As a medium for cold data storage, DNA stands out as it promises significant gains in storage capacity and lifetime. However, it comes with its own data processing challenges to overcome. Constrained codes over the DNA alphabet {A, T, G, C} have been used to design DNA sequences that are free of long homopolymers to increase stability, yet effective error detection and error correction are required to achieve reliability in data retrieval. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely DNA LOCO (D-LOCO) codes, with error detection. In this paper, we equip our D-LOCO codes with error correction for substitution errors via syndrome-like decoding, designated as residue decoding. We only use D-LOCO codewords of indices divisible by a suitable redundancy metricR(m) > 0, wheremis the code length, for error correction. The idea is that the residue, which is the index moduloR(m), of the received word index equals the residue of the index error, i.e., the index difference. We find an exhaustive list of index differences due to single-substitution errors, and we are able to recover the index of the original codeword from the residue of the received word index. This requires storing a table for index errors and their residues. Having this decoding algorithm in hand, we provide the community with a construction of constrained codes forbidding runs of length higher than fixed ℓ ∈ {1,2,3} andGC-content in [0.5−1/2K,0.5+1/2K] that correct K segmented substitution errors, one per codeword. We call the proposed codes error-correction (EC) D-LOCO codes. We also give a list-decoding procedure with near-quadratic time-complexity inmto correct double-substitution errors within EC D-LOCO codewords, which has > 98.20% average success rate. The redundancy metric is projected to require 2log2(m)+O(1)-bit allocation, i.e., 2log2(m)+O(1) reduction in message bits, for a length-mcodeword. Hence, our EC D-LOCO codes are projected to be capacity-approaching with respect to the error-free constrained system.

Low-density parity-check (LDPC) codes are among the most prominent error-correction schemes in today’s data-driven world. They find application to fortify various modern storage, communication, and computing systems. Protograph-based (PB) LDPC codes offer many degrees of freedom in the code design and enable fast encoding and decoding. In particular, spatially-coupled (SC) and multi-dimensional (MD) circulant-based codes are PB-LDPC codes with excellent performance. Efficient finite-length (FL) algorithms are required in order to effectively exploit the available degrees of freedom offered by SC partitioning, lifting, and MD relocations. In this paper, we propose a novel Markov chain Monte Carlo (MCMC or MC2) method to perform this FL optimization, addressing the removal of short cycles. While we focus on partitioning and lifting of SC codes, our MC2approach can effectively work for other procedures and/or other code designs. While iterating, we draw samples from a defined distribution where the probability decreases as the number of short cycles from the previous iteration increases. We analyze our MC2method theoretically as we prove the invariance of the Markov chain where each state represents a possible partitioning or lifting arrangement, i.e., sample, that has a specific probability. Via our simulations, we then fit the distribution of the number of cycles resulting from a given arrangement on a Gaussian distribution. By analyzing the mean, we derive estimates for cycle counts that are close to the actual counts. Furthermore, we derive the order of the expected number of iterations required by our MC2approach to reach a local minimum as well as the size of the Markov chain recurrent class through approximating the probability of getting arbitrarily close to the local minimum. Our approach is compatible with code design techniques based on gradient-descent. Experimental results show that our MC2method generates SC codes with remarkably fewer short cycles and substantial gains in error/erasure-rate performance compared with the current state-of-the-art. Moreover, to reach the same number of cycles, our MC2method requires orders of magnitude less overall time compared with the available literature methods.

Because of their excellent asymptotic and finite-length performance, spatially-coupled (SC) codes are a class of low-density parity-check codes that is gaining increasing attention. Multi-dimensional (MD) SC codes are constructed by connecting copies of an SC code via relocations in order to mitigate various sources of non-uniformity and improve performance in many data storage and data transmission systems. As the number of degrees of freedom in the MD-SC code design increases, appropriately exploiting them becomes more difficult because of the complexity growth of the design process. In this paper, we propose a probabilistic framework for the MD-SC code design, which is based on the gradient-descent (GD) algorithm, to design better MD codes and address this challenge. In particular, we express the expected number of short cycles, which we seek to minimize, in the graph representation of the code in terms of entries of a probability-distribution matrix that characterizes the MD-SC code design. We then find a locally-optimal probability distribution, which serves as the starting point of a finite-length algorithmic optimizer that produces the final MD-SC code. We offer the theoretical analysis as well as the algorithms, and we present experimental results demonstrating that our MD codes, conveniently called GD-MD codes, have notably lower short cycle numbers compared with the available state-of-the-art. Moreover, our algorithms converge on solutions in few iterations, which confirms the complexity reduction as a result of limiting the search space via the locally-optimal GD-MD distributions.

The two-dimensional magnetic recording (TDMR) technology promises a remarkable increase in data storage density using the already-existing magnetic materials. Advanced TD signal processing techniques are required for this technology to fulfill its potential. Constrained coding, which prevents TD error-prone data patterns from being written, is among those techniques. In this paper, we propose lexicographically-ordered constrained (LOCO) coding schemes that offer low complexity, low latency, and low error propagation for TDMR systems with wide read heads, where coding can be applied on groups of 3 down tracks each. In particular, we introduce simple plus LOCO (SP-LOCO) and simple T LOCO (ST-LOCO) coding schemes, where codes defined over GF(2) and GF(4), respectively, are used instead of codes defined over GF (8), allowing the separation of uncoded streams. To better understand the behavior of the constrained-coded system, we derive the probability of transition, both in the horizontal and vertical directions, for the case of SP-LOCO and ST-LOCO coding, and we compare them to the case of optimal, rate-wise, LOCO codes. Moreover, we introduce simulation results on a practical TDMR model that demonstrate the performance gains achieved via the proposed schemes.

Magnetic recording devices are still competitive in the storage density race with solid-state devices thanks to new technologies such as two-dimensional magnetic recording (TDMR). TDMR offers remarkable storage density increase without the need for new magnetic materials; however, advanced data processing schemes are needed to guarantee reliability. Data patterns where a bit is surrounded by complementary bits at the four positions with Manhattan distance 1 on the TDMR grid are called plus isolation (PIS) patterns, and they are error-prone. Recently, we introduced lexicographically-ordered constrained (LOCO) codes, namely optimal plus LOCO (OP-LOCO) codes, with minimal redundancy that prevent these patterns from being written in a TDMR device. However, in the high-density regime or the low-energy regime (as the device ages), additional error-prone patterns emerge, specifically data patterns where a bit is surrounded by complementary bits at only three positions with Manhattan distance 1, and we call them incomplete plus isolation (IPIS) patterns. In this paper, we present capacity-achieving codes that forbid both PIS and IPIS patterns in TDMR systems with wide read heads. Because of their shape, we collectively call the PIS and IPIS patterns rotated T isolation (RTIS) patterns, and we call the new codes optimal T LOCO (OT-LOCO) codes. We analyze OT-LOCO codes and derive their simple encoding-decoding rule that allows reconfigurability. We also present a novel bridging idea for these codes to further increase the rate. Our simulation results demonstrate that OT-LOCO codes not only remarkably outperform OP-LOCO codes, but also entirely eliminate media noise effects, resulting from error-prone data patterns, at practical TD densities in the range [0.6,0.8) with high rates in the range [0.81,0.83]. At the TD density of 0.8, the OT-LOCO code of rate 0.8267 achieves a frame error rate (bit error rate) performance gain of about 1.15 orders (1.23 orders) of magnitude for all TDMR down (horizontal) tracks compared with the uncoded setting. To further preserve the storage capacity, we suggest using OP-LOCO codes, which have higher rates than OT-LOCO codes, early in the device lifetime, then employing the reconfiguration property to switch to OT-LOCO codes later in the device lifetime. While the point of reconfiguration on the density/energy axis is decided manually at the moment, the next step is to use machine learning to make that decision based on the TDMR device status. Moreover, we introduce another coding scheme to remove RTIS patterns in TDMR systems which offers lower complexity, lower error propagation, and track separation, at the expense of a limited rate loss.

The pivotal storage density win achieved by solid-state devices over magnetic devices in 2015 is a result of multiple innovations in physics, architecture, and signal processing. One of the most important innovations in that regard is enabling the storage of more than one bit per cell in the Flash device, i.e., having more than two charge levels per cell. Constrained coding is used in Flash devices to increase reliability via mitigating inter-cell interference that stems from charge propagation among cells. Recently, capacity-achieving constrained codes were introduced to serve that purpose in modern Flash devices, which have more than two levels per cell. While these codes result in minimal redundancy via exploiting the underlying physics, they result in non-negligible complexity increase and access speed limitation since pages cannot be read separately. In this paper, we suggest new constrained coding schemes that have low-complexity and preserve the desirable high access speed in modern Flash devices. The idea is to eliminate error-prone patterns by coding data either only on the left-most page (binary coding) or only on the two left-most pages (4-ary coding) while leaving data on all the remaining pages uncoded. Our coding schemes work for any number of levels$q \geq 4$per cell, offer systematic encoding and decoding, and are capacity-approaching. Since the proposed schemes enable the separation of pages, except the two left-most pages in the case of 4-ary coding, we refer to them as read-and-run (RR) constrained coding schemes as opposed to schemes adopting read-and-wait for other pages. The 4-ary RR coding scheme is introduced in order to limit the rate loss incurred by the binary RR coding schemes, and we show that our 4-ary RR coding scheme is also competitive when it comes to complexity and error propagation. We analyze the new RR coding schemes and discuss their impact on the probability of occurrence of different charge levels. We also demonstrate the performance improvement achieved via RR coding on a practical triple-level cell Flash device.

Spatially-coupled (SC) codes, known for their threshold saturation phenomenon and low-latency windowed decoding algorithms, are ideal for streaming applications and data storage systems. SC codes are constructed by partitioning an underlying block code, followed by rearranging and concatenating the partitioned components in a convolutional manner. The number of partitioned components determines the memory of SC codes. In this paper, we investigate the relation between the performance of SC codes and the density distribution of partitioning matrices. While adopting higher memories results in improved SC code performance, obtaining finite-length, high-performance SC codes with high memory is known to be computationally challenging. We break this computational bottleneck by developing a novel probabilistic framework that obtains (locally) optimal density distributions via gradient descent. Starting from random partitioning matrices abiding by the obtained distribution, we perform low-complexity optimization algorithms that minimize the number of detrimental objects to construct high-memory, high-performance quasi-cyclic SC codes. We apply our framework to various objects of interest, from the simplest short cycles, to more sophisticated objects such as concatenated cycles aiming at finer-grained optimization. Simulation results show that codes obtained through our proposed method notably outperform state-of-the-art SC codes with the same constraint length and optimized SC codes with uniform partitioning. The performance gain is shown to be universal over a variety of channels, from canonical channels such as additive white Gaussian noise and binary symmetric channels, to practical channels underlying flash memory and magnetic recording systems.

The pivotal storage density win achieved by solid-state devices over magnetic devices in 2015 is a result of multiple innovations in physics, architecture, and signal processing. One of the most important innovations in that regard is enabling the storage of more than one bit per cell in the Flash device, i.e., having more than two charge levels per cell. Constrained coding is used in Flash devices to increase reliability via mitigating inter-cell interference that stems from charge propagation among cells. Recently, capacity-achieving constrained codes were introduced to serve that purpose in modern Flash devices, which have more than two levels per cell. While these codes result in minimal redundancy via exploiting the underlying physics, they result in non-negligible complexity increase and access speed limitation since pages cannot be read separately. In this paper, we suggest new constrained coding schemes that have low-complexity and preserve the desirable high access speed in modern Flash devices. The idea is to eliminate error-prone patterns by coding data only on the left-most page while leaving data on all the remaining pages uncoded. Our coding schemes work for any number of levels per cell, offer systematic encoding and decoding, and are capacity-approaching. Since the proposed schemes enable the separation of pages, we refer to them as read-and-run (RR) constrained coding schemes as opposed to schemes adopting read-and-wait for other pages. We analyze the new RR coding schemes and discuss their impact on the probability of occurrence of different charge levels. We also demonstrate the performance improvement achieved via RR coding on a practical triple-level cell Flash device.

Constrained codes are used to prevent errors from occurring in various data storage and data transmission systems. They can help in increasing the storage density of magnetic storage devices, in managing the lifetime of solid-state storage devices, and in increasing the reliability of data transmission over wires. Over the years, designing practical (complexity-wise) capacity-achieving constrained codes has been an area of research gaining significant interest. We recently designed various constrained codes based on lexicographic indexing. We introduced binary symmetric lexicographically-ordered constrained (S-LOCO) codes,$q$-ary asymmetric LOCO (QA-LOCO) codes, and a class of two-dimensional LOCO (TD-LOCO) codes. These families of codes achieve capacity with simple encoding and decoding, and they are easy to reconfigure. We demonstrated that these codes can contribute to notable density and lifetime gains in magnetic recording (MR) and Flash systems, and they find application in other systems too. In this paper, we generalize our work on LOCO codes by presenting a systematic method that guides the code designer to build any constrained code based on lexicographic indexing once the finite set of data patterns to forbid is known. In particular, we connect the set of forbidden patterns directly to the cardinality of the LOCO code and most importantly to the rule that uncovers the index associated with a LOCO codeword. By doing that, we reveal the secret arithmetic of patterns, and make the design of such constrained codes significantly easier. We give examples illustrating the method via codes based on lexicographic indexing from the literature. We then design optimal (rate-wise) constrained codes for the new two-dimensional magnetic recording (TDMR) technology. Over a practical TDMR model, we show notable performance gains as a result of solely applying the new codes. Moreover, we show how near-optimal constrained codes for TDMR can be designed and used to further reduce complexity and error propagation. All the newly introduced LOCO codes are designed using the proposed general method, and they inherit all the desirable properties in our previously designed LOCO codes.

In order to accommodate the ever-growing data from various, possibly independent, sources and the dynamic nature of data usage rates in practical applications, modern cloud data storage systems are required to be scalable, flexible, and heterogeneous. The recent rise of the blockchain technology is also moving various information systems towards decentralization to achieve high privacy at low costs. While codes with hierarchical locality have been intensively studied in the context of centralized cloud storage due to their effectiveness in reducing the average reading time, those for decentralized storage networks (DSNs) have not yet been discussed. In this paper, we propose a joint coding scheme where each node receives extra protection through the cooperation with nodes in its neighborhood in a heterogeneous DSN with any given topology. This work extends and subsumes our prior work on coding for centralized cloud storage. In particular, our proposed construction not only preserves desirable properties such as scalability and flexibility, which are critical in dynamic networks, but also adapts to arbitrary topologies, a property that is essential in DSNs but has been overlooked in existing works.

Spatially-coupled (SC) codes, known for their threshold saturation phenomenon and low-latency windowed decoding algorithms, are ideal for streaming applications and data storage systems. SC codes are constructed by partitioning an underlying block code, followed by rearranging and concatenating the partitioned components in a “convolutional” manner. The number of partitioned components determines the “memory” of SC codes. While adopting higher memories results in improved SC code performance, obtaining optimal SC codes with high memory is known to be hard. In this paper, we investigate the relation between the performance of SC codes and the density distribution of partitioning matrices. We propose a probabilistic framework that obtains (locally) optimal density distributions via gradient descent. Starting from random partitioning matrices abiding by the obtained distribution, we perform low complexity optimization algorithms over the cycle properties to construct high memory, high performance quasi-cyclic SC codes. Simulation results show that codes obtained through our proposed method notably outperform state-of-the-art SC codes with the same constraint length and codes with uniform partitioning.

In various practical systems, certain data patterns are prone to errors if written or transmitted. In magnetic recording and communication over transmission lines, data patterns causing consecutive transitions that are not sufficiently separated are prone to errors. In Flash memory with two levels per cell, data patterns causing high–low–high charge levels on adjacent cells are prone to errors. Constrained codes are used to eliminate error-prone patterns, and they can also achieve other goals. Recently, we introduced efficient binary symmetric lexicographically-ordered constrained (LOCO) codes and asymmetric LOCO (A-LOCO) codes to increase density in magnetic recording systems and lifetime in Flash systems by eliminating the relevant detrimental patterns. Due to their application, LOCO and A-LOCO codes are associated with level-based signaling. Studying the power spectrum of a random signal with certain properties is principal for any storage or transmission system. It reveals important properties such as the average signal power at DC, the bandwidth of the signal, and whether there are discrete power components at certain frequencies. In this paper, we first modify a framework from the literature in order to introduce a method to derive the power spectrum of a sequence of constrained data associated with level-based signaling. We apply our method to infinitely long sequences satisfying symmetric and asymmetric constraints. Next, we show how to generalize the method such that it works for a stream of finite-length codewords as well, thus demonstrating how to overcome the associated finite-length challenges. We use the generalized method to devise closed forms for the spectra of finite-length LOCO and A-LOCO codes from their transition diagrams. Our LOCO and A-LOCO spectral derivations can be performed for any code length and can be extended to other constrained codes. We plot these power spectra, and discuss various important spectral properties for both LOCO and A-LOCO codes. We also briefly discuss an alternative method for deriving the power spectrum and introduce an idea towards reaching the spectra of self-clocked codes.

Flash memory devices are winning the competition for storage density against magnetic recording devices. This outcome results from advances in physics that allow storage of more than one bit per cell, coupled with advances in signal processing that reduce the effect of physical instabilities. Constrained codes are used in storage to avoid problematic patterns, and thus prevent errors from happening. Recently, we introduced binary symmetric lexicographically-ordered constrained codes (LOCO codes) for data storage and data transmission. LOCO codes are capacity-achieving, simple, and can be easily reconfigured. This paper introduces simple constrained codes that support non-binary physical gates in multi, triple, quad, and the currently-in-development penta-level cell (M/T/Q/P-LC) Flash memories. The new codes can be easily modified if problematic patterns change with time. These codes are designed to mitigate inter-cell interference, which is a critical source of error in Flash devices. The occurrence of errors is a consequence of parasitic capacitances in and across floating-gate transistors, resulting in charge propagation from cells being programmed to the highest charge level to neighboring cells being programmed to lower levels or unprogrammed/erased. This asymmetric nature of error-prone patterns distinguishes Flash memories. The new codes are called$q$-ary asymmetric LOCO codes (QA-LOCO codes), and the construction subsumes codes previously designed for single-level cell (SLC) Flash devices (A-LOCO codes). QA-LOCO codes work for a Flash device with any number,$q$, of levels per cell. For$q \geq 4$, we show that QA-LOCO codes can achieve rates greater than$0.95 \log _{2} \!q$input bits per coded symbol. The complexity of encoding and decoding is modest, and reconfiguring a code is as easy as reprogramming an adder. Capacity-achieving rates, affordable encoding-decoding complexity, and ease of reconfigurability support the growing improvement of M/T/Q/P-LC Flash memory devices, as well as lifecycle management as the characteristics of these devices change with time, which increases their lifetime.

Flash memory devices are winning the competition for storage density against magnetic recording devices. This outcome results from advances in physics that allow storage of more than one bit per cell, coupled with advances in signal processing that reduce the effect of physical instabilities. Constrained codes are used in storage to avoid problematic patterns. Recently, we introduced binary symmetric lexicographically-ordered constrained codes (LOCO codes) for data storage and transmission. This paper introduces simple constrained codes that support non-binary physical gates in multi, triple, quad, and the currently-in-development penta-level cell (M/T/Q/P-LC) Flash memories. The new codes can be easily modified if problematic patterns change with time. These codes are designed to mitigate inter-cell interference, which is a critical source of error in Flash devices. The new codes are called q-ary asymmetric LOCO codes (QA-LOCO codes), and the construction subsumes codes previously designed for single-level cell (SLC) Flash devices (ALOCO codes). QA-LOCO codes work for a Flash device with any number, q, of levels per cell. For q ≥ 4, we show that QA-LOCO codes can achieve rates greater than 0.95log2q information bits per coded symbol. Capacity-achieving rates, affordable encoding-decoding complexity, and ease of reconfigurability support the growing improvement of M/T/Q/P-LC Flash memory devices, as well as lifecycle management as the characteristics of these devices change with time.

The continuous rise of the blockchain technology is moving various information systems towards decentralization. Blockchain-based decentralized storage networks (DSNs) offer significantly higher privacy and lower costs to customers compared with centralized cloud storage associated with specific vendors. Coding is required to retrieve data stored on failing components. While coding solutions for centralized storage have been intensely studied, those for DSNs have not yet been discussed. In this paper, we propose a coding scheme where each node receives extra protection through cooperation with nodes in its neighborhood in a heterogeneous DSN with any given topology. Our scheme can achieve faster recovery speed compared with existing network coding methods, and can correct more erasure patterns compared with our previous work.

The increasing demand for access to data has led to dramatic increases in data storage densities, and as densities increase, new sources of error appear. Multi-dimensional (MD) graph-based codes are capable of mitigating error sources like interference and channel non-uniformity in dense storage devices. A recent innovation improves the performance of MD spatially-coupled codes that are based on circulants by carefully relocating some circulants to minimize the number of short cycles. However, cycles become more detrimental when they combine together to form more advanced objects, e.g., absorbing sets, including low-weight codewords. In this paper, we show how MD relocations can be exploited to minimize the number of detrimental objects in the graph of an MD code. Moreover, we demonstrate the savings in the number of relocation arrangements earned by focusing on objects rather than their constituent cycles. Our technique is applicable to a wide variety of one-dimensional (OD) codes. Simulation results demonstrate significant lifetime gains achieved by the proposed MD codes on an industry-recommended model for Flash systems, and signal-to-noise ratio gains on an industry-recommended model for magnetic recording systems, both with respect to OD codes with similar parameters. The second order analysis of MD relocations relies on conditions and options for an object, called a pattern, to form a bigger cycle after MD relocations, which are discussed in this paper.

Line codes make it possible to mitigate interference, to prevent short pulses, and to generate streams of bipolar signals with no direct-current (DC) power content through balancing. They find application in magnetic recording (MR) devices, in Flash devices, in optical recording devices, and in some computer standards. This paper introduces a new family of fixed-length, binary constrained codes, named lexicographically-ordered constrained codes (LOCO codes), for bipolar non-return-to-zero signaling. LOCO codes are capacity-achieving, the lexicographic indexing enables simple, practical encoding and decoding, and this simplicity is demonstrated through analysis of circuit complexity. LOCO codes are easy to balance, and their inherent symmetry minimizes the rate loss with respect to unbalanced codes having the same constraints. Furthermore, LOCO codes that forbid certain patterns can be used to alleviate inter-symbol interference in MR systems and inter-cell interference in Flash systems. Numerical results demonstrate a gain of up to 10% in rate achieved by LOCO codes with respect to other practical constrained codes, including run-length-limited codes, designed for the same purpose. Simulation results suggest that it is possible to achieve a channel density gain of about 20% in MR systems by using a LOCO code to encode only the parity bits, limiting the rate loss, of a low-density parity-check code before writing.

Because of their capacity-approaching performance and their complexity/latency advantages, spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern dense data storage systems. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for additive white Gaussian noise (AWGN) and Flash channels. The OO stage operates on the protograph of the SC code to derive the optimal partitioning that minimizes the number of graphical objects that undermine the performance of SC codes under iterative decoding. Then, the CPO optimizes the circulant powers to further reduce this number. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the proposed OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by up to around 3 orders of magnitude in frame error rate (FER) and 1.1 dB in signal-to-noise ratio (SNR). More intriguingly, our SC codes outperform structured block codes of the same length and rate by up to around 1.8 orders of magnitude in FER and 0.4 dB in SNR. The performance advantage of SC codes designed using the devised OO-CPO approach over block codes of the same parameters is not only pronounced in the error floor region, but also in the waterfall region.

In order to accommodate the ever-growing data from various, possibly independent, sources and the dynamic nature of data usage rates in practical applications, modern cloud data storage systems are required to be scalable, flexible, and heterogeneous. Codes with hierarchical locality have been intensively studied due to their effectiveness in reducing the average reading time in cloud storage. In this paper, we present the first codes with hierarchical locality that achieve scalability and flexibility in heterogeneous cloud storage using small field size. We propose a double- level construction utilizing so-called Cauchy Reed-Solomon codes. We then develop a triple-level construction based on this double-level code; this construction can be easily generalized into any hierarchical structure with a greater number of layers since it naturally achieves scalability in the cloud storage systems.

Line codes make it possible to mitigate interference, to prevent short pulses, and to generate streams of bipolar signals with no direct-current (DC) power content through balancing. They find application in magnetic recording (MR) devices, in Flash devices, and in optical recording devices. This paper introduces a new family of fixed-length, binary constrained codes, named lexicographically-ordered constrained codes (LOCO codes), for bipolar non-return-to-zero signaling. LOCO codes are capacity achieving, the lexicographic indexing enables simple, practical encoding and decoding, and this simplicity is demonstrated through analysis of circuit complexity. Experimental results demonstrate a gain of up to 10% in rate achieved by LOCO codes with respect to practical run-length-limited codes designed for the same purpose. Simulation results suggest that it is possible to achieve channel density gains of about 20% in MR systems by using a LOCO code to encode only the parity bits of a low-density parity-check code before writing.

In order to meet the demands of data-hungry applications, data storage devices are required to be increasingly denser. Various sources of error appear with this increase in density. Multi-dimensional (MD) graph-based codes are capable of mitigating error sources like interference and channel non-uniformity in dense storage devices. Recently, a technique was proposed to enhance the performance of MD spatially-coupled codes that are based on circulants. The technique carefully relocates circulants to minimize the number of short cycles. However, cycles become more detrimental when they combine together to form more advanced objects, e.g., absorbing sets, including low-weight codewords. In this paper, we show how MD relocations can be exploited to minimize the number of detrimental objects in the graph of an MD code. Moreover, we demonstrate the savings in the number of relocation arrangements earned by focusing on objects rather than cycles. Our technique is applicable to a wide variety of one-dimensional (OD) codes. Simulation results reveal significant lifetime gains in practical Flash systems achieved by MD codes designed using our technique compared with OD codes having similar parameters.

Spatially-coupled (SC) codes are a family of graph-based codes that have attracted significant attention, thanks to their capacity approaching performance and low decoding latency. An SC code is constructed by partitioning an underlying block code into a number of components and coupling their copies together. In this paper, we first introduce a general approach for the enumeration of detrimental combinatorial objects in the graph of finite-length SC codes. Our approach is general in the sense that it effectively works for SC codes with various partitioning schemes, column weights, and memories. Next, we present a two-stage framework for the construction of high performance binary SC codes optimized for the additive white Gaussian noise channels; we aim at minimizing the number of detrimental combinatorial objects in the error floor region. In the first stage, we deploy a novel partitioning scheme, called the optimal overlap partitioning, to produce the optimal partitioning corresponding to the smallest number of detrimental objects. In the second stage, we apply a new circulant power optimizer to further reduce the number of detrimental objects in the lifted graph. SC codes constructed by our new framework have up to two orders of magnitude error floor performance improvement and up to 0.6 dB SNR gain compared to prior state-of-the-art SC codes.

Non-binary (NB) low-density parity-check (LDPC) codes are graph-based codes that are increasingly being considered as a powerful error correction tool for modern dense storage devices. Optimizing NB-LDPC codes to overcome their error floor is one of the main code design challenges facing storage engineers upon deploying such codes in practice. Furthermore, the increasing levels of asymmetry incorporated by the channels underlying modern dense storage systems, e.g., multi-level Flash systems, exacerbate the error floor problem by widening the spectrum of problematic objects that contribute to the error floor of an NB-LDPC code. In a recent research, the weight consistency matrix (WCM) framework was introduced as an effective combinatorial NB-LDPC code optimization methodology that is suitable for modern Flash memory and magnetic recording (MR) systems. The WCM framework was used to optimize codes for asymmetric Flash channels, MR channels that have intrinsic memory, in addition to canonical symmetric additive white Gaussian noise channels. In this paper, we provide an in-depth theoretical analysis needed to understand and properly apply the WCM framework. We focus on general absorbing sets of type two (GASTs) as the detrimental objects of interest. In particular, we introduce a novel tree representation of a GAST called the unlabeled GAST tree, using which we prove that the WCM framework is optimal in the sense that it operates on the minimum number of matrices, which are the WCMs, to remove a GAST. Then, we enumerate WCMs and demonstrate the significance of the savings achieved by the WCM framework in the number of matrices processed to remove a GAST. Moreover, we provide a linear-algebraic analysis of the null spaces of WCMs associated with a GAST. We derive the minimum number of edge weight changes needed to remove a GAST via its WCMs, along with how to choose these changes. In addition, we propose a new set of problematic objects, namely oscillating sets of type two (OSTs), which contribute to the error floor of NB-LDPC codes with even column weights on asymmetric channels, and we show how to customize the WCM framework to remove OSTs. We also extend the domain of the WCM framework applications by demonstrating its benefits in optimizing column weight 5 codes, codes used over Flash channels with additional soft information, and spatially coupled codes. The performance gains achieved via the WCM framework range between 1 and nearly 2.5 orders of magnitude in the error floor region over interesting channels.

Spatially-coupled (SC) codes are among the most attractive error-correcting codes for use in modern storage devices. SC codes are constructed by partitioning an underlying block code and coupling the partitioned components. Here, we focus on circulant-based SC codes. Recently, the optimal overlap (OO), circulant power optimizer (CPO) approach was introduced to construct high performance SC codes for AWGN and Flash channels. The OO partitioning stage operates on the protograph of the SC code, while the CPO optimizes the circulant powers, in order to minimize the number of detrimental objects. Since the nature of detrimental objects in the graph of a code critically depends on the characteristics of the channel of interest, extending the OO-CPO approach to construct SC codes for channels with intrinsic memory is not a straightforward task. In this paper, we tackle one relevant extension; we construct high performance SC codes for practical 1-D magnetic recording channels, i.e., partial-response (PR) channels. Via combinatorial techniques, we carefully build and solve the optimization problem of the OO partitioning, focusing on the objects of interest in the case of PR channels. Then, we customize the CPO to further reduce the number of these objects in the graph of the code. SC codes designed using the OO-CPO approach for PR channels outperform prior state-of-the-art SC codes by around 3 orders of magnitude in FER and 1.1 dB in SNR, and more intriguingly, outperform structured block codes of the same length by around 1.6 orders of magnitude in FER and 0.4 dB in SNR.

Spatially-coupled (SC) codes are a family of graph-based codes that have attracted significant attention thanks to their capacity approaching performance. An SC code is constructed by partitioning an underlying block code into a number of components, and coupling their copies together. The number of components is determined by the memory parameter. In this paper, we study a finite length construction for the circulant-based SC codes. We introduce a new partitioning scheme, which we call minimum overlap partitioning, that outperforms previous methods. We also present a general approach for the enumeration of problematic objects in the error-floor regime that can be applied to any circulant-based SC code and to a variety of partitioning schemes. Compared to the uncoupled block codes, an SC code constructed by the new approach has more than 1.5 and 3 orders of magnitude performance improvement for the memory 1 and 2, respectively. Additionally, it outperforms the existing method of partitioning via cutting vectors by at least half an order of magnitude; this performance advantage becomes more pronounced for SC codes with higher memories.

Modern dense Flash memory devices operate at very low error rates, which require powerful error correcting coding (ECC) techniques. An emerging class of graph-based ECC techniques that has broad applications is the class of spatially-coupled (SC) codes, where a block code is partitioned into components that are then rewired multiple times to construct an SC code. Here, our focus is on SC codes with the underlying circulant-based structure. In this paper, we present a three-stage approach for the design of high performance non-binary SC (NB-SC) codes optimized for practical Flash channels; we aim at minimizing the number of detrimental general absorbing sets of type two (GASTs) in the graph of the designed NB-SC code. In the first stage, we deploy a novel partitioning mechanism, called the optimal overlap partitioning, which acts on the protograph of the SC code to produce optimal partitioning corresponding to the smallest number of detrimental objects. In the second stage, we apply a new circulant power optimizer to further reduce the number of detrimental GASTs. In the third stage, we use the weight consistency matrix framework to manipulate edge weights to eliminate as many as possible of the GASTs that remain in the NB-SC code after the first two stages (that operate on the unlabeled graph of the code). Simulation results reveal that NB-SC codes designed using our approach outperform state-of-the-art NB-SC codes when used over Flash channels.

Transmission channels underlying modern memory systems, e.g., Flash memories, possess a significant amount of asymmetry. While existing LDPC codes optimized for symmetric, AWGN-like channels are being actively considered for Flash applications, we demonstrate that, due to channel asymmetry, such approaches are fairly inadequate. We propose a new, general, combinatorial framework for the analysis and design of non-binary LDPC (NB-LDPC) codes for asymmetric channels. We introduce a refined definition of absorbing sets, which we call general absorbing sets (GASs), and an important subclass of GASs, which we refer to as general absorbing sets of type two (GASTs). Additionally, we study the combinatorial properties of GASTs. We then present the weight consistency matrix (WCM), which succinctly captures key properties in a GAST. Based on these new concepts, we then develop a general code optimization framework, and demonstrate its effectiveness on the realistic highly-asymmetric normal-Laplace mixture (NLM) Flash channel. Our optimized codes enjoy over one order (resp., half of an order) of magnitude performance gain in the uncorrectable BER (UBER) relative to the unoptimized codes (resp. the codes optimized for symmetric channels).

Transmission channels underlying modern dense storage systems, e.g., Flash memory and magnetic recording (MR) systems, significantly differ from canonical channels, like additive white Gaussian noise (AWGN) channels. While existing low-density parity-check (LDPC) codes optimized for symmetric, AWGN-like channels are being actively considered for Flash applications, we demonstrate that, due to channel asymmetry, such approaches are inadequate. We introduce a refined definition of absorbing sets, which we callgeneral absorbing sets of type two (GASTs), and study the combinatorial properties of GASTs. We then present theweight consistency matrix (WCM), which succinctly captures key properties in a GAST. Furthermore, we show how to customize the WCM definition such that it suits other special subclasses of GASTs. Based on these new concepts, we then develop a new, general combinatorial code optimization framework, which we call theWCM framework, and demonstrate its effectiveness on the realistic highly-asymmetric normal-Laplace mixture (NLM) Flash channel. Moreover, we show that our framework can be customized to optimize non-binary LDPC (NB-LDPC) codes for other asymmetric channels, channels with memory (incorporated in MR systems), and canonical symmetric channels. For all the channels we have simulated NB-LDPC codes over, the codes optimized using the WCM framework enjoy at least 1 order, and up to nearly 2 orders of magnitude performance gain in the uncorrectable bit error rate (UBER) or the frame error rate (FER) relative to the unoptimized codes. Our simulations also show that codes optimized for symmetric channels are not the best choice for asymmetric channels.

In this paper, we provide a comprehensive analysis of the error floor along with code optimization guidelines for structured and regular non-binary low-density parity-check (NB-LDPC) codes in magnetic recording (MR) applications. While the topic of the error floor performance of binary LDPC codes over additive white Gaussian noise (AWGN) channels has recently received considerable attention, very little is known about the error floor performance of NB-LDPC codes over other types of channels, despite the early results demonstrating superior characteristics of NB-LDPC codes relative to their binary counterparts. We first show that, due to the outer looping between the detector and the decoder in the receiver, the error profile of NB-LDPC codes over partial-response (PR) channels is qualitatively different from the error profile over AWGN channels—this observation motivates us to introduce new combinatorial objects aimed at capturing decoding errors that dominate the PR channel error floor region. We call these objects balanced absorbing sets (BASs), which are viewed as a special subclass of previously introduced absorbing sets (ASs). Aided by these new objects (BASs), we develop a method that combines analytical equations and biased simulations to predict the error floor performance of NB-LDPC codes over PR channels without the need to execute extensive Monte Carlo (MC) simulations. We show that explicitly incorporating the inter-symbol interference of MR channels into our prediction method makes the accuracy of the error floor estimate within 0.2 of an order of magnitude from the traditional MC simulation. In addition, we prove that, due to the more restrictive definition of BASs (relative to the more general class of ASs), an additional degree of freedom can be exploited in the code design for PR channels. We then demonstrate that the proposed code optimization aimed at removing dominant BASs offers performance improvements in the frame error rate in the error floor region by up to 2.5 orders of magnitude over the unoptimized designs. Our code optimization technique carefully, yet provably, removes BASs from the code while preserving its overall structure (node degree, quasi-cyclic property, regularity, and so forth). The resulting codes outperform the existing binary and NB-LDPC solutions for PR channels by about 2.5 and 1.25 orders of magnitude, respectively.

In this paper, we analyze and optimize non- binary low-density parity-check (NB-LDPC) codes for magnetic recording applications. While the topic of the error floor performance of binary LDPC codes over additive white Gaussian noise (AWGN) channels has recently received considerable attention, very little is known about the error floor performance of NB-LDPC codes over other types of channels, despite the early results demonstrating superior characteristics of NB-LDPC codes relative to their binary counterparts. We first show that, due to outer looping between detector and decoder in the receiver, the error profile of NB-LDPC codes over partial-response (PR) channels is qualitatively different from the error profile over AWGN channels - this observation motivates us to introduce new combinatorial definitions aimed at capturing decoding errors that dominate PR channel error floor region. We call these errors (or objects) balanced absorbing sets (BASs), which are viewed as a special subclass of previously introduced absorbing sets (ASs). Additionally, we prove that due to the more restrictive definition of BASs (relative to the more general class of ASs), an additional degree of freedom can be exploited in code design for PR channels. We then demonstrate that the proposed code optimization aimed at removing dominant BASs offers improvements in the frame error rate (FER) in the error floor region by up to 2.5 orders of magnitude over the uninformed designs. Our code optimization technique carefully yet provably removes BASs from the code while preserving its overall structure (node degree, quasi-cyclic property, regularity, etc.). The resulting codes outperform existing binary and NB-LDPC solutions for PR channels by about 2.5 and 1.5 orders of magnitude, respectively.

With the growing importance of error correction in different communication systems, using an efficient and easily implementable code is always appreciated. One of the most important codes is the low‐density parity‐check (LDPC) code. Two main iterative decoding algorithms are usually used, namely the sum‐product (SP) algorithm (also referred to as belief propagation) and the min–sum (MS). The SP algorithm is more accurate but suffers from very high complexity. On the other hand, the MS algorithm has a much lower complexity at the expense of some performance degradation. To handle this performance degradation, many algorithms were presented in the literature as improvements for the MS, like the scaled MS and the offset MS. However, all those improved algorithms are more complex than the traditional MS. In this study, an efficient and low complexity LDPC decoding algorithm, called selective max–min (SMM), is proposed. The SMM performance is closer to SP than to MS as long as the average number of ones per column in the parity check matrix is around or less than 4 (which is the case for most of the communication systems using LDPC). On the other hand, the SMM exhibits only a minor complexity increase over traditional MS making it suitable for practical implementation.