Codes for Asymmetric Limited-Magnitude Errors With Application to Multilevel Flash Memories

Several physical effects that limit the reliability and performance of multilevel flash memories induce errors that have low magnitudes and are dominantly asymmetric. This paper studies block codes for asymmetric limited-magnitude errors over q-ary channels. We propose code constructions and bounds for such channels when the number of errors is bounded by t and the error magnitudes are bounded by ℓ. The constructions utilize known codes for symmetric errors, over small alphabets, to protect large-alphabet symbols from asymmetric limited-magnitude errors. The encoding and decoding of these codes are performed over the small alphabet whose size depends only on the maximum error magnitude and is independent of the alphabet size of the outer code. Moreover, the size of the codes is shown to exceed the sizes of known codes (for related error models), and asymptotic rate-optimality results are proved. Extensions of the construction are proposed to accommodate variations on the error model and to include systematic codes as a benefit to practical implementation

플래시 메모리를 위한 양방향 비대칭 오류 정정 부호 및 간섭 완화 기법

학위논문 (박사)-- 서울대학교 대학원 : 전기·컴퓨터공학부, 2014. 2. 이정우.Recently, NAND multi-level cell (MLC) flash memories are now widely used due to low cost and high capacity. However, when the number of cell levels increases, cell-to-cell interference (C2CI) which shifts threshold voltage may degrades the error rate in reading process. There are several approaches to alleviate the errors caused by the threshold voltage shift and we discuss error correcting codes and message encoding schemes. First, we propose error correcting codes that are effective for multi-level cell flash memory and non-binary WOM (write once memory) codes. In particular, we focus on bidirectional error correction codes. The errors in MLC flash memories tend to be directional and limited-magnitude. Many related works focus on asymmetric errors, but bidirectional errors also occur because of the bidirectional interference and the adjustment of the hard-decision reference voltages. The code treats both upward and downward errors when the error magnitude in each direction differs. The maximum magnitudes of the upward error and downward error are lu and ld, respectively. One of proposed codes extends the technique of the distinct sum sets to the bidirectional error correction codes. The other code is bidirectional limited magnitude error correction codes based on modulo operation and uses non-binary conventional error correction codes. These proposed codes can reduce the parity size, and have better error correction performance than the conventional error correction codes when the code rate is equal. Furthermore, error correcting schemes for non-binary WOM codes are discussed. WOM codes is a coding scheme that allows information to be written in a memory cell multiple times without erasure, and conventional error correction codes cannot be directly applied to WOM codes. The advantages of the proposed methods are that these are practical and systematic codes, and the complexity of encoding and decoding processes are low. We also introduce effective error locating limited-magnitude parity check error correction codes for the MLC flash memory error with lower complexity. Second, we introduce coding schemes to lower the generated interferences by cell to cell interference. It is known that C2CI is caused by the threshold voltage change of neighbor cells in writing operation. The amount of threshold voltage change is proportional to the magnitude. To minimize the generated interference, the average magnitude needs to be decreased. We propose two new C2CI reduction coding schemes that adjust the average magnitude to reduce C2CI. The proposed coding scheme deals with q-ary message codes, and generates fixed length codes. Message codewords are divided into several blocks, and are modified by modulo addition with proper values to minimize the average magnitude. We also propose low energy Huffman codes based on entropy coding when the frequency of symbols is not distributed uniformly. This scheme produces variable-length codes without redundancy. We modified Huffman codes to minimize average number of high bits ('1' bits). We show that proposed codes generate optimal codewords which have minimum high bits with minimum average codeword length.Chapter 1 Introduction 1 1.1 Backgrounds 1 1.2 Scope and Organization 5 Chapter 2 MLC Flash Memory Interference and Mitigation Techniques for Reliability 9 2.1 MLC flash memory and interference 9 2.2 Signal processing based interference mitigation in MLC flash memories 15 2.3 WOM codes 22 2.4 Asymmetric limited-magitude error correction codes based on distinct sum set 27 Chapter 3 Error Correction Codes for Flash Memories 29 3.1 Introduction 29 3.2 Bidirectional error correction codes for non-binary WOM codes based on distinct sum sets 30 3.2.1 Bidirectional error correction codes based on distinct sum sets 30 3.2.2 Error correction coding schemes for WOM codes based on distinct sum sets 41 3.3 Bidirectional error correction codes for WOM codes based on modulo operation 44 3.3.1 Bidirectional error correction codes based on modulo operation 44 3.3.2 Performance simulation of bidirectional error correction codes based on modulo operation 54 3.3.3 Error correction coding schemes for WOM codes based on modulo operation 58 3.4 Performance of error correction coding schemes for WOM code 61 3.5 Error locating parity check codes for errors with limited magnitude 68 3.6 Summary 77 Chapter 4 On Interference Mitigating Codes for Multi-level Flash Memories 79 4.1 Introduction 79 4.2 The modeling of generated interference in flash memory 80 4.3 Coding schemes for interference mitigation 83 4.3.1 Minimum energy coding 83 4.3.2 Module shift coding 85 4.3.3 Low energy Huffman code 89 4.4 Performance analysis of proposed coding schemes 91 4.4.1 Performance analysis of ME codes 91 4.4.2 Performance analysis of MS codes 93 4.4.3 Performance of low-energy Huffman codes 97 4.4.4 C2CI reduction performance 99 4.5 Summary 102 Chapter 5 Conclusions 105 Appendix A 109 A.1 Performance analysis of MS coding with eta=2 case in chap. 4.4.2. 109 Bibliography 113 Abstract in Korean 120Docto

Asymmetric Error Correction and Flash-Memory Rewriting using Polar Codes

We propose efficient coding schemes for two communication settings: 1. asymmetric channels, and 2. channels with an informed encoder. These settings are important in non-volatile memories, as well as optical and broadcast communication. The schemes are based on non-linear polar codes, and they build on and improve recent work on these settings. In asymmetric channels, we tackle the exponential storage requirement of previously known schemes, that resulted from the use of large Boolean functions. We propose an improved scheme, that achieves the capacity of asymmetric channels with polynomial computational complexity and storage requirement. The proposed non-linear scheme is then generalized to the setting of channel coding with an informed encoder, using a multicoding technique. We consider specific instances of the scheme for flash memories, that incorporate error-correction capabilities together with rewriting. Since the considered codes are non-linear, they eliminate the requirement of previously known schemes (called polar write-once-memory codes) for shared randomness between the encoder and the decoder. Finally, we mention that the multicoding scheme is also useful for broadcast communication in Marton's region, improving upon previous schemes for this setting.Comment: Submitted to IEEE Transactions on Information Theory. Partially presented at ISIT 201

EQUALISATION TECHNIQUES FOR MULTI-LEVEL DIGITAL MAGNETIC RECORDING

A large amount of research has been put into areas of signal processing, medium design, head and servo-mechanism design and coding for conventional longitudinal as well as perpendicular magnetic recording. This work presents some further investigation in the signal processing and coding aspects of longitudinal and perpendicular digital magnetic recording. The work presented in this thesis is based upon numerical analysis using various simulation methods. The environment used for implementation of simulation models is C/C + + programming. Important results based upon bit error rate calculations have been documented in this thesis. This work presents the new designed Asymmetric Decoder (AD) which is modified to take into account the jitter noise and shows that it has better performance than classical BCJR decoders with the use of Error Correction Codes (ECC). In this work, a new method of designing Generalised Partial Response (GPR) target and its equaliser has been discussed and implemented which is based on maximising the ratio of the minimum squared euclidean distance of the PR target to the noise penalty introduced by the Partial Response (PR) filter. The results show that the new designed GPR targets have consistently better performance in comparison to various GPR targets previously published. Two methods of equalisation including the industry's standard PR, and a novel Soft-Feedback- Equalisation (SFE) have been discussed which are complimentary to each other. The work on SFE, which is a novelty of this work, was derived from the problem of Inter Symbol Interference (ISI) and noise colouration in PR equalisation. This work also shows that multi-level SFE with MAP/BCJR feedback based magnetic recording with ECC has similar performance when compared to high density binary PR based magnetic recording with ECC, thus documenting the benefits of multi-level magnetic recording. It has been shown that 4-level PR based magnetic recording with ECC at half the density of binary PR based magnetic recording has similar performance and higher packing density by a factor of 2. A novel technique of combining SFE and PR equalisation to achieve best ISI cancellation in a iterative fashion has been discussed. A consistent gain of 0.5 dB and more is achieved when this technique is investigated with application of Maximum Transition Run (MTR) codes. As the length of the PR target in PR equalisation increases, the gain achieved using this novel technique consistently increases and reaches up to 1.2 dB in case of EEPR4 target for a bit error rate of 10-5

Trellis coded modulation techniques

The subject of this thesis is an investigation of various trellis coded modulation (TCM) techniques that have potential for out-performing conventional methods. The primary advantage of TCM over modulation schemes employing traditional error-correction coding is the ability to achieve increased power efficiency without the normal expansion of bandwidth introduced by the coding process. Thus, channels that are power limited and bandwidth limited are an ideal application for TCM. In this thesis, four areas of interest are investigated. These include: signal constellation design, multilevel convolutional coding, adaptive TCM and finally low-complexity implementation of TCM. An investigation of the effect of signal constellation design on probability of error has led to the optimisation of constellation angles for a given channel signal to noise ratio and a given code. The use of multilevel convolutional codes based on rings of integers and multi-dimensional modulation is presented. The potential benefits of incorporating several modulation schemes with adaptive TCM which require a single decoder are also investigated. The final area of investigation has been the development of an algorithm for decoding of convolutional codes with a low complexity decoder. The research described in this thesis investigated the use of trellis coded modulation to develop various techniques applicable to digital data transmission systems. Throughout this work, emphasis has been placed on enhancing the performance or complexity of conventional communication systems by simple modifications to the existing structures

Algorithms and Data Representations for Emerging Non-Volatile Memories

The evolution of data storage technologies has been extraordinary. Hard disk drives that fit in current personal computers have the capacity that requires tons of transistors to achieve in 1970s. Today, we are at the beginning of the era of non-volatile memory (NVM). NVMs provide excellent performance such as random access, high I/O speed, low power consumption, and so on. The storage density of NVMs keeps increasing following Moore’s law. However, higher storage density also brings significant data reliability issues. When chip geometries scale down, memory cells (e.g. transistors) are aligned much closer to each other, and noise in the devices will become no longer negligible. Consequently, data will be more prone to errors and devices will have much shorter longevity. This dissertation focuses on mitigating the reliability and the endurance issues for two major NVMs, namely, NAND flash memory and phase-change memory (PCM). Our main research tools include a set of coding techniques for the communication channels implied by flash memory and PCM. To approach the problems, at bit level we design error correcting codes tailored for the asymmetric errors in flash and PCM, we propose joint coding scheme for endurance and reliability, error scrubbing methods for controlling storage channel quality, and study codes that are inherently resisting to typical errors in flash and PCM; at higher levels, we are interested in analyzing the structures and the meanings of the stored data, and propose methods that pass such metadata to help further improve the coding performance at bit level. The highlights of this dissertation include the first set of write-once memory code constructions which correct a significant number of errors, a practical framework which corrects errors utilizing the redundancies in texts, the first report of the performance of polar codes for flash memories, and the emulation of rank modulation codes in NAND flash chips

Performance Analysis of Quantum Error-Correcting Codes via MacWilliams Identities

One of the main challenges for an efficient implementation of quantum information technologies is how to counteract quantum noise. Quantum error correcting codes are therefore of primary interest for the evolution towards quantum computing and quantum Internet. We analyze the performance of stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors of stabilizer codes based on the quantum MacWilliams identities. The WE is then used to evaluate the error rate of quantum codes under maximum likelihood decoding or, in the case of surface codes, under minimum weight perfect matching (MWPM) decoding. Our findings lead to analytical formulas for the performance of generic stabilizer codes, including the Shor code, the Steane code, as well as surface codes. For example, on a depolarizing channel with physical error rate $\rho \to 0$ it is found that the logical error rate $\rho_\mathrm{L}$ is asymptotically $\rho_\mathrm{L} \to 16.2 \rho^2$ for the $[[9,1,3]]$ Shor code, $\rho_\mathrm{L} \to 16.38 \rho^2$ for the $[[7,1,3]]$ Steane code, $\rho_\mathrm{L} \to 18.74 \rho^2$ for the $[[13,1,3]]$ surface code, and $\rho_\mathrm{L} \to 149.24 \rho^3$ for the $[[41,1,5]]$ surface code.Comment: 25 pages, 5 figures, submitted to an IEEE journal. arXiv admin note: substantial text overlap with arXiv:2302.1301