Bounds on the Threshold of Linear Programming Decoding

A design of robust error-correcting codes that achieve reliable communication over various channels is of great theoretical and practical interest. Such codes are termed universal. This paper considers the universality of low-density parity-check (LDPC) code ensembles over families of memoryless binary-input output-symmetric (MBIOS) channels. Universality is considered both under belief-propagation (BP) and maximum-likelihood (ML) decoding. For the BP decoding case, we derive a density-evolution-based analytical method for designing LDPC code ensembles that are universal over various families of MBIOS channels. We also derive a necessary condition for universality of LDPC code ensembles under BP decoding; this condition is used to provide bounds on the universally achievable fraction of capacity. These results enable us to provide conditions for reliable/unreliable communications under BP decoding that are based on the Bhattacharyya parameter of the channel. For the ML decoding case, we prove that properly selected regular LDPC code ensembles are universally capacity-achieving for the set of equi-capacity MBIOS channels and extend this result to punctured regular LDPC code ensembles.

We detail a field-programmable gate array (FPGA) based implementation of linear programming (LP) decoding. LP decoding frames error correction as an optimization problem. This is in contrast to variants of belief propagation (BP) that view error correction as a problem of graphical inference. LP decoding, when implemented with standard LP solvers, does not easily scale to the block-lengths of modern error-correction codes. This is the main challenge we surmount in this paper. In earlier work we demonstrated how to draw on decomposition methods from optimization theory to build an LP decoding solver competitive with BP, in terms of both performance and speed, but only in double-precision floating point. In this paper we translate the novel computational primitives of our new LP decoding technique into fixed-point. Using our FPGA implementation, we demonstrate that error-rate performance very close to double-precision is possible with 10-bit fixed-point messages.

We show that for low-density parity-check (LDPC) codes whose Tanner graphs have sufficient expansion, the linear programming (LP) decoder of Feldman, Karger, and Wainwright can correct a constant fraction of errors. A random graph will have sufficient expansion with high probability, and recent work shows that such graphs can be constructed efficiently. A key element of our method is the use of a dual witness: a zero-valued dual solution to the decoding linear program whose existence proves decoding success. We show that as long as no more than a certain constant fraction of the bits are flipped by the channel, we can find a dual witness. This new method can be used for proving bounds on the performance of any LP decoder, even in a probabilistic setting. Our result implies that the word error rate of the LP decoder decreases exponentially in the code length under the binary-symmetric channel (BSC). This is the first such error bound for LDPC codes using an analysis based on "pseudocodewords." Recent work by Koetter and Vontobel shows that LP decoding and min-sum decoding of LDPC codes are closely related by the "graph cover" structure of their pseudocodewords; in their terminology, our result implies that that there exist families of LDPC codes where the minimum BSC pseudoweight grows linearly in the block length

The problem of low-complexity linear programming (LP) decoding of nonbinary low-density parity-check (LDPC) codes is considered, and an iterative LP decoding algorithm is presented. Results that were previously derived for binary LDPC codes are extended to the nonbinary case. Both simple and generalized nonbinary LDPC codes are considered. It is shown how the algorithm can be implemented efficiently using a finite-field fast Fourier transform. Then, the convergence rate of the algorithm is analyzed. The complexity of the algorithm scales linearly in the block length, and it can approximate, up to an arbitrarily small relative error, the objective function of the exact LP solution. When applied to a typical code from an appropriate nonbinary LDPC code ensemble, the algorithm can correct a constant fraction of errors in linear (in the block length) computational complexity. Computer experiments with the new iterative LP decoding algorithm show that, in the error floor region, it can have better performance compared to belief propagation decoding, with similar computational requirements.

We consider the problem of Linear Programming (LP) decoding of binary linear codes. The LP excess lemma was introduced by the first author, B. Ghazi, and R. Urbanke (IEEE Trans. Inf. Th., 2014) as a technique to trade crossover probability for “LP excess” over the Binary Symmetric Channel. We generalize the LP excess lemma to discrete, binary-input, Memoryless, Symmetric and LLR-Bounded (MSB) channels. As an application, we extend a result by the first author and H. Audah (IEEE Trans. Inf. Th., 2015) on the impact of redundant checks on LP decoding to discrete MSB channels.

We consider coded data transmission over a binary-input output-symmetric memoryless channel using a binary linear code. In order to understand the performance of maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's attention needs to be shifted to the pseudo-codewords, in particular, to the minimal pseudo-codewords and their pseudo-weights. In this paper, we investigate some families of codes that have good properties under LP decoding, namely certain families of low-density parity-check (LDPC) codes that are derived from projective and Euclidean planes: we study the structure of their minimal pseudo-codewords and give lower bounds on their pseudo-weight. Besides this main focus, we also present some results that hold for pseudo-codewords and minimal pseudo-codewords of any Tanner graph, and we highlight how the importance of minimal pseudo-codewords under LP decoding varies depending on which binary-input output-symmetric memoryless channel is used.

Channel coding linear programming decoding (CC- LPD) and compressed sensing linear programming decoding (CS- LPD) are two setups that are formally tightly related. Recently, a connection between CC-LPD and CS-LPD was exhibited that goes beyond this formal relationship. The main ingredient was a lemma that allowed one to map vectors in the nullspace of some zero-one measurement matrix into vectors of the fundamental cone defined by that matrix. The aim of the present paper is to extend this connection along several directions. In particular, the above-mentioned lemma is extended from real measurement matrices where every entry is equal to either zero or one to complex measurement matrices where the absolute value of every entry is a non-negative integer. Moreover, this lemma and its generalizations are used to translate performance guarantees from CC-LPD to CS-LPD. In addition, the present paper extends the formal relationship between CC-LPD and CS-LPD with the help of graph covers. First, this graph-cover viewpoint is used to obtain new connec- tions between, on the one hand, CC-LPD for binary parity- check matrices, and, on the other hand, CS-LPD for complex measurement matrices. Secondly, this graph-cover viewpoint is used to see CS-LPD not only as a well-known relaxation of some zero-norm minimization problem but (at least in the case of real measurement matrices with only zeros, ones, and minus ones) also as a relaxation of a problem we call the zero-infinity operator minimization problem.

When linear programming is used to decode low-density parity-check (LDPC) codes, the outcome is a codeword or a pseudocodeword that contains fractional symbol values. It is possible to make pseudocodewords infeasible and increase the performance of linear programming decoders by generating redundant parity check equations. In this paper, redundant parity check equations that can eliminate pseudocodewords are searched using an integer programming based optimization approach. We show that the generated parity check equations increase the performance of the adaptive linear programming decoder and that its performance can converge that of a maximum-likelihood decoder.

The average finite-length (elementary) fully absorbing set enumerators for binary protograph-based low-density parity-check (LDPC) code ensembles are derived. The method relies on generating functions, and it provides an efficient mean to obtain the normalized logarithmic asymptotic distribution of (elementary) fully absorbing sets. Using these results, the asymptotic distribution of (elementary) fully absorbing sets is evaluated for some protograph-based LDPC code ensembles. Numerical results show how the distributions can be used to estimate the error floor performance of protograph LDPC codes.

We consider the decoding problem for low-density parity-check codes, and apply nonlinear programming methods. This extends previous work using linear programming (LP) to decode linear block codes. First, a multistage LP decoder based on the branch-and-bound method is proposed. This decoder makes use of the maximum-likelihood-certificate property of the LP decoder to refine the results when an error is reported. Second, we transform the original LP decoding formulation into a box-constrained quadratic programming form. Efficient linear-time parallel and serial decoding algorithms are proposed and their convergence properties are investigated. Extensive simulation studies are performed to assess the performance of the proposed decoders. It is seen that the proposed multistage LP decoder outperforms the conventional sum-product (SP) decoder considerably for low-density parity-check (LDPC) codes with short to medium block length. The proposed box-constrained quadratic programming decoder has less complexity than the SP decoder and yields much better performance for LDPC codes with regular structure

A universal design of low-density parity-check (LDPC) code ensembles which enables to operate reliably over various channels is of great theoretical and practical interest. This paper considers the universality of LDPC code ensembles over multitude memoryless binary-input output-symmetric (MBIOS) channels, addressing their universality under belief-propagation (BP) decoding. Based on the density evolution approach, analytical results related to the universality of LDPC code ensembles under BP decoding are derived; these results are expressed in closed form, and are easy to calculate. The full paper version related to this work [11] provides further results, full proofs, additional discussions on the theorems, and it also considers the universality issue under ML decoding.

The linear programming (LP) decoding of Low-Density Parity-Check (LDPC) codes is widely concerned for its maximum likelihood (ML) features. It is that the ML decoding rule for LDPC codes can be relaxed to LP optimization problem. Therefore, this paper focus on an efficient algorithm called infeasible primal-dual interior-point (IPDIP) to solve the LP problem. In each iteration, the IPDIP algorithm obtains the predictor and corrector steps by solving the Karush-Kuhn-Tucker (KKT) equation twice. The predictor term is used to responsible for optimal solution and the corrector term keeps the current iteration away from the boundary of the feasible region. Furthermore, a modification of the centering parameter is developed to accelerate the convergence speed for the IPDIP algorithm. Simulation results of LP decoding demonstrate that the proposed IPDIP algorithm achieves beautiful bit error rate (BER) performance and good global convergence properties with less iteration number and time than other algorithms which only solve the KKT equation once by Newton method or use the not modified centering parameter.

The alternating direction method of multipliers (ADMM) with over-relaxation and the accelerated ADMM are two effective methods to improve the convergence rate of ADMM. In this paper, an efficient linear programming (LP) decoder based on accelerated ADMM for low-density parity-check (LDPC) codes is proposed. Then it is shown that the ADMM update steps for decoding LDPC codes using the accelerated ADMM and the over-relaxed ADMM are quite similar. Therefore, it is impossible to obtain a more efficient ADMM decoder by combining the two improved ADMM methods. Simulation results also validate that the average number of iterations for LDPC decoding using the accelerated ADMM and the over-relaxed ADMM are close to each other.

Decoding by alternating direction method of multipliers (ADMM) is a promising linear programming decoder for low-density parity-check (LDPC) codes. In this paper, we propose a two-step scheme to lower the error floor of LDPC codes with ADMM penalized decoder. For the undetected errors that cannot be avoided at the decoder side, we modify the code structure slightly to eliminate low-weight code words. For the detected errors induced by small error-prone structures, we propose a post-processing method for the ADMM penalized decoder. Simulation results show that the error floor can be reduced significantly over three illustrated LDPC codes by the proposed two-step scheme.

This paper presents a novel approach for constructing structured regular QC-LDPC codes based on a special type of combinatorial designs,known as the balanced incomplete block design (BIBD).The code design approach based on the technology of block cyclic shift and dispertion (BCSD) for correcting erasure-bursts over the binary bursts erasure channel (BBEC). Furthermore, the Tanner graph of the constructed codes is free of 4cycles, hence the BCSD-QC-LDPC codes have girth at least 6. The simulation results show that the BCSD-QC-LDPC codes perform well over the BBEC channel with iterative message-passing decoding.