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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

A Note on the Linear Programming Decoding of Binary Linear Codes for Multiple-Access Channel

Shunsuke Horii, Toshiyasu Matsushima, Shigeichi Hirasawa

Published Jan 1, 2011• 0 citations
Publisher

Abstract

In this paper, we develop linear-programming (LP) decoding for multiple-access channels with binary linear codes. For single-user channels, LP decoding has attracted much attention in recent years as a good approximation to maximum-likelihood (ML) decoding. We demonstrate how the ML decoding problem for multiple-access channels with binary linear codes can be formulated as an LP problem. This is not straightforward, because the objective function of the problem is generally a non-linear function of the codeword symbols. We introduce auxiliary variables such that the objective function is a linear function of these variables. The ML decoding problem then reduces to the LP problem. As in the case for single-user channels, we formulate the relaxed LP problem to reduce the complexity for practical implementation, and as a result propose a decoder that has the desirable property known as the ML certificate property (i.e., if the decoder outputs an integer solution, the solution is guaranteed to be the ML codeword). Although the computational complexity of the proposed algorithm is exponential in the number of users, we can reduce this complexity for Gaussian multiple-access channels. Furthermore, we compare the performance of the proposed decoder with a decoder based on the sum-product algorithm.

DOI
10.1587/transfun.e94.a.1230

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In this paper, we develop linear-programming (LP) decoding for multiple-access channels with binary linear codes. For single-user channels, LP decoding has attracted much attention in recent years as a good approximation to maximum-likelihood (ML) decoding. We demonstrate how the ML decoding problem for multiple-access channels with binary linear codes can be formulated as an LP problem. This is not straightforward, because the objective function of the problem is generally a non-linear function of the codeword symbols. We introduce auxiliary variables such that the objective function is a linear function of these variables. The ML decoding problem then reduces to the LP problem. As in the case for single-user channels, we formulate the relaxed LP problem to reduce the complexity for practical implementation, and as a result propose a decoder that has the desirable property known as the ML certificate property (i.e., if the decoder outputs an integer solution, the solution is guaranteed to be the ML codeword). Although the computational complexity of the proposed algorithm is exponential in the number of users, we can reduce this complexity for Gaussian multiple-access channels. Furthermore, we compare the performance of the proposed decoder with a decoder based on the sum-product algorithm.
Abstract — Given a linear code and observations from a noisy channel, the decoding problem is to determine the most likely (ML) codeword. We describe a method for approximate ML decoding of an arbitrary binary linear code, 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, which generalizes our previous work on turbo-like codes [FK02, FWK02], has the ML certificate property: it either outputs the ML codeword with a guarantee of correctness, or acknowledges an error. We provide a precise characterization of when the LP decoder succeeds, based on the cost of pseudocodewords associated with the factor graph. We introduce the notion of the fractional distance δ of a code, defined with respect to a particular LP relaxation, and prove that the LP decoder will correct up to [δ/2] − 1 errors. For the BEC, we prove that the performance of LP decoding is equivalent to standard iterative decoding. I.
Abstract. Error-correcting codes are fundamental tools used to transmit digital infor-mation over unreliable channels. Their study goes back to the work of Hamming [Ham50] and Shannon [Sha48], who used them as the basis for the field of information theory. The problem of decoding the original information up to the full error-correcting potential of the system is often very complex, especially for modern codes that approach the theoretical limits of the communication channel. In this thesis we investigate the application of linear programming (LP) relaxation to the problem of decoding an error-correcting code. Linear programming relaxation is a standard technique in approximation algorithms and operations research, and is central to the study of efficient algorithms to find good (albeit suboptimal) solutions to very difficult optimization problems. Our new "LP decoders " have tight combinatorial characterizations of decoding success that can be used to analyze error-correcting performance. Furthermore, LP decoders have the desirable (and rare) property that whenever they output a result, it is guaranteed to be the optimal result: the most likely (ML) information sent over the channel. We refer to this property as the ML certificate property.
In this letter, we investigate the application of the subgradient method to design efficient algorithm for linear programming (LP) decoding of binary linear codes. A major drawback of the original formulation of LP decoding is that the description complexity of the feasible region is exponential in the check node degrees of the code. In order to tackle the problem, we propose a processing technique for LP decoding with the subgradient method, whose complexity is linear in the check node degrees. Consequently, a message-passing type decoding algorithm can be obtained, whose per-iteration complexity is extremely low. Moreover, if the algorithm converges to a valid codeword, it is guaranteed to be a maximum likelihood codeword. Simulation results on several binary linear codes with short lengths suggest that the performances of LP decoding based on the subgradient method and the state-of-art LP decoding implementation approach are comparable.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has been derived for both binary and nonbinary decoding frameworks. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation for the nonbinary codes as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. The complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80, 48] LDPC code defined over ℤ4using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has also been extended to nonbinary linear codes by Flanagan et al. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation proposed by Flanagan et al. as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. In contrast to the binary low-complexity LP decoding algorithm, our proposed algorithm is not directly related to the nonbinary SP algorithm. Nevertheless, the complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80,48][80,48][80,48] LDPC code defined over Z4\mathbb{Z}_4Z4​ using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
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.