  A general method of coding over expansions is proposed, which allows one to
reduce the highly non-trivial problem of coding over continuous channels to a
much simpler discrete ones. More specifically, the focus is on the additive
exponential noise (AEN) channel, for which the (binary) expansion of the
(exponential) noise random variable is considered. It is shown that each of the
random variables in the expansion corresponds to independent Bernoulli random
variables. Thus, each of the expansion levels (of the underlying channel)
corresponds to a binary symmetric channel (BSC), and the coding problem is
reduced to coding over these parallel channels while satisfying the channel
input constraint. This optimization formulation is stated as the achievable
rate result, for which a specific choice of input distribution is shown to
achieve a rate which is arbitrarily close to the channel capacity in the high
SNR regime. Remarkably, the scheme allows for low-complexity capacity-achieving
codes for AEN channels, using the codes that are originally designed for BSCs.
Extensions to different channel models and applications to other coding
problems are discussed.
