Stopping Sets Research Articles

On the Pairwise Error Probability of Linear Programming Decoding on Independent Rayleigh Flat-Fading Channels

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> In this paper, we consider the pairwise error probability (PEP) of a linear programming (LP) decoder for a general binary linear code as formulated by Feldman <etal/> (<emphasis emphasistype="boldital">IEEE Trans. Inf. Theory</emphasis>, Mar. 2005) on an independent (or memoryless) Rayleigh flat-fading channel with coherent detection and perfect channel state information (CSI) at the receiver. Let <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">${\bf H}$</tex></formula></emphasis> be a parity-check matrix of a binary linear code and consider LP decoding based on <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">${\bf H}$</tex></formula></emphasis>. The output of the LP decoder is always a <emphasis emphasistype="boldital">pseudocodeword</emphasis>. We will show that the PEP of decoding to a pseudocodeword <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">${\mmb \omega}$</tex></formula></emphasis> when the all-zero codeword is transmitted on the above-mentioned channel, behaves asymptotically as <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$K({\mmb \omega}) \cdot (E_s/N_0)^{-\vert\chi({\mmb \omega})\vert}$</tex></formula></emphasis>, where <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$\chi({\mmb \omega})$</tex></formula></emphasis> is the support set of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">${\mmb \omega}$</tex></formula></emphasis>, i.e., the set of nonzero coordinates, <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$E_s/N_0$</tex></formula></emphasis> is the average signal-to-noise ratio (SNR), and <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$K({\mmb \omega})$</tex></formula></emphasis> is a constant independent of the SNR. Note that the support set <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">$\chi({\mmb \omega})$</tex></formula></emphasis> of <emphasis emphasistype="italic"><formula formulatype="inline"><tex Notation="TeX">${\mmb \omega}$</tex></formula></emphasis> is a <emphasis emphasistype="italic">stopping set</emphasis>. Thus, the asymptotic decay rate of the error probability with the average SNR is determined by the size of the smallest nonempty stopping set in the Tanner graph of <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">${\bf H}$</tex></formula></emphasis>. As an example, we analyze the well-known <emphasis emphasistype="italic"><formula formulatype="inline"> <tex Notation="TeX">$(155,64)$</tex></formula></emphasis> Tanner code and present performance curves on the independent Rayleigh flat-fading channel. </para>

Eliminating small stopping sets in irregular low-density parity-check codes

We present a novel scheme to eliminate small stopping sets in irregular low-density parity-check (LDPC) codes. By adding several new check nodes and having their edges connected to small stopping sets in the original code, the improved code decreases the number of small stopping sets efficiently. Simulation results show that the proposed scheme decreases the frame error rate (FER) significantly in the error floor region, whereas having almost the same rate as the original code.

Finding All Small Error-Prone Substructures in LDPC Codes

It is proven in this work that it is NP-complete to exhaustively enumerate small error-prone substructures in arbitrary, finite-length low-density parity-check (LDPC) codes. Two error-prone patterns of interest include stopping sets for binary erasure channels (BECs) and trapping sets for general memoryless symmetric channels. Despite the provable hardness of the problem, this work provides an exhaustive enumeration algorithm that is computationally affordable when applied to codes of practical short lengths <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ap 500. By exploiting the sparse connectivity of LDPC codes, the stopping sets of size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">13</i> and the trapping sets of size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">les11</i> can be exhaustively enumerated. The central theorem behind the proposed algorithm is a new provably tight <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">upper</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bound</i> on the error rates of iterative decoding over BECs. Based on a tree-pruning technique, this upper bound can be iteratively sharpened until its asymptotic order equals that of the error floor. This feature distinguishes the proposed algorithm from existing non-exhaustive ones that correspond to finding <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">lower</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">bounds</i> of the error floor. The upper bound also provides a worst case performance guarantee that is crucial to optimizing LDPC codes when the target error rate is beyond the reach of Monte Carlo simulation. Numerical experiments on both randomly and algebraically constructed LDPC codes demonstrate the efficiency of the search algorithm and its significant value for finite-length code optimization.

Construction of Near-Optimum Burst Erasure Correcting Low-Density Parity-Check Codes

In this paper, a simple and effective tool for the design of low-density parity-check (LDPC) codes for iterative correction of bursts of erasures is presented. The design method consists of starting from the parity-check matrix of an LDPC code and developing an optimized parity-check matrix, with the same performance over the memoryless erasure channel, and suitable also for the iterative correction of single erasure bursts. The parity-check matrix optimization is performed by an algorithm called pivot searching and swapping (PSS) algorithm. It executes permutations of carefully chosen columns of the parity-check matrix, after a local analysis of particular variable nodes called stopping set pivots. This algorithm can be in principle applied to any LDPC code. If the input parity-check matrix is designed to achieve a good performance over the memoryless erasure channel, then the code obtained after the application of the algorithm provides a good joint correction of independent erasures and single erasure bursts. Numerical results are provided in order to show the algorithm effectiveness when applied to different categories of LDPC codes.

Burst erasure correcting LDPC codes

In this paper low-density parity-check (LDPC) codes are designed for burst erasure channels. Firstly, lower bounds for the maximum length erasure burst that can always be corrected with message-passing decoding are derived as a function of the parity-check matrix properties. We then show how parity-check matrices for burst erasure correcting LDPC codes can be constructed using superposition, where the burst erasure correcting performance of the resulting codes is derived as a property of the stopping set size of the base matrices and the choice of permutation matrices for the superposition. This result is then used to design both single burst erasure correcting LDPC codes which are also resilient to the presence of random erasures in the received bits and LDPC codes which can correct multiple erasure bursts in the same codeword.

Bachelier-Version of Russian Option with a Finite Time Horizon

We consider an optimal stopping problem for the Russian option in the Bachelier model with a finite time horizon. We obtain an integral equation, which yields us a border between stopping and continuation sets. Also the asymptotic behavior of this border at 0 and infinity is found.

Average Stopping Set Weight Distributions of Redundant Random Ensembles

In this paper, redundant random ensembles are defined and their average stopping set (SS) weight distributions are analyzed. A redundant random ensemble consists of a set of binary matrices with linearly dependent rows. These linearly dependent rows significantly reduce the number of stopping sets (SS) of small size. Upper and lower bounds on the average SS weight distribution of the redundant random ensembles are proved based on a combinatorial argument. Asymptotic forms of these bounds reveal asymptotic behavior of the average SS weight distributions. From these bounds, a tradeoff between the number of redundant rows (corresponding to decoding complexity of belief propagation on binary erasure channel) and the average SS weight distribution (corresponding to decoding performance) can be derived.

Linear Time Encoding of LDPC Codes

In this paper, we propose a linear complexity encoding method for arbitrary LDPC codes. We start from a simple graph-based encoding method ``label-and-decide.'' We prove that the ``label-and-decide'' method is applicable to Tanner graphs with a hierarchical structure--pseudo-trees-- and that the resulting encoding complexity is linear with the code block length. Next, we define a second type of Tanner graphs--the encoding stopping set. The encoding stopping set is encoded in linear complexity by a revised label-and-decide algorithm--the ``label-decide-recompute.'' Finally, we prove that any Tanner graph can be partitioned into encoding stopping sets and pseudo-trees. By encoding each encoding stopping set or pseudo-tree sequentially, we develop a linear complexity encoding method for general LDPC codes where the encoding complexity is proved to be less than $4 \cdot M \cdot (\overline{k} - 1)$, where $M$ is the number of independent rows in the parity check matrix and $\overline{k}$ represents the mean row weight of the parity check matrix.

Design of regular (2,d/sub c/)-LDPC codes over GF(q) using their binary images

In this paper, a method to design regular (2, dc)- LDPC codes over GF(q) with both good waterfall and error floor properties is presented, based on the algebraic properties of their binary image. First, the algebraic properties of rows of the parity check matrix H associated with a code are characterized and optimized to improve the waterfall. Then the algebraic properties of cycles and stopping sets associated with the underlying Tanner graph are studied and linked to the global binary minimum distance of the code. Finally, simulations are presented to illustrate the excellent performance of the designed codes.

A Combined Matrix Ensemble of Low-Density Parity-Check Codes for Correcting a Solid Burst Erasure

A new ensemble of low-density parity-check (LDPC) codes for correcting a solid burst erasure is proposed. This ensemble is an instance of a combined matrix ensemble obtained by concatenating some LDPC matrices. We derive a new bound on the critical minimum span ratio of stopping sets for the proposed code ensemble by modifying the bound for ordinary code ensemble. By calculating this bound, we show that the critical minimum span ratio of stopping sets for the proposed code ensemble is better than that of the conventional one with keeping the same critical exponent of stopping ratio for both ensemble. Furthermore from experimental results, we show that the average minimum span of stopping sets for a solid burst erasure of the proposed codes is larger than that of the conventional ones.

On the Stopping Distance and Stopping Redundancy of Product Codes

Stopping distance and stopping redundancy of product binary linear block codes is studied. The relationship between stopping sets in a few parity-check matrices of a given product code C and those in the parity-check matrices for the component codes is determined. It is shown that the stopping distance of a particular parity-check matrix of C, denoted HP, is equal to the product of the stopping distances of the associated constituent parity-check matrices. Upper bounds on the stopping redundancy of C is derived. For each minimum distance d = 2r, r ≥ 1, a sequence of [n,k,d] optimal stopping redundancy binary codes is given such k/n tends to 1 as n tends to infinity.

Return Vane Installed in Multistage Centrifugal Pump

To optimize the stationary components in the multistage centrifugal pump, the effects of the return vane profile on the performances of the multistage centrifugal pump were investigated experimentally, taking account of the inlet flow conditions for the next stage impeller. The return vane, whose trailing edge is set at the outer wall position of the annular channel downstream of the vane and which discharges the swirl-less flow, gives better pump performances. By equipping such return vane with the swirl stop set from the trailing edge to the main shaft position, the unstable head characteristics can be also suppressed successfully at the lower discharge. Taking the pump performances and the flow conditions into account, the impeller blade was modified so as to get the shock-free condition where the incidence angle is zero at the inlet.

Small stopping sets in Steiner triple systems

The size of the smallest stopping set in a low-density parity check (LDPC) code determines, to an extent, the performance of iterative decoding methods over the binary erasure channel. In an LDPC code arising from a block design, a stopping set is a subset of its blocks with the property that every point belonging to one of the selected blocks belongs to at least two of them. While some Steiner triple systems have no stopping sets of size 5 or less, it is shown that every Steiner triple system has a stopping set of size at most 7. The proof can be viewed as an application of polynomial identities among configuration counts that generalize the family of known linear identities. While no example of a Steiner triple system whose smallest stopping set has size seven is known, it is shown that a partial Steiner triple system on v points with c·v 1.8 triples that has no stopping set of size less than 7 exists.

Stopping Set Analysis of Iterative Row-Column Decoding of Product Codes

In this paper, we introduce stopping sets for iterative row-column decoding of product codes using optimal constituent decoders. When transmitting over the binary erasure channel (BEC), iterative row-column decoding of product codes using optimal constituent decoders will either be successful, or stop in the unique maximum-size stopping set that is contained in the (initial) set of erased positions. Let C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> denote the product code of two binary linear codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> of minimum distances dc and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and second generalized Hamming weights d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ), respectively. We show that the size s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> of the smallest noncode- word stopping set is at least mm(d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ),d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> )) > d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , where the inequality follows from the Griesmer bound. If there are no codewords in C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> with support set S, where S is a stopping set, then S is said to be a noncodeword stopping set. An immediate consequence is that the erasure probability after iterative row-column decoding using optimal constituent decoders of (finite-length) product codes on the BEC, approaches the erasure probability after maximum-likelihood decoding as the channel erasure probability decreases. We also give an explicit formula for the number of noncodeword stopping sets of size s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , which depends only on the first nonzero coefficient of the constituent (row and column) first and second support weight enumerators, for the case when d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ) < 2d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) < 2d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> . Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.

Turbo Codes in Binary Erasure Channel

In this correspondence, the stopping set of turbo codes with iterative decoding in the binary erasure channel is defined. Block and bit erasure probabilities of turbo codes are studied by using the stopping set analysis. It is found that block and bit erasure probabilities of turbo codes with iterative decoding are higher than those with maximum-likelihood decoding, where the differences are negligible in the error floor region. It is shown that in the error floor region, block and bit erasure probabilities of turbo codes with iterative decoding are dominated by small stopping sets and are asymptotically dominated by low weight codewords.

Improved Probabilistic Bounds on Stopping Redundancy

For a linear code C, the stopping redundancy of C is defined as the minimum number of check nodes in a Tanner graph T for C such that the size of the smallest stopping set in T is equal to the minimum distance of C. Han and Siegel recently proved an upper bound on the stopping redundancy of general linear codes, using probabilistic analysis. For most code parameters, this bound is the best currently known. In this correspondence, we present several improvements upon this bound.

Results on Parity-Check Matrices With Optimal Stopping And/Or Dead-End Set Enumerators

The performance of iterative decoding techniques for linear block codes correcting erasures depends very much on the sizes of the stopping sets associated with the underlying Tanner graph, or, equivalently, the parity-check matrix representing the code. In this correspondence, we introduce the notion of dead-end sets to explicitly demonstrate this dependency. The choice of the parity-check matrix entails a tradeoff between performance and complexity. We give bounds on the complexity of iterative decoders achieving optimal performance in terms of the sizes of the underlying parity-check matrices. Further, we fully characterize codes for which the optimal stopping set enumerator equals the weight enumerator.

Generalized LDPC codes and generalized stopping sets

A generalized low-density parity check code (GLDPC) is a low-density parity check code in which the constraint nodes of the code graph are block codes, rather than single parity checks. In this paper, we study GLDPC codes which have BCH or Reed-Solomon codes as subcodes under bounded distance decoding (BDD). The performance of the proposed scheme is investigated in the limit case of an infinite length (cycle free) code used over a binary erasure channel (BEC) and the corresponding thresholds for iterative decoding are derived. The performance of the proposed scheme for finite code lengths over a BEC is investigated as well. Structures responsible for decoding failures are defined and a theoretical analysis over the ensemble of GLDPC codes which yields exact bit and block error rates of the ensemble average is derived. Unfortunately this study shows that GLDPC codes do not compare favorably with their LDPC counterpart over the BEC. Fortunately, it is also shown that under certain conditions, objects identified in the analysis of GLDPC codes over a BEC and the corresponding theoretical results remain useful to derive tight lower bounds on the performance of GLDPC codes over a binary symmetric channel (BSC). Simulation results show that the proposed method yields competitive performance with a good decoding complexity trade-off for the BSC.

Turbo Decoding on the Binary Erasure Channel: Finite-Length Analysis and Turbo Stopping Sets

This paper is devoted to the finite-length analysis of turbo decoding over the binary erasure channel (BEC). The performance of iterative belief-propagation decoding of low-density parity-check (LDPC) codes over the BEC can be characterized in terms of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stopping sets</i> . We describe turbo decoding on the BEC which is simpler than turbo decoding on other channels. We then adapt the concept of stopping sets to turbo decoding and state an exact condition for decoding failure. Apply turbo decoding until the transmitted codeword has been recovered, or the decoder fails to progress further. Then the set of erased positions that will remain when the decoder stops is equal to the unique maximum-size <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">turbo stopping set</i> which is also a subset of the set of erased positions. Furthermore, we present some improvements of the basic turbo decoding algorithm on the BEC. The proposed improved turbo decoding algorithm has substantially better error performance as illustrated by the given simulation results. Finally, we give an expression for the turbo stopping set size enumerating function under the uniform interleaver assumption, and an efficient enumeration algorithm of small-size turbo stopping sets for a particular interleaver. The solution is based on the algorithm proposed by Garello <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">et al.</i> in 2001 to compute an exhaustive list of all low-weight codewords in a turbo code.

Efficient Puncturing Method for Rate-Compatible Low-Density Parity-Check Codes

In this paper, we propose an efficient puncturing method for LDPC codes. The proposed algorithm provides the order of variable nodes for puncturing based on the proposed cost function. The proposed cost function tries to maximize the minimum reliability among those provided from all check nodes. Also, it tries to allocate survived check nodes evenly to all punctured variable nodes. Furthermore, the proposed algorithm prevents the formation of a stopping set from the punctured variable nodes even when the amount of puncturing is quite large. Simulation results show that the proposed punctured LDPC codes perform better than existing punctured LDPC codes.