. We adopt a Pearson code that has codewords with at least one '0' symbol and at least one 'q − 1' symbol. ...
Preprint
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We investigate machine learning based on clustering techniques that are suitable for the detection of encoded strings of q-ary symbols transmitted over a noisy channel with partially unknown characteristics. We consider the detection of the q-ary data as a classification problem, where objects are recognized from a corrupted vector, which is obtained by an unknown corruption process. We first evaluate the error performance of k-means clustering technique without constrained coding. Secondly, we apply constrained codes that create an environment that improves the detection reliability and it allows a wider range of channel uncertainties.
... We denote this code by T (n, q). It is a member of the class of Tconstrained codes [3], consisting of sequences in which T pre-determined reference symbols each appear at least once. • The set of all q-ary sequences of length n having at least one symbol '0', at least one symbol not equal to '0', and having the greatest common divisor of the sequence symbols equal to '1'. ...
... We denote this code by Z(n, q). It is also a member of the class of T -constrained codes [3]. Due to the presence of the reference symbol '0' it is resistant against offset mismatch. ...
... As an example, we consider scheme Z VF (3, 2) for a memoryless binary source producing zeroes and ones with equal probability. The seven codewords of Z (3,2) are then used with probabilities as indicated in Table II, and thus the average redundancy is 1/4. This result can be obtained by applying (2), i.e., 3 + 6 × (1/8) log 2 (1/8) + (1/4) log 2 (1/4) = 1/4, or by directly applying Theorem 1, i.e, (1 − 1/2) 2 = 1/4. ...
Conference Paper
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The recently proposed Pearson codes offer immunity against channel gain and offset mismatch. These codes have very low redundancy, but efficient coding procedures were lacking. In this paper, systematic Pearson coding schemes are presented. The redundancy of these schemes is analyzed for memoryless uniform sources. It is concluded that simple coding can be established at only a modest rate loss.
... Jiang et al. [4] addressed a q-ary balanced coding technique, called rank modulation, for circumventing the difficulties with flash memories having aging offset levels. Zhou et al. [5], Sala et al. [6], and Immink [7] investigated the usage of balanced codes for enabling 'dynamic' reading thresholds in nonvolatile memories. ...
... It will be shown that the Pearson distance measure can only be applied to codebooks with special properties, and constrained coding is therefore required. To that end, we propose q-ary Tconstrained codes, where T , 0 < T ≤ q, preferred or reference symbols must appear at least once in every codeword [7]. The redundancy of T -constrained codes is much lower than that of prior art q-ary balanced codes, which makes Minimum Pearson Distance detection in conjunction with T -constrained codes an attractive alternative for practical applications. ...
... T -CONSTRAINED CODES We will show that the Minimum Pearson Distance detector can only operate unambiguously in case the codebook, S, used satisfies specific constraints, and thus constrained codes are required. To that end, we will introduce a class of codes, called T -constrained codes, that satisfy the requirements of unambiguous detection using the new Pearson-distance-based detector [7]. First we will start with a presentation of the Minimum Pearson Distance detection method. ...
Article
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The performance of certain transmission and storage channels, such as optical data storage and nonvolatile memory (flash), is seriously hampered by the phenomena of unknown offset (drift) or gain. We will show that minimum Pearson distance (MPD) detection, unlike conventional minimum Euclidean distance detection, is immune to offset and/or gain mismatch. MPD detection is used in conjunction with (T) -constrained codes that consist of (q) -ary codewords, where in each codeword (T) reference symbols appear at least once. We will analyze the redundancy of the new (q) -ary coding technique and compute the error performance of MPD detection in the presence of additive noise. Implementation issues of MPD detection will be discussed, and results of simulations will be given.
... Nevertheless, if the offset changes very rapidly, an AGC may be sub-optimal. A similar approach is accomplished by using training sequences or reference memory cells to estimate the unknown channel offset [27] and then adjusting the detector settings to match the actual values. Estimated values may be inaccurate because they lag behind actual values. ...
... There are usually two approaches to address the physical-related offset issues. One approach uses pilot sequences to estimate the unknown channel offset [27], which is often considered too expensive concerning redundancy. Other approaches are errorcorrecting techniques. ...
... Pearson codes can be regarded as a type of T -constrained code where T pre-defined symbols each appear at least once in each codeword [11]. As discussed in [4], a known construction method for Pearson codes is to ensure that every q-ary codeword contains at least one symbol "0" and one symbol "1". ...
... As N → ∞, the FSM of binary Pearson codes extends to an infinite number of states. The capacity C of this FSM is [11] ...
Article
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Sequences encoded with Pearson codes are immune to channel gain and offset mismatch that cause performance loss in communication systems. In this paper, we introduce an efficient method of constructing capacity-approaching variable-length Pearson codes. We introduce a finite state machine (FSM) description of Pearson codes, and present a variable-length code construction process based on this FSM. We then analyze the code rate, redundancy and the convergence property of our codes. We show that our proposed codes have less redundancy than codes recently described in the literature and that they can be implemented in a straightforward fashion.
... Jiang et al. [3] addressed a qary balanced coding technique, called rank modulation, for circumventing the difficulties with flash memories having aging offset levels. Zhou et al. [4], Sala et al. [5], and Immink [6] investigated the usage of balanced codes for Kees enabling 'dynamic' reading thresholds in non-volatile memories. Alternative detection methods, such as minimum Pearson distance detection, that are immune to offset and gain mismatch, have been presented in [7]. ...
... Before we proceed with a description of the new detector, we will define T -constrained codes. T -constrained codes, presented in [6] for enabling simple dynamic threshold detection of q-ary codewords, consist of q-ary n-length codewords, where T , 0 < T ≤ q, preferred or reference symbols must appear at least once in a codeword. The code, denoted by S 1 , comprises codewords where the symbol '0' appears at least once. ...
Conference Paper
The error performance of optical storage and Non-Volatile Memory (Flash) is susceptible to unknown offset of the retrieved signal. Balanced codes offer immunity against unknown offset at the cost of a significant code redundancy, while minimum Pearson distance detection offers immunity with low-redundant codes at the price of lessened noise margin. We will present a hybrid detection method, where the distance measure is a weighted sum of the Euclidean and Pearson distance, so that the system designer may trade noise margin versus amount of immunity to unknown offset.
... Jiang et al. [3] addressed a q-ary balanced coding technique, called rank modulation, for circumventing the difficulties with flash memories having aging offset levels. Zhou et al. [4], Immink [5], and Sala et al. [6], [7] investigated constrained codes that enable dynamic reading thresholds in non-volatile memories. Immink & Weber [8] advocated Pearson-distance-based detection, which is intrinsically resistant to offset and gain mismatch. ...
... T -constrained codes, presented in [5] for enabling simple dynamic threshold detection of q-ary codewords, consist of q-ary n-length codewords, where T , 0 < T ≤ q, prescribed symbols must each appear at least once in a codeword. ...
Article
The reliability of mass storage systems, such as optical data recording and non-volatile memory (Flash), is seriously hampered by uncertainty of the actual value of the offset (drift) or gain (amplitude) of the retrieved signal. The recently introduced minimum Pearson distance detection is immune to unknown offset or gain, but this virtue comes at the cost of a lessened noise margin at nominal channel conditions. We will present a novel hybrid detection method, where we combine the outputs of the minimum Euclidean distance and Pearson distance detectors so that we may trade detection robustness versus noise margin. We will compute the error performance of hybrid detection in the presence of unknown channel mismatch and additive noise.
... For integers T satisfying 1 ≤ T ≤ q, T -constrained codes [1], denoted by S q,n (a 1 , . . . , a T ), consist of q-ary codewords of length n, where T preferred or reference symbols a 1 , . . . ...
... The number of q-ary sequences of length n, N T (q, n), where T distinct pre-defined symbols occur at least once in every sequence, equals [1] ...
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The Pearson distance has been advocated for improving the error performance of noisy channels with unknown gain and offset. The Pearson distance can only fruitfully be used for sets of $q$-ary codewords, called Pearson codes, that satisfy specific properties. We will analyze constructions and properties of optimal Pearson codes. We will compare the redundancy of optimal Pearson codes with the redundancy of prior art $T$-constrained codes, which consist of $q$-ary sequences in which $T$ pre-determined reference symbols appear at least once. In particular, it will be shown that for $q\le 3$ the $2$-constrained codes are optimal Pearson codes, while for $q\ge 4$ these codes are not optimal.
... The Pearson constraint that is immune to unknown channel gain and offset can be regarded as a type of T -constrained code where each of the T pre-defined symbols appears at least once in every codeword [45]. As discussed in [14], [18], [24], a known construction for q-ary Pearson codes is to ensure that every q-ary codeword has at least one symbol "0" and one symbol "1". ...
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We study the ability of recently developed variable-length constrained sequence codes to determine codeword boundaries in the received sequence upon initial receipt of the sequence and if errors in the received sequence cause synchronization to be lost.We first investigate construction of these codes based on the finite state machine description of a given constraint, and develop new construction criteria to achieve high synchronization probabilities. Given these criteria, we propose a guided partial extension algorithm to construct variable-length constrained sequence codes with high synchronization probabilities. With this algorithm we construct new codes and determine the number of codewords and coded bits that are needed to recover synchronization once synchronization is lost.We consider a large variety of constraints including the runlength limited (RLL) constraint, the DC-free constraint, the Pearson constraint and constraints for inter-cell interference mitigation in flash memories. Simulation results show that the codes we construct exhibit excellent synchronization properties, often resynchronizing within a few bits.
... To address the physical-related offset issues, two approaches are usually investigated and applied in storage systems. One approach uses pilot sequences to estimate the unknown channel offset [3]. The method is often considered too expensive with respect to redundancy. ...
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Data storage systems may not only be disturbed by noise. In some cases, the error performance can also be seriously degraded by offset mismatch. Here, channels are considered for which both the noise and offset are bounded. For such channels, Euclidean distance-based decoding, Pearson distance-based decoding, and Maximum Likelihood decoding are considered. In particular, for each of these decoders, bounds are determined on the magnitudes of the noise and offset intervals which lead to a word error rate equal to zero. Case studies with simulation results are presented confirming the findings.
... Generating functions are used to keep a count of such codes. In [5], it is shown that q-ary code designs (q ≥ 2) where T reference symbols appear at least once in each codeword, offer resistance against offset and gain uncertainties, with a small loss in the noise detection margin. ...
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Principles of Optical Disc SystemsCoding Methods for High-Density Optical Recording
  • J Bouwhuis
  • A Braat
  • J Huijser
  • G Pasman
  • K A S Van Rosmalen
  • Immink
Bouwhuis, J. Braat, A. Huijser, J. Pasman, G. van Rosmalen, and K.A.S. Immink, Principles of Optical Disc Systems, Adam Hilger Ltd, Bristol and Boston, 1985. 3 K.A.S. Immink, 'Coding Methods for High-Density Optical Recording', Philips J. Res., vol. 41, pp. 410-430, 1986.