  A ternary message passing (TMP) decoding algorithm for low-density
parity-check codes is developed. All messages exchanged between variable and
check nodes have a ternary alphabet, and the variable nodes exploit soft
information from the channel. A density evolution analysis is developed for
unstructured and protograph-based ensembles. For unstructured ensembles the
stability condition is derived. Optimized ensembles for TMP decoding show
asymptotic gains of up to 0.6 dB with respect to ensembles optimized for binary
message passing decoding. Finite length simulations of codes from TMP-optimized
ensembles show gains of up to 0.5 dB under TMP compared to protograph-based
codes designed for unquantized belief propagation decoding.
