Analysis of Message-Passing Decoding of Finite-Length Concatenated Codes
ABSTRACT We analyze the performance of message-passing decoding of finite-length concatenated codes. We first show that the message-passing decoder is closely related to a dual optimization decoder. The connections between these two decoders are further elucidated by proving that both of them attain the same objective function value of a generalized linear programming decoder in the limit as the signal-to-noise ratio (SNR) goes to infinity. Consequently, the framework of pseudo-weight analysis, which was originally proposed for analyzing the linear programming decoder, can be extended to analyze the performance of the message-passing decoder for finite-length codes. We then derive lower bounds to the pseudo-weights of general concatenated codes by utilizing the special structure of their parity-check matrices. We finally present a method to increase the max-fractional weight by adding redundant parity-check constraints and thereby improving the decoding performance. Simulation studies are carried out to assess the performance of the proposed algorithms and substantiate the theoretic claims.
Conference Proceeding: Cascaded Formulation of the Fundamental Polytope of General Linear Block Codes[show abstract] [hide abstract]
ABSTRACT: We propose a new linear programming formulation for the decoding of general linear block codes. Different from the original formulation given in , the number of total variables to characterize a parity-check constraint in our formulation is less than twice the degree of the corresponding check node. The equivalence between our new formulation and the original formulation is proven. Moreover, we show that any fundamental polytope is simply the intersection of a group of so-called minimum polytopes. Based on this, we propose a branch-and-bound method to compute a non-trivial lower bound to the minimum distance of a linear block code with affordable complexity.Information Theory, 2007. ISIT 2007. IEEE International Symposium on; 07/2007
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ABSTRACT: 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</sub> 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 ┌ d<sub>frac</sub>/2 ┐ -1 errors and that there are codes with d<sub>frac</sub>=Ω(n<sup>1-ε</sup>). 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.IEEE Transactions on Information Theory 04/2005; · 2.62 Impact Factor
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ABSTRACT: An analogy is examined between serially concatenated codes and parallel concatenations whose interleavers are described by bipartite graphs with good expanding properties. In particular, a modified expander code construction is shown to behave very much like Forney's classical concatenated codes, though with improved decoding complexity. It is proved that these new codes achieve the Zyablov bound δ<sub>Z</sub> on the minimum distance. For these codes, a soft-decision, reliability-based, linear-time decoding algorithm is introduced, that corrects any fraction of errors up to almost δ<sub>Z</sub>/2. For the binary-symmetric channel, this algorithm's error exponent attains the Forney bound previously known only for classical (serial) concatenations.IEEE Transactions on Information Theory 06/2005; · 2.62 Impact Factor