  For any given short code (referred to as the basic code), block Markov
superposition transmission (BMST) provides a simple way to obtain predictable
extra coding gain by spatial coupling the generator matrix of the basic code.
This paper presents a systematic design methodology for BMST systems to
approach the channel capacity at any given target bit-error-rate (BER) of
interest. To simplify the design, we choose the basic code as the Cartesian
product of a short block code. The encoding memory is then inferred from the
genie-aided lower bound according to the performance gap of the short block
code to the corresponding Shannon limit at the target BER. In addition to the
sliding-window decoding algorithm, we propose to perform one more phase
decoding to remove residual (rare) errors. A new technique that assumes a noisy
genie is proposed to upper bound the performance. Under some mild assumptions,
these genie-aided bounds can be used to predict the performance of the proposed
two-phase decoding algorithm in the extremely low BER region. Using the
Cartesian product of a repetition code as the basic code, we construct a BMST
system with an encoding memory 30 whose performance at the BER of $10^{-15}$
can be predicted within one dB away from the Shannon limit over the
binary-input additive white Gaussian noise channel (BI-AWGNC).
