  In this paper, we present the block Markov superposition transmission of BCH
(BMST-BCH) codes, which can be constructed to obtain a very low error floor. To
reduce the implementation complexity, we design a low complexity iterative
sliding-window decoding algorithm, in which only binary and/or erasure messages
are processed and exchanged between processing units. The error floor can be
predicted by a genie-aided lower bound, while the waterfall performance can be
analyzed by the density evolution method. To evaluate the error floor of the
constructed BMST-BCH codes at a very low bit error rate (BER) region, we
propose a fast simulation approach. Numerical results show that, at a target
BER of $10^{-15}$, the hard-decision decoding of the BMST-BCH codes with
overhead $25\%$ can achieve a net coding gain (NCG) of $10.55$ dB. Furthermore,
the soft-decision decoding can yield an NCG of $10.74$ dB. The construction of
BMST-BCH codes is flexible to trade off latency against performance at all
overheads of interest and may find applications in optical transport networks
as an attractive~candidate.
