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Paul H. Siegel

Publications and source records attributed to Paul H. Siegel.

At least 55 records · Page 3Linked to original sources
arXiv2013

Quantized Iterative Message Passing Decoders with Low Error Floor for LDPC Codes

The error floor phenomenon observed with LDPC codes and their graph-based, iterative, message-passing (MP) decoders is commonly attributed to the existence of error-prone substructures -- variously referred to as near codewords, trapping sets, absorbing sets, or pseudocodewords -- in a Tanner graph representation of the code. Many approaches have been proposed to lower the error floor by designing new LDPC codes with fewer such substructures or by modifying the decoding algorithm. Using a theoretical analysis of iterative MP decoding in an idealized trapping set scenario, we show that a contributor to the error floors observed in the literature may be the imprecise implementation of decoding algorithms and, in particular, the message quantization rules used. We then propose a new quantization method -- (q+1)-bit quasi-uniform quantization -- that efficiently increases the dynamic range of messages, thereby overcoming a limitation of conventional quantization schemes. Finally, we use the quasi-uniform quantizer to decode several LDPC codes that suffer from high error floors with traditional fixed-point decoder implementations. The performance simulation results provide evidence that the proposed quantization scheme can, for a wide variety of codes, significantly lower error floors with minimal increase in decoder complexity.

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arXiv2013

Error Floor Approximation for LDPC Codes in the AWGN Channel

This paper addresses the prediction of error floors of low-density parity-check (LDPC) codes with variable nodes of constant degree in the additive white Gaussian noise (AWGN) channel. Specifically, we focus on the performance of the sum-product algorithm (SPA) decoder formulated in the log-likelihood ratio (LLR) domain. We hypothesize that several published error floor levels are due to the manner in which decoder implementations handled the LLRs at high SNRs. We employ an LLR-domain SPA decoder that does not saturate near-certain messages and find the error rates of our decoder to be lower by at least several orders of magnitude. We study the behavior of trapping sets (or near-codewords) that are the dominant cause of the reported error floors. We develop a refined linear model, based on the work of Sun and others, that accurately predicts error floors caused by elementary tapping sets for saturating decoders. Performance results of several codes at several levels of decoder saturation are presented.

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arXiv2013

Generalized Sharp Bounds on the Spectral Radius of Digraphs

The spectral radius ρ(G) of a digraph G is the maximum modulus of the eigenvalues of its adjacency matrix. We present bounds on ρ(G) that are often tighter and are applicable to a larger class of digraphs than previously reported bounds. Calculating the final bound pair is particularly suited to sparse digraphs. For strongly connected digraphs, we derive equality conditions for the bounds, relating to the outdegree regularity of the digraph. We also prove that the bounds hold with equality only if ρ(G) is the r-th root of an integer, where r divides the index of imprimitivity of G.

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arXiv2013

Bounds on the Minimum Distance of Punctured Quasi-Cyclic LDPC Codes

Recent work by Divsalar et al. has shown that properly designed protograph-based low-density parity-check (LDPC) codes typically have minimum (Hamming) distance linearly increasing with block length. This fact rests on ensemble arguments over all possible expansions of the base protograph. However, when implementation complexity is considered, the expansions are frequently selected from a smaller class of structured expansions. For example, protograph expansion by cyclically shifting connections generates a quasi-cyclic (QC) code. Other recent work by Smarandache and Vontobel has provided upper bounds on the minimum distance of QC codes. In this paper, we generalize these bounds to punctured QC codes and then show how to tighten these for certain classes of codes. We then evaluate these upper bounds for the family of protograph codes known as AR4JA codes that have been recommended for use in deep space communications in a standard established by the Consultative Committee for Space Data Systems (CCSDS). At block lengths larger than 4400 bits, these upper bounds fall well below the ensemble lower bounds.

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arXiv2013

Perspectives on Balanced Sequences

We examine and compare several different classes of "balanced" block codes over q-ary alphabets, namely symbol-balanced (SB) codes, charge-balanced (CB) codes, and polarity-balanced (PB) codes. Known results on the maximum size and asymptotic minimal redundancy of SB and CB codes are reviewed. We then determine the maximum size and asymptotic minimal redundancy of PB codes and of codes which are both CB and PB. We also propose efficient Knuth-like encoders and decoders for all these types of balanced codes.

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arXiv2012

Windowed Decoding of Spatially Coupled Codes

Spatially coupled codes have been of interest recently owing to their superior performance over memoryless binary-input channels. The performance is good both asymptotically, since the belief propagation thresholds approach capacity, as well as for finite lengths, since degree-2 variables that result in high error floors can be completely avoided. However, to realize the promised good performance, one needs large blocklengths. This in turn implies a large latency and decoding complexity. For the memoryless binary erasure channel, we consider the decoding of spatially coupled codes through a windowed decoder that aims to retain many of the attractive features of belief propagation, while trying to reduce complexity further. We characterize the performance of this scheme by defining thresholds on channel erasure rates that guarantee a target erasure rate. We give analytical lower bounds on these thresholds and show that the performance approaches that of belief propagation exponentially fast in the window size. We give numerical results including the thresholds computed using density evolution and the erasure rate curves for finite-length spatially coupled codes.

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arXiv2012

Rewriting Codes for Flash Memories

Flash memory is a non-volatile computer memory comprising blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes - known as floating codes (or flash codes) and buffer codes - have been designed in order to maximize the number of times that information stored in a flash memory can be written (and re-written) prior to incurring a block erasure. An (n,k,t)q flash code C is a coding scheme for storing k information bits in $n$ cells in such a way that any sequence of up to t writes can be accommodated without a block erasure. The total number of available level transitions in n cells is n(q-1), and the write deficiency of C, defined as δ(C) = n(q-1)-t, is a measure of how close the code comes to perfectly utilizing all these transitions. In this paper, we show a construction of flash codes with write deficiency O(qk\log k) if q \geq \log_2k, and at most O(k\log^2 k) otherwise. An (n,r,\ell,t)q buffer code is a coding scheme for storing a buffer of r \ell-ary symbols such that for any sequence of t symbols it is possible to successfully decode the last r symbols that were written. We improve upon a previous upper bound on the maximum number of writes t in the case where there is a single cell to store the buffer. Then, we show how to improve a construction by Jiang et al. that uses multiple cells, where n\geq 2r.

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arXiv2012

Numerical Issues Affecting LDPC Error Floors

Numerical issues related to the occurrence of error floors in floating-point simulations of belief propagation (BP) decoders are examined. Careful processing of messages corresponding to highly-certain bit values can sometimes reduce error floors by several orders of magnitude. Computational solutions for properly handling such messages are provided for the sum-product algorithm (SPA) and several variants.

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arXiv2012

Time-Space Constrained Codes for Phase-Change Memories

Phase-change memory (PCM) is a promising non-volatile solid-state memory technology. A PCM cell stores data by using its amorphous and crystalline states. The cell changes between these two states using high temperature. However, since the cells are sensitive to high temperature, it is important, when programming cells, to balance the heat both in time and space. In this paper, we study the time-space constraint for PCM, which was originally proposed by Jiang et al. A code is called an \emph{$(α,β,p)$-constrained code} if for any $α$ consecutive rewrites and for any segment of $β$ contiguous cells, the total rewrite cost of the $β$ cells over those $α$ rewrites is at most $p$. Here, the cells are binary and the rewrite cost is defined to be the Hamming distance between the current and next memory states. First, we show a general upper bound on the achievable rate of these codes which extends the results of Jiang et al. Then, we generalize their construction for $(α\geq 1, β=1,p=1)$-constrained codes and show another construction for $(α= 1, β\geq 1,p\geq1)$-constrained codes. Finally, we show that these two constructions can be used to construct codes for all values of $α$, $β$, and $p$.

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arXiv2012

Adaptive Cut Generation Algorithm for Improved Linear Programming Decoding of Binary Linear Codes

Linear programming (LP) decoding approximates maximum-likelihood (ML) decoding of a linear block code by relaxing the equivalent ML integer programming (IP) problem into a more easily solved LP problem. The LP problem is defined by a set of box constraints together with a set of linear inequalities called "parity inequalities" that are derived from the constraints represented by the rows of a parity-check matrix of the code and can be added iteratively and adaptively. In this paper, we first derive a new necessary condition and a new sufficient condition for a violated parity inequality constraint, or "cut," at a point in the unit hypercube. Then, we propose a new and effective algorithm to generate parity inequalities derived from certain additional redundant parity check (RPC) constraints that can eliminate pseudocodewords produced by the LP decoder, often significantly improving the decoder error-rate performance. The cut-generating algorithm is based upon a specific transformation of an initial parity-check matrix of the linear block code. We also design two variations of the proposed decoder to make it more efficient when it is combined with the new cut-generating algorithm. Simulation results for several low-density parity-check (LDPC) codes demonstrate that the proposed decoding algorithms significantly narrow the performance gap between LP decoding and ML decoding.

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arXiv2011

Enhancing Binary Images of Non-Binary LDPC Codes

We investigate the reasons behind the superior performance of belief propagation decoding of non-binary LDPC codes over their binary images when the transmission occurs over the binary erasure channel. We show that although decoding over the binary image has lower complexity, it has worse performance owing to its larger number of stopping sets relative to the original non-binary code. We propose a method to find redundant parity-checks of the binary image that eliminate these additional stopping sets, so that we achieve performance comparable to that of the original non-binary LDPC code with lower decoding complexity.

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arXiv2010

Write Channel Model for Bit-Patterned Media Recording

We propose a new write channel model for bit-patterned media recording that reflects the data dependence of write synchronization errors. It is shown that this model accommodates both substitution-like errors and insertion-deletion errors whose statistics are determined by an underlying channel state process. We study information theoretic properties of the write channel model, including the capacity, symmetric information rate, Markov-1 rate and the zero-error capacity.

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arXiv2010

On Distance Properties of Quasi-Cyclic Protograph-Based LDPC Codes

Recent work has shown that properly designed protograph-based LDPC codes may have minimum distance linearly increasing with block length. This notion rests on ensemble arguments over all possible expansions of the base protograph. When implementation complexity is considered, the expansion is typically chosen to be quite orderly. For example, protograph expansion by cyclically shifting connections creates a quasi-cyclic (QC) code. Other recent work has provided upper bounds on the minimum distance of QC codes. In this paper, these bounds are expanded upon to cover puncturing and tightened in several specific cases. We then evaluate our upper bounds for the most prominent protograph code thus far, one proposed for deep-space usage in the CCSDS experimental standard, the code known as AR4JA.

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arXiv2009

Storage Coding for Wear Leveling in Flash Memories

Flash memory is a non-volatile computer memory comprised of blocks of cells, wherein each cell is implemented as either NAND or NOR floating gate. NAND flash is currently the most widely used type of flash memory. In a NAND flash memory, every block of cells consists of numerous pages; rewriting even a single page requires the whole block to be erased and reprogrammed. Block erasures determine both the longevity and the efficiency of a flash memory. Therefore, when data in a NAND flash memory are reorganized, minimizing the total number of block erasures required to achieve the desired data movement is an important goal. This leads to the flash data movement problem studied in this paper. We show that coding can significantly reduce the number of block erasures required for data movement, and present several optimal or nearly optimal data-movement algorithms based upon ideas from coding theory and combinatorics. In particular, we show that the sorting-based (non-coding) schemes require at least O(nlogn) erasures to move data among n blocks, whereas coding-based schemes require only O(n) erasures. Furthermore, coding-based schemes use only one auxiliary block, which is the best possible, and achieve a good balance between the number of erasures in each of the n+1 blocks.

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arXiv2009

A Nearly Optimal Construction of Flash Codes

Flash memory is a non-volatile computer memory comprised of blocks of cells, wherein each cell can take on q different values or levels. While increasing the cell level is easy, reducing the level of a cell can be accomplished only by erasing an entire block. Since block erasures are highly undesirable, coding schemes - known as floating codes or flash codes - have been designed in order to maximize the number of times that information stored in a flash memory can be written (and re-written) prior to incurring a block erasure. An (n,k,t)_q flash code C is a coding scheme for storing k information bits in n cells in such a way that any sequence of up to t writes (where a write is a transition 0 -> 1 or 1 -> 0 in any one of the k bits) can be accommodated without a block erasure. The total number of available level transitions in n cells is n(q-1), and the write deficiency of C, defined as δ(C) = n(q-1) - t, is a measure of how close the code comes to perfectly utilizing all these transitions. For k > 6 and large n, the best previously known construction of flash codes achieves a write deficiency of O(qk^2). On the other hand, the best known lower bound on write deficiency is Ω(qk). In this paper, we present a new construction of flash codes that approaches this lower bound to within a factor logarithmic in k. To this end, we first improve upon the so-called "indexed" flash codes, due to Jiang and Bruck, by eliminating the need for index cells in the Jiang-Bruck construction. Next, we further increase the number of writes by introducing a new multi-stage (recursive) indexing scheme. We then show that the write deficiency of the resulting flash codes is O(qk\log k) if q \geq \log_2k, and at most O(k\log^2 k) otherwise.

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arXiv2009

Multidimensional Flash Codes

Flash memory is a non-volatile computer memory comprised of blocks of cells, wherein each cell can take on q different levels corresponding to the number of electrons it contains. Increasing the cell level is easy; however, reducing a cell level forces all the other cells in the same block to be erased. This erasing operation is undesirable and therefore has to be used as infrequently as possible. We consider the problem of designing codes for this purpose, where k bits are stored using a block of n cells with q levels each. The goal is to maximize the number of bit writes before an erase operation is required. We present an efficient construction of codes that can store an arbitrary number of bits. Our construction can be viewed as an extension to multiple dimensions of the earlier work of Jiang and Bruck, where single-dimensional codes that can store only 2 bits were proposed.

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arXiv2009

Efficient implementation of linear programming decoding

While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.

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