  High-rate product codes (PCs) and staircase codes (SCs) are ubiquitous codes
in high-speed optical communication achieving near-capacity performance on the
binary symmetric channel (BSC). Their success is mostly due to very efficient
iterative decoding algorithm that require very little complexity. In this
paper, we extend the density evolution (DE) analysis for PCs and SCs to a
channel with ternary output and ternary message passing, where the third symbol
marks an erasure. We investigate the performance of a standard
error-and-erasure decoder and of two simplifications using DE. The proposed
analysis can be used to find component code configurations and quantizer levels
for the channel output. We also show how the use of even-weight BCH subcodes as
component codes can improve the decoding performance at high rates. The DE
results are verified by Monte-Carlo simulations, which show that additional
coding gains of up to 0.6 dB are possible by ternary decoding, at only a small
additional increase in complexity compared to traditional binary message
passing.
