Using Linear Programming to Decode Linear Codes

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