VL-ECC: Variable Data-Length Error Correction Code for Embedded Memory in DSP Applications

In this paper, efficient implementation of error correction code (ECC) processing circuits based on single error correction and double error detection (SEC-DED) code with check bit pre-computation is proposed for memories. During the write operation of memory, check bit pre-computation eliminates the overall bits computation required to detect a double error, thereby reducing the complexity of the ECC processing circuits. In order to implement the ECC processing circuits using the check bit precomputation more efficiently, the proper SEC-DED codes are proposed. The H-matrix of the proposed SEC-DED code is the same as that of the odd-weightcolumn code during the write operation and is designed by replacing 0’s with 1’s at the last row of the H-matrix of the odd-weight-column code during the read operation. When compared with a conventional implementation utilizing the oddweight-column code, the implementation based on the proposed SEC-DED code with check bit precomputation achieves reductions in the number of gates, latency, and power consumption of the ECC processing circuits by up to 9.3%, 18.4%, and 14.1% for 64 data bits in a word.

Error-correcting codes (ECC) are widely used during data transfer in wireless communication systems as well as in computer memory architectures. The error-correcting process is based on sending data with extra parity bits and decoding the received data for error correction. The first error detection and correction code, introduced in 1950, Hamming Code (7,4) is a linear error-correcting code able to detect and correct a single-bit error by encoding 7-bit data from 4-bit data, including 3 parity bits. Previous efforts using unipolar resistive random access memory (ReRAM) based in-memory computation of Hamming Code (7,4) resulted in 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> times lower power consumption compared to GPU and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> times less than CPU-based computations. However further reduction of power consumption can be achieved by vector-matrix multiplication (VMM) using bipolar ReRAM arrays. In the VMM based approach, an encoding or decoding code matrix is stored in the array where it leverages the nonvolatile properties of ReRAM. With the VMM approach, the total number of computation cycles is not limited by the endurance of the ReRAM devices. Here we report the first experimental results of encoding and decoding Hamming code (7,4) using 1 transistor 1 hafnium oxide-based ReRAM (1T1R) arrays fabricated using 65nm CMOS technology. Our results show bipolar 1T1R arrays can correctly encode 4-bit message data to 7 bit encoded data as well as error position detection with overall 3 fold less power consumption than previously reported unipolar ReRAM crossbar array-based computation. Furthermore, we propose and simulate a peripheral circuit to convert the analog column output from a 1T1R array to single-bit binary output using the Cadence Spectre simulator. Our results pave the way for using a memristor-based fast and scalable hardware solution for encoding decoding of error-correcting codes

Error correction codes (ECCs) are commonly used in computer systems to protect information from errors. For example, single error correction (SEC) codes are frequently used for memory protection. Due to continuous technology scaling, soft errors on registers have become a major concern, and ECCs are required to protect them. Nevertheless, using an ECC increases delay, area and power consumption. In this way, ECCs are traditionally designed focusing on minimizing the number of redundant bits added. This is important in memories, as these bits are added to each word in the whole memory. However, this fact is less important in registers, where minimizing the encoding and decoding delay can be more interesting. This paper proposes a method to develop codes with 1-gate delay encoders and 4-gate delay decoders, independently of the word length. These codes have been designed to correct single errors only in data bits to reduce the overhead.

Due to the rapid technology scaling and increasing program/erase cycling, the raw bit error rate (RBER) in NAND flash memory keeps increasing rapidly. Conventional error correction codes (ECCs) with stronger protection capability are usually equipped for all flash pages as a solution to maintain the mandatory yield and reliability levels. However, the growth of RBER induced by increasing P/E cycles will lead to uneven distribution of errors. Applying uniform ECC protection capability for all flash pages might incur unnecessary hardware and latency overhead. Moreover, the overlong ECC check bits might be stored in two different flash pages. Therefore, two flash reads are required to retrieve a codeword. In this paper, ECC caching (E <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> C) techniques are proposed to cure these drawbacks of conventional uniform protection techniques. The main idea is to upgrade the ECC protection levels for flash pages when their correction slack is below the specified threshold. According to experimental results, we can enhance the reliability of flash memories significantly with negligible hardware cost.

The increasing demand for high-density and high-speed digital optical recording systems has made the development of advanced coding and signal processing techniques for fast and reliable data recovery increasingly important. In recent years, the parity-check (PC)-code-based reception technique has been widely studied for magnetic recording systems, and it is projected to be highly promising for high-density optical recording. The PC code is an inner error correction code (ECC), which can detect dominant short error events of the system, using only a few parity bits. This reduces the loss in error correction capability of the outer ECC due to random short errors and results in a simple and efficient solution to improve the overall performance. This thesis is dedicated to the design and analysis of PCcode- based recording systems to achieve higher recording capacity with low implementation complexity, for high-density blue laser disc systems. In particular, most of the key components of the PC-code-based optical recording system have been designed and optimized for different recording densities, and different proportions of white noise and media noise. During the development of advanced coding and detection techniques, it becomes necessary to investigate the system’s performance with different coding schemes and recording densities. In Chapter 2 of the thesis, we propose a generalized Braat-Hopkins model for optical recording channels, which provides a fair basis for the performance comparison of detection schemes over different coding schemes and recording densities, for channels with additive, white Gaussian noise (AWGN) and media noise. Various basic issues associated with the PC-code-based systems are investigated in Chapter 3. These include bounds for bit error rates and error event probabilities, the dominant error events at channel detector output, different PC codes, simple and efficient post-processors, the impact of error events that are split across data block boundaries and the corresponding remedy, as well as the effects of code rates and recording densities. Simulation results show that a 4-bit PC code achieves the best performance. The corresponding bit error rates (BERs) are very close to the performance bounds, at both nominal and high densities. Constrained codes, which serve as a key component in the read-write channel of data storage systems, are desired to have a high code rate with low-complexity encoder/decoder. In Chapter 4, we investigate the design of certain capacity-approaching constrained codes for optical recording systems. In particular, we derive analytically the relationship between the number of encoder states and the maximum size of these codes. We identify the minimum number of encoder states that maximizes the code rate, for any desired codeword length. The design of constrained PC codes is key to the development of PCcode- based systems, and the systematic design of efficient constrained PC codes remains a challenging problem. In Chapter 5, we propose a general and systematic code design technique for constructing capacity-approaching constrained PC codes, which can detect any type of error events of the system, with the minimum code rate loss. Compared to the rate 2/3 code without parity, the newly designed constrained 4-bit PC code achieves a performance gain of 2 dB at nominal density, and 1.5 dB at high density, at BER = 10-5. As the dominant noise for high-density optical recording systems, media noise severely degrades the performance of channel detectors and postprocessors that are designed for AWGN. In Chapter 6, we propose two novel modifications to the bit detector to combat media noise. We further develop a data-dependent post-processing scheme for error correction. Compared to the system designed without considering media noise and without PC codes, the overall performance gain of the developed scheme can be more than 11 dB at high media noise levels. In data storage systems, the ECC failure rate (EFR) serves as the ultimate measure of the data recovery performance. In Chapter 7, we develop semi-analytical approaches for estimating the EFR, and analyze EFRs for the developed PC-code-based systems, for the cases without and with interleaving. Our analysis shows that with the optimum interleaving degrees, compared to the rate 9/13 code without parity, the 2-bit PC code achieves a gain of around 0.5 dB, and the 4-bit PC code gains 0.6 dB, at high record ing density and EFR = 10-16. The thesis concludes in Chapter 8 with some suggestions for further work.

The write operation asymmetry of many memory technologies causes different write failure rates at 0 →1 and 1 → 0 bit-flipping's. Conventional error correction codes (ECCs) spend the same efforts on both bit-flipping directions, leading to very unbalanced write reliability enchantment over different bit-flipping distributions of codewords (i.e., the number of 0 →1 or 1 → 0 bit-flipping's). In this work, we developed an analytic asymmetric write channel (AWC) model to analyze the asymmetric write errors in spin-transfer torque random access memory (STT-RAM) designs. A new ECC design concept, namely, content-dependent ECC (CD-ECC), is proposed to achieve balanced error correction at both bit-flipping directions. Two CD-ECC schemes - typical-corner-ECC (TCE) and worst-corner-ECC (WCE), are designed for the codewords with different bit-flipping distributions. Our simulation results show that compared to the common ECC schemes utilized in embedded applications like Hamming code, CD-ECCs can improve the STT-RAM write reliability by 10 - 30x with low hardware overhead and very marginal impact on system performance.

Error control coding improves reliability and efficiency of wireless communication systems. This research delves into the latest advancements in error control coding schemes for wireless communication systems, with a specific focus on their application within the domain of Low Power Wide Area Networks (LPWANs) such as Narrowband Internet of Things (NB-IoT) systems, LoRa energy-efficient hardware design, random access control systems, and wireless sensor networks. In the context of LPWANs, particularly NB-IoT systems, we investigate the adaptive coding and modulation (ACM) in NB-IoT systems, which adapts coding rates and modulation techniques to changing channel circumstances. Comprehensive analysis of the literature review shows that the proposed approaches are more energy efficient and less error-prone than fixed coding and modulation schemes. Using Cell Design Methodology (CDM), Single Error Correction (SEC) codes are optimized for energy efficiency. Power and energy efficiency have improved in standard CMOS and CDM logic systems. This impacts IoT devices, memory storage, and security systems. Next, Forward Error Correction (FEC) is examined for ALOHA-based random-access control systems wherein packet loss rate and spectral efficiency metrics are mathematically calculated. FEC is beneficial, especially when time and frequency synchronization diverge. Monte Carlo simulations studied in the literature shows that FEC improves satellite communication systems. This review establishes benefits of FEC, notably in satellite communications systems. In addition, Wireless Sensor Networks (WSNs) error control techniques and network reliability and performance are examined along with Convolutional, Turbo, Low density parity check codes(LDPC), and Polar error correction coding (ECC) methods. We learn about their benefits, drawbacks, and implementation challenges by assessing their efficacy, efficiency, and applicability in wireless sensor networks (WSN) applications. This study further highlights advancements of energy-efficient error control coding when, integrated into network designs alongwith the trade-offs between efficacy and intricacy to help improve wireless communication networks. Finally, this study summarizes current error control coding approaches for wireless communication systems. We provide insights into improving reliability, energy conservation, and efficacy by examining ACM, CDM, FEC, and error correction coding (ECC) across varied domains including LPWAN and IoT technologies with earnest hope to facilitate researchers, engineers, and professionals working in the field of error control coding in wireless communication.

Brain-inspired Hyperdimensional Computing (HDC) is emerging as a compelling paradigm for learning at the edge because of its one-shot learning capability, inherent scalability, and exceptionally low computational overhead. While HDC is robust to noise, soft and hard memory faults in the memory components of HDC accelerator can still significantly degrade accuracy. Conventional error correction codes (ECC) are commonly used to mitigate such faults, but their associated overhead make them impractical for resource-constrained edge devices. In this paper, we present a novel memory-free error correction technique to enhance the fault tolerance of HDC systems without requiring any dedicated memory to store check-bits. This way, not only is the memory overhead and its associated constraints eliminated, but also the possibility of errors occurring within the Error-Correcting Code (ECC) check-bits themselves is omitted. Additionally, the proposed method is highly scalable, with minimal hardware overhead, and is therefore suitable for edge implementations. We validate the approach on an FPGA, demonstrating its practicality and effectiveness. Compared to the state-of-the-art correction methods, our memory-free design achieves $27 \times$ lower LUT utilization, more than $80 \times$ fewer registers and no DSP, BRAM, or latency at all. At the same time, it is capable of preserving inference accuracy under a $\mathbf{1 2} \boldsymbol{\times}$ higher fault possibility.

Fault tolerant encoders for Single Error Correction and Double Adjacent Error Correction codes

This chapter discusses fundamentals of information theory and channel coding (error correction coding ECC, Forward error correction FEC). First, key results of Shannon, including channel capacity and the relationship between transmit power and bandwidth, are discussed. Block codes multiply a block of data with an encoding matrix. For systematic codes, the resulting codeword consists of systematic bits and parity check bits. Decoding is achieved by multiplication with a parity check matrix and determination of the syndrome vector. Correction of errors within the correction sphere can be done, e.g., through lookup tables. Convolutional codes send the source data stream through shift registers and adders, a process described by a trellis diagram. Maximum-likelihood sequence estimation (Viterbi algorithm) determines the (transmit) bit sequence with best distance metric (Hamming or Euclidean) from the received signal. Methods for combining coding and higher-order modulation include trellis coded modulation (using set partitioning for code design) and bit-interleaved coded modulation (BICM). Turbo codes are close to capacity-achieving. They are based on transmitting differently interleaved convolutional codes; receivers use soft decoding of the constituent codes and iterative exchange of soft information (log-likelihood ratios). Low-density parity check codes (LDPCC) are decoded by belief propagation (message passing), where variable nodes and constraint nodes exchange beliefs about the codeword. In fading channels, (block or convolutional) interleaving is essential to provide diversity. From an information-theoretic point of view, in fading channels we can distinguish between the ergodic capacity and the outage capacity. If the channel state information is known at the transmitter, waterfilling is the optimum power allocation strategy. Automatic Repeat Request ARQ) and Hybrid ARQ (including chase combining or incremental redundancy) are described in the appendix.

To improve the reliability of static random access memory (SRAM), error-correcting codes (ECC) are typically used to protect SRAM in the cache. While improving the reliability, we also need additional circuits to support ECC, including encoding and decoding logic. In a high-speed circuit such as a CPU, the L1 cache maintains the same frequency as the CPU, and the decoding of the ECC codes in the cache consumes considerable combinational logic, resulting in limited frequency and performance. This study proposes a high-performance and flexible design scheme with ECC protection in the cache, in which the cache has two working modes: a high-performance mode and a high-reliability mode. The high-performance mode uses simple ECC codes, which can maintain high frequency with low access latency. The high-reliability mode uses more complex ECC codes, which improves the error correction capability and enhances the reliability of the SRAM. To meet the application requirements of different scenarios, the proposed scheme supports the software in switching between the above two modes by configuring the register, which improves the flexibility of the system. The results of synthesis show that the theoretical maximum frequency of proposed ECC design scheme increased from approximately 1.4 GHz in the conventional ECC design scheme to approximately 2.2 GHz. Some of the error correction capability of the high-performance mode is traded off against a 57% increase in frequency. In the high-reliability mode, the error correction capability of the SRAM is enhanced; however, the latency of accessing the cache increases by one cycle.

The existing charge-based memories like DRAM and flash are reaching their scaling limits. Phase change memory (PCM) is one of the most promising emerging memory technologies due to its good scalability and low leakage power. Multi-level cell (MLC) operation in PCM, which stores two or more bits in a single cell, is necessary to achieve a high storage density. However, a reduced resistance range in multiple storage levels of MLC PCM, introduces a lot of soft errors because of the resistance drift phenomenon. The poor reliability of MLC PCM requires strong error correction codes (ECC) which could severely degrade the storage density and performance. In this paper, we propose a cost-effective reliable MLC PCM architecture for improving MLC PCM reliability, storage density and performance. The proposed architecture exploits the data-dependent nature of resistance drift problem to reduce the ECC overhead. A simple state mapping is used to generate virtual data, which is half of the actual data size. ECC parity bits are generated based on virtual data bits instead of actual message bits, thus resulting in a reduced number of cells for parity bits. This improves the reliability and storage density of MLC PCM. The performance is also improved by minimizing the ECC overhead. Simulation results show an improvement of about 10 4 times in reliability and 10.9% in storage density compared to a conventional MLC PCM which uses a typical error correction scheme. Performance and energy efficiency are also improved up to 13.7% and 10%, respectively, by the proposed architecture.

Chapter 9 - Introduction to information theory and coding

Soft errors caused by radiation particles are becoming an increasingly important issue for memory reliability. Traditionally, Single Error Correction (SEC) codes have been used to correct soft errors. Such codes can correct one error per memory word, but as errors become more frequent they may be insufficient. More complex error correction codes (ECCs) could be used, but they typically require complex decoders which impact memory area and speed. In this paper, the use of Euclidean geometry (EG) Codes for memory applications is studied. The results show that these codes, which provide powerful error correction capabilities, can be implemented efficiently in terms of area and latency.

The conventional error correcting code (ECC) schemes for caches are based on a fixed mapping between cache data words and ECC check bits, and fixed ECC word granularity. This leads to inefficient usage of the ECC check bits. We propose to manage the check bits flexibly for low-cost error protection of last-level caches. The proposed ECC schemes work at the word level, whereas the conventional ECC schemes work at the cache line or set level. The proposed schemes protect only dirty words with ECC check bits using a flexible mapping. Moreover, the proposed schemes utilize variable ECC word granularities. Dirty (modified) words that are unlikely to be modified further before being evicted are collectively protected with a larger ECC word granularity. The proposed schemes reduce DRAM and data bus energy overheads by 28% and 45%, respectively, with the same area overhead as previously proposed competitive schemes. Our schemes show more energy reduction results for multicore systems without noticeable performance degradation.