Abstract: Message-passing iterative decoders for low-density parity-check (LDPC) block codes (linear binary codes in the nullspace of a sparse matrix or, equivalently, with an associated bipartite graph with very few edges) are known to be subject to decoding failures due to so-called pseudo-codewords. These are real-valued vectors that can be loosely described as error patterns that cause non-convergence in iterative decoding due to the fact that the algorithm works locally and can give priority to a vector that fulfills the equations of a graph cover rather than the graph itself. The function that measures the effect that decoding failures have on the performance of the code is given not by the minimum Hamming weight but by the minimum pseudo-weight. This is defined, for the case of an additive white Gaussian noise channel (AWGNC), as where P is the set of all pseudo-codewords of the code.

In this talk we address the pseudo-codeword problem from the convolutional-code perspective. In particular, we compare the performance of LDPC convolutional codes, (linear codes over the rational field A = F (D)), with that of their “wrapped" quasicyclic block versions (linear codes over some ring of the form F [Y ]/(Y r – 1)), and we show that the minimum pseudo-weight of an LDPC convolutional code is at least as large as the minimum pseudo-weight of an underlying quasi-cyclic code. This result, which parallels a well-known relationship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudo-codeword in the convolutional code induces a pseudo-codeword in the block code with pseudo-weight no larger than that of the convolutional code's pseudo-codeword. This difference in the weight spectra leads to improved performance at low-to-moderate signal-to-noise ratios for the convolutional code.