We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA( $2$ ) codes.We examine regular and irregular repeat-accumulate (RA) codes with repetition degrees which are all even. For these codes and with a particular choice of an interleaver, we give an upper bound on the decoding error probability of a linear-programming based decoder which is an inverse polynomial in the block length. Our bound is valid for any memoryless, binary-input, output-symmetric (MBIOS) channel. This result generalizes the bound derived by Feldman et al., which was for regular RA($2$) codes.

Advances in Mathematics of CommunicationsJan 1, 20117 citations

Improved Bounds on the Word Error Probability of $RA(2)$ Codes With Linear-Programming-Based DecodingImproved Bounds on the Word Error Probability of $RA(2)$ Codes With Linear-Programming-Based Decoding

Nissim Halabi, Guy Even

This paper deals with the linear-programming-based decoding algorithm of Feldman and Karger for repeat-accumulate "turbo-like" codes. We present a new structural characterization that captures the event that decoding fails. Based on this structural characterization, we develop polynomial algorithms that, given an RA(2) code, compute upper and lower bounds on the word error probability P/sub w/ for the binary-symmetric and the additive white Gaussian noise (AWGN) channels. Our experiments with an implementation of these algorithms for bounding P/sub w/ demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by Feldman et al.. The experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases. The computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound.This paper deals with the linear-programming-based decoding algorithm of Feldman and Karger for repeat-accumulate "turbo-like" codes. We present a new structural characterization that captures the event that decoding fails. Based on this structural characterization, we develop polynomial algorithms that, given an RA(2) code, compute upper and lower bounds on the word error probability P/sub w/ for the binary-symmetric and the additive white Gaussian noise (AWGN) channels. Our experiments with an implementation of these algorithms for bounding P/sub w/ demonstrate in many interesting cases an improvement in the upper bound on the word error probability by a factor of over 1000 compared to the bounds by Feldman et al.. The experiments also indicate that the improvement in upper bound increases as the codeword length increases and the channel noise decreases. The computed lower bounds on the word error probability in our experiments are roughly ten times smaller than the upper bound.

IEEE Transactions on Information TheoryJan 1, 200520 citations

Decoding turbo-like codes via linear programming

J. Feldman, David Ron Karger

We introduce a novel algorithm for decoding turbo-like codes based on linear programming. We prove that for the case of repeat-accumulate (RA) codes, under the binary symmetric channel with a certain constant threshold bound on the noise, the error probability of our algorithm is bounded by an inverse polynomial in the code length. Our linear program (LP) minimizes the distance between the received bits and binary variables representing the code bits. Our LP is based on a representation of the code where code words are paths through a graph. Consequently, the LP bears a strong resemblance to the min-cost flow LP. The error bounds are based on an analysis of the probability, over the random noise of the channel, that the optimum solution to the LP is the path corresponding to the original transmitted code word.We introduce a novel algorithm for decoding turbo-like codes based on linear programming. We prove that for the case of repeat-accumulate (RA) codes, under the binary symmetric channel with a certain constant threshold bound on the noise, the error probability of our algorithm is bounded by an inverse polynomial in the code length. Our linear program (LP) minimizes the distance between the received bits and binary variables representing the code bits. Our LP is based on a representation of the code where code words are paths through a graph. Consequently, the LP bears a strong resemblance to the min-cost flow LP. The error bounds are based on an analysis of the probability, over the random noise of the channel, that the optimum solution to the LP is the path corresponding to the original transmitted code word.

Annual Symposium on Foundations of Computer ScienceJun 26, 200322 citations

LP decoding of regular LDPC codes in memoryless channels

Nissim Halabi, Guy Even

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20106 citations

LP Decoding of Regular LDPC Codes in Memoryless Channels

Nissim Halabi, Guy Even

We study error bounds for linear programming decoding of regular LDPC codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly-exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of $σ=0.735$ (upper bound of $\frac{Eb}{N_0}=2.67$ dB) on the threshold of $(3,6)$ -regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.We study error bounds for linear programming decoding of regular LDPC codes. For memoryless binary-input output-symmetric channels, we prove bounds on the word error probability that are inverse doubly-exponential in the girth of the factor graph. For memoryless binary-input AWGN channel, we prove lower bounds on the threshold for regular LDPC codes whose factor graphs have logarithmic girth under LP-decoding. Specifically, we prove a lower bound of $σ=0.735$ (upper bound of $\frac{Eb}{N_0}=2.67$dB) on the threshold of $(3,6)$-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.

arXiv:1002.3117Feb 16, 20100 citations

2

) codes.

Improved Bounds on the Word Error Probability of

RA(2)

Codes With Linear-Programming-Based Decoding

σ=0.735

(upper bound of

\frac{Eb}{N_0}=2.67

dB) on the threshold of

(3,6)

-regular LDPC codes whose factor graphs have logarithmic girth. Our proof is an extension of a recent paper of Arora, Daskalakis, and Steurer [STOC 2009] who presented a novel probabilistic analysis of LP decoding over a binary symmetric channel. Their analysis is based on the primal LP representation and has an explicit connection to message passing algorithms. We extend this analysis to any MBIOS channel.