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arXiv:1105.4665

Improved Linear Programming Decoding using Frustrated Cycles

Shrinivas Kudekar, Jason K. Johnson, Misha Chertkov

Published May 24, 2011• 4 citations
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Abstract

We consider transmission over a binary-input additive white Gaussian noise channel using low-density parity-check codes. One of the most popular techniques for decoding low-density parity-check codes is the linear programming decoder. In general, the linear programming decoder is suboptimal. I.e., the word error rate is higher than the optimal, maximum a posteriori decoder. In this paper we present a systematic approach to enhance the linear program decoder. More precisely, in the cases where the linear program outputs a fractional solution, we give a simple algorithm to identify frustrated cycles which cause the output of the linear program to be fractional. Then adding these cycles, adaptively to the basic linear program, we show improved word error rate performance.

DOI
10.48550/arxiv.1105.4665

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We consider transmission over a binary-input additive white Gaussian noise channel using low-density parity-check codes. One of the most popular techniques for decoding low-density parity-check codes is the linear programming decoder. In general, the linear programming decoder is suboptimal. I.e., the word error rate is higher than the optimal, maximum a posteriori decoder. In this paper we present a systematic approach to enhance the linear program decoder. More precisely, in the cases where the linear program outputs a fractional solution, we give a simple algorithm to identify frustrated cycles which cause the output of the linear program to be fractional. Then adding these cycles, adaptively to the basic linear program, we show improved word error rate performance.
We consider data transmission over a binary-input additive white Gaussian noise channel using low-density parity-check codes. One of the most popular techniques for decoding low-density parity-check codes is the linear programming decoder. In general, the linear programming decoder is suboptimal. In this paper we present a systematic approach to enhance the linear programming decoder. More precisely, in the cases where the linear program outputs a fractional solution, we give a simple algorithm to identify frustrated cycles which cause the output of the linear program to be fractional. Then adding these cycles, adaptively to the basic linear program, we show improved word error rate performance.
A new method is given for performing approximate maximum-likelihood (ML) decoding of an arbitrary binary linear code based on observations received from any discrete memoryless symmetric channel. The decoding algorithm is based on a linear programming (LP) relaxation that is defined by a factor graph or parity-check representation of the code. The resulting "LP decoder" generalizes our previous work on turbo-like codes. A precise combinatorial characterization of when the LP decoder succeeds is provided, based on pseudocodewords associated with the factor graph. Our definition of a pseudocodeword unifies other such notions known for iterative algorithms, including "stopping sets," "irreducible closed walks," "trellis cycles," "deviation sets," and "graph covers." The fractional distance d/sub frac/ of a code is introduced, which is a lower bound on the classical distance. It is shown that the efficient LP decoder will correct up to /spl lceil/d/sub frac//2/spl rceil/-1 errors and that there are codes with d/sub frac/=/spl Omega/(n/sup 1-/spl epsi//). An efficient algorithm to compute the fractional distance is presented. Experimental evidence shows a similar performance on low-density parity-check (LDPC) codes between LP decoding and the min-sum and sum-product algorithms. Methods for tightening the LP relaxation to improve performance are also provided.
The problem of improving the performance of linear programming (LP) decoding of low-density parity-check (LDPC) codes is considered in this paper. A multistep linear programming (MLP) algorithm was developed for decoding LDPC codes that includes a slight increase in computational complexity. The MLP decoder adaptively adds new constraints which are compatible with a selected check node to refine the results when an error is reported by the original LP decoder. The MLP decoder result is shown to have the maximum-likelihood (ML) certificate property. Simulations with moderate block length LOpe codes suggest that the M LP decoder gives better performance than both the original LP decoder and the conventional sum-product (SP) decoder.
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.
Abstract—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. Index Terms—Iterative decoding, linear codes, linear programming (LP) decoding, low-density parity-check (LDPC) codes, maximum-likelihood (ML) decoding, pseudocodewords. I.
Linear programming (LP) decoding for low-density parity-check codes (and related domains such as compressed sensing) has received increased attention over recent years because of its practical performance-coming close to that of iterative decoding algorithms-and its amenability to finite-blocklength analysis. Several works starting with the work of Feldman showed how to analyze LP decoding using properties of expander graphs. This line of analysis works for only low error rates, about a couple of orders of magnitude lower than the empirically observed performance. It is possible to do better for the case of random noise, as shown by Daskalakis and Koetter and Vontobel. Building on work of Koetter and Vontobel, we obtain a novel understanding of LP decoding, which allows us to establish a 0.05 fraction of correctable errors for rate-½ codes; this comes very close to the performance of iterative decoders and is significantly higher than the best previously noted correctable bit error rate for LP decoding. Our analysis exploits an explicit connection between LP decoding and message-passing algorithms and, unlike other techniques, directly works with the primal linear program. An interesting byproduct of our method is a notion of a “locally optimal” solution that we show to always be globally optimal (i.e., it is the nearest codeword). Such a solution can in fact be found in near-linear time by a “reweighted” version of the min-sum algorithm, obviating the need for LP. Our analysis implies, in particular, that this reweighted version of the min-sum decoder corrects up to a 0.05 fraction of errors.