Linear Programming-Based Decoding of Turbo-Like Codes and its Relation to Iterative Approaches

In recent work (Feldman and Karger [8]), we introduced a new approach to decoding turbo-like codes based on linear programming (LP). We gave a precise characterization of the noise patterns that cause decoding error under the binary symmetric and additive white Gaussian noise channels. We used this characterization to prove that the word error rate is bounded by an inverse polynomial in the code length. Furthermore, for any turbo-like code, our algorithm has the ML certificate property: whenever it outputs a code word, it is guaranteed to be the maximum-likelihood (ML) code word. In this paper we extend these results and give an iterative decoder whose output is equivalent to that of the LP decoder. We also extend the ML certificate property to the more efficient iterative tree reweighted max-product message-passing algorithm developed by Wainwright, Jaakkola, and Willsky [13]: we show that whenever this algorithm converges to a code word, it must be the ML code word. Finally, we demonstrate experimentally that the noise patterns that cause decoding error in the LP decoder also cause decoding error in the standard iterative sum-product and max-product (min-sum) message-passing algorithms. Consequently, the deterministically constructible interleaver used by the LP decoder to achieve its bounds on error rate is useful in practice not only for the LP decoder, but for these standard iterative decoders as well.