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Paul H. Siegel

Publications and source records attributed to Paul H. Siegel.

At least 37 records · Page 2Linked to original sources
arXiv2020

Coding over Sets for DNA Storage

In this paper we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where a data set is represented by an unordered set of $M$ sequences, each of length $L$. Errors within that model are a loss of whole sequences and point errors inside the sequences, such as insertions, deletions and substitutions. We derive Gilbert-Varshamov lower bounds and sphere packing upper bounds on achievable cardinalities of error-correcting codes within this storage model. We further propose explicit code constructions than can correct errors in such a storage system that can be encoded and decoded efficiently. Comparing the sizes of these codes to the upper bounds, we show that many of the constructions are close to optimal.

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arXiv2020

Rate-Constrained Shaping Codes for Structured Sources

Shaping codes are used to encode information for use on channels with cost constraints. Applications include data transmission with a power constraint and, more recently, data storage on flash memories with a constraint on memory cell wear. In the latter application, system requirements often impose a rate constraint. In this paper, we study rate-constrained fixed-to-variable length shaping codes for noiseless, memoryless costly channels and general i.i.d. sources. The analysis relies on the theory of word-valued sources. We establish a relationship between the code expansion factor and minimum average symbol cost. We then determine the expansion factor that minimizes the average cost per source symbol (total cost), corresponding to a conventional optimal source code with cost. An equivalence is established between codes minimizing average symbol cost and codes minimizing total cost, and a separation theorem is proved, showing that optimal shaping can be achieved by a concatenation of optimal compression and optimal shaping for a uniform i.i.d. source. Shaping codes often incorporate, either explicitly or implicitly, some form of non-equiprobable signaling. We use our results to further explore the connections between shaping codes and codes that map a sequence of i.i.d. source symbols into an output sequence of symbols that are approximately independent and distributed according to a specified target distribution, such as distribution matching (DM) codes. Optimal DM codes are characterized in terms of a new performance measure - generalized expansion factor (GEF) - motivated by the costly channel perspective. The GEF is used to study DM codes that minimize informational divergence and normalized informational divergence.

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arXiv2019

Anchor-Based Correction of Substitutions in Indexed Sets

Motivated by DNA-based data storage, we investigate a system where digital information is stored in an unordered set of several vectors over a finite alphabet. Each vector begins with a unique index that represents its position in the whole data set and does not contain data. This paper deals with the design of error-correcting codes for such indexed sets in the presence of substitution errors. We propose a construction that efficiently deals with the challenges that arise when designing codes for unordered sets. Using a novel mechanism, called anchoring, we show that it is possible to combat the ordering loss of sequences with only a small amount of redundancy, which allows to use standard coding techniques, such as tensor-product codes to correct errors within the sequences. We finally derive upper and lower bounds on the achievable redundancy of codes within the considered channel model and verify that our construction yields a redundancy that is close to the best possible achievable one. Our results surprisingly indicate that it requires less redundancy to correct errors in the indices than in the data part of vectors.

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arXiv2018

Coding over Sets for DNA Storage

In this paper, we study error-correcting codes for the storage of data in synthetic deoxyribonucleic acid (DNA). We investigate a storage model where data is represented by an unordered set of $M$ sequences, each of length $L$. Errors within that model are losses of whole sequences and point errors inside the sequences, such as substitutions, insertions and deletions. We propose code constructions which can correct these errors with efficient encoders and decoders. By deriving upper bounds on the cardinalities of these codes using sphere packing arguments, we show that many of our codes are close to optimal.

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arXiv2017

Multi-Erasure Locally Recoverable Codes Over Small Fields

Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for storage arrays. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. We first develop upper and lower bounds on the minimum distance of ME-LRCs. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting properties. A decoding algorithm tailored for erasure recovery is given, and correctable erasure patterns are identified. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters, and present some explicit ME-LRCs over small fields. Finally, we show that generalized integrated interleaving (GII) codes can be treated as a subclass of generalized tensor product codes, thus defining the exact relation between these codes.

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arXiv2017

Syndrome-Coupled Rate-Compatible Error-Correcting Codes

Rate-compatible error-correcting codes (ECCs), which consist of a set of extended codes, are of practical interest in both wireless communications and data storage. In this work, we first study the lower bounds for rate-compatible ECCs, thus proving the existence of good rate-compatible codes. Then, we propose a general framework for constructing rate-compatible ECCs based on cosets and syndromes of a set of nested linear codes. We evaluate our construction from two points of view. From a combinatorial perspective, we show that we can construct rate-compatible codes with increasing minimum distances. From a probabilistic point of view, we prove that we are able to construct capacity-achieving rate-compatible codes.

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arXiv2017

Performance of Optimal Data Shaping Codes

Data shaping is a coding technique that has been proposed to increase the lifetime of flash memory devices. Several data shaping codes have been described in recent work, including endurance codes and direct shaping codes for structured data. In this paper, we study information-theoretic properties of a general class of data shaping codes and prove a separation theorem stating that optimal data shaping can be achieved by the concatenation of optimal lossless compression with optimal endurance coding. We also determine the expansion factor that minimizes the total wear cost. Finally, we analyze the performance of direct shaping codes and establish a condition for their optimality.

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arXiv2017

Multi-Erasure Locally Recoverable Codes Over Small Fields For Flash Memory Array

Erasure codes play an important role in storage systems to prevent data loss. In this work, we study a class of erasure codes called Multi-Erasure Locally Recoverable Codes (ME-LRCs) for flash memory array. Compared to previous related works, we focus on the construction of ME-LRCs over small fields. We first develop upper and lower bounds on the minimum distance of ME-LRCs. These bounds explicitly take the field size into account. Our main contribution is to propose a general construction of ME-LRCs based on generalized tensor product codes, and study their erasure-correcting property. A decoding algorithm tailored for erasure recovery is given. We then prove that our construction yields optimal ME-LRCs with a wide range of code parameters. Finally, we present several families of ME-LRCs over different fields.

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arXiv2016

On the Capacity of the Beta-Binomial Channel Model for Multi-Level Cell Flash Memories

The beta-binomial (BBM) channel model was recently proposed to model the overdispersed statistics of empirically observed bit errors in multi-level cell (MLC) flash memories. In this paper, we study the capacity of the BBM channel model for MLC flash memories. Using the compound channel approach, we first show that the BBM channel model capacity is zero. However, through empirical observation, this appears to be a very pessimistic estimate of the flash memory channel capacity. We propose a refined channel model called the truncated-support beta-binomial (TS-BBM) channel model and derive its capacity. Using empirical error statistics from 1X-nm and 2Y-nm MLC flash memories, we numerically estimate the TS-BBM channel model capacity as a function of the program/erase (P/E) cycling stress. The capacity of the 2-TS-BBM channel model provides an upper bound on the coding rates for the flash memory chip assuming a single binary error correction code is used.

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arXiv2016

Channel Models for Multi-Level Cell Flash Memories Based on Empirical Error Analysis

We propose binary discrete parametric channel models for multi-level cell (MLC) flash memories that provide accurate ECC performance estimation by modeling the empirically observed error characteristics under program/erase (P/E) cycling stress. Through a detailed empirical error characterization of 1X-nm and 2Y-nm MLC flash memory chips from two different vendors, we observe and characterize the overdispersion phenomenon in the number of bit errors per ECC frame. A well studied channel model such as the binary asymmetric channel (BAC) model is unable to provide accurate ECC performance estimation. Hence we propose a channel model based on the beta-binomial probability distribution (2-BBM channel model) which is a good fit for the overdispersed empirical error characteristics and show through statistical tests and simulation results for BCH, LDPC and polar codes, that the 2-BBM channel model provides accurate ECC performance estimation in MLC flash memories.

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arXiv2016

Performance of Multilevel Flash Memories with Different Binary Labelings: A Multi-User Perspective

In this work, we study the performance of different decoding schemes for multilevel flash memories where each page in every block is encoded independently. We focus on the multi-level cell (MLC) flash memory, which is modeled as a two-user multiple access channel suffering from asymmetric noise. The uniform rate regions and sum rates of Treating Interference as Noise (TIN) decoding and Successive Cancelation (SC) decoding are investigated for a Program/Erase (P/E) cycling model and a data retention model. We examine the effect of different binary labelings of the cell levels, as well as the impact of further quantization of the memory output (i.e., additional read thresholds). Finally, we extend our analysis to the three-level cell (TLC) flash memory.

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arXiv2016

Multihead Multitrack Detection with Reduced-State Sequence Estimation

To achieve ultra-high storage capacity, the data tracks are squeezed more and more on the magnetic recording disks, causing severe intertrack interference (ITI). The multihead multitrack (MHMT) detector is proposed to better combat ITI. Such a detector, however, has prohibitive implementation complexity. In this paper we propose to use the reduced-state sequence estimation (RSSE) algorithm to significantly reduce the complexity, and render MHMT practical. We first consider a commonly used symmetric two-head two-track (2H2T) channel model. The effective distance between two input symbols is redefined. It provides a better distance measure and naturally leads to an unbalanced set partition tree. Different trellis configurations are obtained based on the desired performance/complexity tradeoff. Simulation results show that the reduced MHMT detector can achieve near maximum-likelihood (ML) performance with a small fraction of the original number of trellis states. Error event analysis is given to explain the behavior of RSSE algorithm on 2H2T channel. Search results of dominant RSSE error events for different channel targets are presented. We also study an asymmetric 2H2T system. The simulation results and error event analysis show that RSSE is applicable to the asymmetric channel.

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arXiv2016

Multihead Multitrack Detection with ITI Estimation in Next Generation Magnetic Recording System

Multitrack detection with array-head reading is a promising technique proposed for next generation magnetic storage systems. The multihead multitrack (MHMT) system is characterized by intersymbol interference (ISI) in the downtrack direction and intertrack interference (ITI) in the crosstrack direction. Constructing the trellis of a MHMT maximum likelihood (ML) detector requires knowledge of the ITI, which is generally unknown at the receiver. In addition, to retain efficiency, the ML detector requires a static estimate of the ITI, whose true value may in reality vary. In this paper we propose a modified ML detector on the $n$-head, $n$-track ($n$H$n$T) channel which could efficiently track the change of ITI, and adapt to new estimates. The trellis used in the proposed detector is shown to be independent of the ITI level. A gain loop structure is used to estimate the ITI. Simulation results show that the proposed detector offers a performance advantage in settings where complexity constraints limit the traditional ML detector to use a static ITI estimate.

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arXiv2015

Binary Linear Locally Repairable Codes

Locally repairable codes (LRCs) are a class of codes designed for the local correction of erasures. They have received considerable attention in recent years due to their applications in distributed storage. Most existing results on LRCs do not explicitly take into consideration the field size $q$, i.e., the size of the code alphabet. In particular, for the binary case, only a few results are known. In this work, we present an upper bound on the minimum distance $d$ of linear LRCs with availability, based on the work of Cadambe and Mazumdar. The bound takes into account the code length $n$, dimension $k$, locality $r$, availability $t$, and field size $q$. Then, we study binary linear LRCs in three aspects. First, we focus on analyzing the locality of some classical codes, i.e., cyclic codes and Reed-Muller codes, and their modified versions, which are obtained by applying the operations of extend, shorten, expurgate, augment, and lengthen. Next, we construct LRCs using phantom parity-check symbols and multi-level tensor product structure, respectively. Compared to other previous constructions of binary LRCs with fixed locality or minimum distance, our construction is much more flexible in terms of code parameters, and gives various families of high-rate LRCs, some of which are shown to be optimal with respect to their minimum distance. Finally, availability of LRCs is studied. We investigate the locality and availability properties of several classes of one-step majority-logic decodable codes, including cyclic simplex codes, cyclic difference-set codes, and $4$-cycle free regular low-density parity-check (LDPC) codes. We also show the construction of a long LRC with availability from a short one-step majority-logic decodable code.

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arXiv2015

On the Capacity of Channels with Timing Synchronization Errors

We consider a new formulation of a class of synchronization error channels and derive analytical bounds and numerical estimates for the capacity of these channels. For the binary channel with only deletions, we obtain an expression for the symmetric information rate in terms of subsequence weights which reduces to a tight lower bound for small deletion probabilities. We are also able to exactly characterize the Markov-1 rate for the binary channel with only replications. For a channel that introduces deletions as well as replications of input symbols, we design approximating channels that parameterize the state space and show that the information rates of these approximate channels approach that of the deletion-replication channel as the state space grows. For the case of the channel where deletions and replications occur with the same probabilities, a stronger result in the convergence of mutual information rates is shown. The numerous advantages this new formulation presents are explored.

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arXiv2014

LDPC Code Density Evolution in the Error Floor Region

This short paper explores density evolution (DE) for low-density parity-check (LDPC) codes at signal-to-noise-ratios (SNRs) that are significantly above the decoding threshold. The focus is on the additive white Gaussian noise channel and LDPC codes in which the variable nodes have regular degree. Prior work, using DE, produced results in the error floor region which were asymptotic in the belief-propagation decoder's log-likelihood ratio (LLR) values. We develop expressions which closely approximate the LLR growth behavior at moderate LLR magnitudes. We then produce bounds on the mean extrinsic check-node LLR values required, as a function of SNR, such that the growth rate of the LLRs exceeds that of a particular trapping set's internal LLRs such that its error floor contribution may be eliminated. We find that our predictions for the mean LLRs to be accurate in the error floor region, but the predictions for the LLR variance to be lacking beyond several initial iterations.

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arXiv2014

Adaptive Linear Programming Decoding of Polar Codes

Polar codes are high density parity check codes and hence the sparse factor graph, instead of the parity check matrix, has been used to practically represent an LP polytope for LP decoding. Although LP decoding on this polytope has the ML-certificate property, it performs poorly over a BAWGN channel. In this paper, we propose modifications to adaptive cut generation based LP decoding techniques and apply the modified-adaptive LP decoder to short blocklength polar codes over a BAWGN channel. The proposed decoder provides significant FER performance gain compared to the previously proposed LP decoder and its performance approaches that of ML decoding at high SNRs. We also present an algorithm to obtain a smaller factor graph from the original sparse factor graph of a polar code. This reduced factor graph preserves the small check node degrees needed to represent the LP polytope in practice. We show that the fundamental polytope of the reduced factor graph can be obtained from the projection of the polytope represented by the original sparse factor graph and the frozen bit information. Thus, the LP decoding time complexity is decreased without changing the FER performance by using the reduced factor graph representation.

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