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Graphcover decoding and finitelength analysis of messagepassing iterative decoding of LDPC codes
 IEEE TRANS. INFORM. THEORY
, 2005
"... The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are comp ..."
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The goal of the present paper is the derivation of a framework for the finitelength analysis of messagepassing iterative decoding of lowdensity paritycheck codes. To this end we introduce the concept of graphcover decoding. Whereas in maximumlikelihood decoding all codewords in a code are competing to be the best explanation of the received vector, under graphcover decoding all codewords in all finite covers of a Tanner graph representation of the code are competing to be the best explanation. We are interested in graphcover decoding because it is a theoretical tool that can be used to show connections between linear programming decoding and messagepassing iterative decoding. Namely, on the one hand it turns out that graphcover decoding is essentially equivalent to linear programming decoding. On the other hand, because iterative, locally operating decoding algorithms like messagepassing iterative decoding cannot distinguish the underlying Tanner graph from any covering graph, graphcover decoding can serve as a model to explain the behavior of messagepassing iterative decoding. Understanding the behavior of graphcover decoding is tantamount to understanding
LinearProgramming Decoding of Nonbinary Linear Codes
, 2009
"... A framework for linearprogramming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linearprogramming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ‘max ..."
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Cited by 21 (11 self)
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A framework for linearprogramming (LP) decoding of nonbinary linear codes over rings is developed. This framework facilitates linearprogramming based reception for coded modulation systems which use direct modulation mapping of coded symbols. It is proved that the resulting LP decoder has the ‘maximumlikelihood certificate ’ property. It is also shown that the decoder output is the lowest cost pseudocodeword. Equivalence between pseudocodewords of the linear program and pseudocodewords of graph covers is proved. It is also proved that if the modulatorchannel combination satisfies a particular symmetry condition, the codeword error rate performance is independent of the transmitted codeword. Two alternative polytopes for use with linearprogramming decoding are studied, and it is shown that for many classes of codes these polytopes yield a complexity advantage for decoding. These polytope representations lead to polynomialtime decoders for a wide variety of classical nonbinary linear codes. LP decoding performance is illustrated for the [11, 6] ternary Golay code with ternary PSK modulation over AWGN, and in this case it is shown that the performance of the LP decoder is comparable to codeworderrorrateoptimum harddecision based decoding. LP decoding is also simulated for mediumlength
Loop calculus helps to improve belief propagation and linear programming decodings of LDPC codes
 1294 INFERENCE ON PLANAR GRAPHS USING LOOP CALCULUS AND BP
, 2006
"... Abstract — We illustrate the utility of the recently developed loop calculus [1], [2] for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code ..."
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Abstract — We illustrate the utility of the recently developed loop calculus [1], [2] for improving the Belief Propagation (BP) algorithm. If the algorithm that minimizes the Bethe free energy fails we modify the free energy by accounting for a critical loop in a graphical representation of the code. The loglikelihood specific critical loop is found by means of the loop calculus. The general method is tested using an example of the Linear Programming (LP) decoding, that can be viewed as a special limit of the BP decoding. Considering the (155, 64, 20) code that performs over AdditiveWhiteGaussian channel we show that the loop calculus improves the LP decoding and corrects all previously found dangerous configurations of loglikelihoods related to pseudocodewords with low effective distance, thus reducing the code’s errorfloor. Belief Propagation (BP) constitutes an efficient approximation,
Minimum pseudoweight and minimum pseudocodewords of LDPC codes
 IEEE Transactions on Information Theory
, 2008
"... In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of lowdensity paritycheck (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudoweight of a pseudocodeword of an LDPC code ..."
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Cited by 8 (3 self)
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In this correspondence, we study the minimum pseudoweight and minimum pseudocodewords of lowdensity paritycheck (LDPC) codes under linear programming (LP) decoding. First, we show that the lower bound of Kelly, Sridhara, Xu and Rosenthal on the pseudoweight of a pseudocodeword of an LDPC code with girth greater than 4 is tight if and only if this pseudocodeword is a real multiple of a codeword. Then, we show that the lower bound of Kashyap and Vardy on the stopping distance of an LDPC code is also a lower bound on the pseudoweight of a pseudocodeword of this LDPC code with girth 4, and this lower bound is tight if and only if this pseudocodeword is a real multiple of a codeword. Using these results we further show that for some LDPC codes, there are no other minimum pseudocodewords except the real multiples of minimum codewords. This means that the LP decoding for these LDPC codes is asymptotically optimal in the sense that the ratio of the probabilities of decoding errors of LP decoding and maximumlikelihood decoding approaches to 1 as the signaltonoise ratio leads to infinity. Finally, some LDPC codes are listed to illustrate these results. Index Terms: LDPC codes, linear programming (LP) decoding, fundamental cone, pseudocodewords, pseudoweight, stopping sets.
The Trapping Redundancy of Linear Block Codes
, 2008
"... We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in a ..."
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We generalize the notion of the stopping redundancy in order to study the smallest size of a trapping set in Tanner graphs of linear block codes. In this context, we introduce the notion of the trapping redundancy of a code, which quantifies the relationship between the number of redundant rows in any paritycheck matrix of a given code and the size of its smallest trapping set. Trapping sets with certain parameter sizes are known to cause errorfloors in the performance curves of iterative belief propagation decoders, and it is therefore important to identify decoding matrices that avoid such sets. Bounds on the trapping redundancy are obtained using probabilistic and constructive methods, and the analysis covers both general and elementary trapping sets. Numerical values for these bounds are computed for the [2640, 1320] Margulis code and the class of projective geometry codes, and compared with some new codespecific trapping set size estimates.
A Cutting Plane Method based on Redundant Rows for Improving Fractional Distance
, 807
"... Abstract — In this paper, an idea of the cutting plane method is employed to improve the fractional distance of a given binary parity check matrix. The fractional distance is the minimum weight (with respect to ℓ1distance) of vertices of the fundamental polytope. The cutting polytope is defined bas ..."
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Abstract — In this paper, an idea of the cutting plane method is employed to improve the fractional distance of a given binary parity check matrix. The fractional distance is the minimum weight (with respect to ℓ1distance) of vertices of the fundamental polytope. The cutting polytope is defined based on redundant rows of the parity check matrix and it plays a key role to eliminate unnecessary fractional vertices in the fundamental polytope. We propose a greedy algorithm and its efficient implementation for improving the fractional distance based on the cutting plane method. I.
A Unified Framework for LinearProgramming Based Communication Receivers
, 2009
"... It is shown that any communication system which admits a sumproduct (SP) receiver also admits a corresponding linearprogramming (LP) receiver. The two receivers have a relationship defined by the local structure of the underlying graphical model, and are inhibited by the same phenomenon, which we ..."
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It is shown that any communication system which admits a sumproduct (SP) receiver also admits a corresponding linearprogramming (LP) receiver. The two receivers have a relationship defined by the local structure of the underlying graphical model, and are inhibited by the same phenomenon, which we call pseudoconfigurations. This concept is a generalization of the concept of pseudocodewords for linear codes. It is proved that the LP receiver has the ‘optimum certificate ’ property, and that the receiver output is the lowest cost pseudoconfiguration. Equivalence of graphcover pseudoconfigurations and linearprogramming pseudoconfigurations is also proved. A concept of system pseudodistance is defined which generalizes the existing concept of pseudodistance for binary and nonbinary linear codes. While the LP receiver generally has a higher complexity than the corresponding SP receiver, the LP receiver and its associated pseudoconfiguration structure provide an interesting tool for the analysis of SP receivers. As an illustrative example, we show in detail how the LP design technique may be applied to the problem of joint equalization and decoding of coded transmissions over a frequency selective channel. A simulationbased analysis of the error events of the resulting LP receiver is also provided.
On the Convex Geometry of Binary Linear Codes
"... A code polytope is defined to be the convex hull in R n of the points in {0, 1} n corresponding to the codewords of a binary linear code. This paper contains a collection of results concerning the structure of such code polytopes. A survey of known results on the dimension and the minimal polyhedr ..."
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A code polytope is defined to be the convex hull in R n of the points in {0, 1} n corresponding to the codewords of a binary linear code. This paper contains a collection of results concerning the structure of such code polytopes. A survey of known results on the dimension and the minimal polyhedral representation of a code polytope is first presented. We show how these results can be extended to obtain the complete facial structure of the polytope determined by the [n, n−1] evenweight code. We then give a result classifying the types of 3faces a general code polytope can have, which shows that the faces of such a polytope cannot be completely arbitrary. Finally, we show how geometrical arguments lead to a simple lower bound on the number of minimal codewords of a code, and characterize the codes for which this bound is attained with equality. This also yields an interesting intermediate result that classifies simple code polytopes. The motivation for our study of code polytopes comes from the formulation by Feldman, Wainwright and Karger of maximumlikelihood decoding as a linear programming problem over the code polytope.