  A concatenated coding scheme over binary memoryless symmetric (BMS) channels
using a polarization transformation followed by outer sub-codes is analyzed.
Achievable error exponents and upper bounds on the error rate are derived. The
first bound is obtained using outer codes which are typical linear codes from
the ensemble of parity check matrices whose elements are chosen independently
and uniformly. As a byproduct of this bound, it determines the required rate
split of the total rate to the rates of the outer codes. A lower bound on the
error exponent that holds for all BMS channels with a given capacity is then
derived. Improved bounds and approximations for finite blocklength codes using
channel dispersions (normal approximation), as well as converse and approximate
converse results, are also obtained. The bounds are compared to actual
simulation results from the literature. For the cases considered, when
transmitting over the binary input additive white Gaussian noise channel, there
was only a small gap between the normal approximation prediction and the actual
error rate of concatenated BCH-polar codes.
