  We discuss and analyze a list-message-passing decoder with verification for
low-density parity-check (LDPC) codes on the q-ary symmetric channel (q-SC).
Rather than passing messages consisting of symbol probabilities, this decoder
passes lists of possible symbols and marks some lists as verified. The density
evolution (DE) equations for this decoder are derived and used to compute
decoding thresholds. If the maximum list size is unbounded, then we find that
any capacity-achieving LDPC code for the binary erasure channel can be used to
achieve capacity on the q-SC for large q. The decoding thresholds are also
computed via DE for the case where each list is truncated to satisfy a maximum
list size constraint. Simulation results are also presented to confirm the DE
results. During the simulations, we observed differences between two
verification-based decoding algorithms, introduced by Luby and Mitzenmacher,
that were implicitly assumed to be identical. In this paper, we provide an
analysis of the node-based algorithms from that paper and verify that it
matches simulation results. The probability of false verification (FV) is also
considered and techniques are discussed to mitigate the FV. Optimization of the
degree distribution is also used to improve the threshold for a fixed maximum
list size. Finally, the proposed algorithm is compared with a variety of other
algorithms using both density evolution thresholds and simulation results.
