  We propose a computationally efficient multilevel coding scheme to achieve
the capacity of an ISI channel using layers of binary inputs. The transmitter
employs multilevel coding with linear mapping. The receiver uses multistage
decoding where each stage performs a separate linear minimum mean square error
(LMMSE) equalization and decoding. The optimality of the scheme is due to the
fact that the LMMSE equalizer is information lossless in an ISI channel when
signal to noise ratio is sufficiently low. The computational complexity is low
and scales linearly with the length of the channel impulse response and the
number of layers. The decoder at each layer sees an equivalent AWGN channel,
which makes coding straightforward.
