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Annual Symposium on Foundations of Computer Science

Decoding turbo-like codes via linear programming

J. Feldman, David Ron Karger

Published Jun 26, 2003• 22 citations
Publisher

Abstract

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.

DOI
10.1109/sfcs.2002.1181948

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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.